{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE EmptyCase #-}
{-# LANGUAGE EmptyDataDecls #-}
{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE NoMonomorphismRestriction #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE PatternSynonyms #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}

{-# OPTIONS_GHC -Wno-overlapping-patterns #-}

module MAlonzo.Code.Data.Integer.Properties where

import MAlonzo.RTE (coe, erased, AgdaAny, addInt, subInt, mulInt,
                    quotInt, remInt, geqInt, ltInt, eqInt, add64, sub64, mul64, quot64,
                    rem64, lt64, eq64, word64FromNat, word64ToNat)
import qualified MAlonzo.RTE
import qualified Data.Text
import qualified MAlonzo.Code.Agda.Builtin.Bool
import qualified MAlonzo.Code.Agda.Builtin.Equality
import qualified MAlonzo.Code.Agda.Builtin.Nat
import qualified MAlonzo.Code.Agda.Builtin.Sigma
import qualified MAlonzo.Code.Agda.Builtin.Unit
import qualified MAlonzo.Code.Agda.Primitive
import qualified MAlonzo.Code.Algebra.Bundles
import qualified MAlonzo.Code.Algebra.Bundles.Raw
import qualified MAlonzo.Code.Algebra.Construct.NaturalChoice.Base
import qualified MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp
import qualified MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp
import qualified MAlonzo.Code.Algebra.Lattice.Bundles
import qualified MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp
import qualified MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinOp
import qualified MAlonzo.Code.Algebra.Lattice.Structures
import qualified MAlonzo.Code.Algebra.Morphism.Structures
import qualified MAlonzo.Code.Algebra.Structures
import qualified MAlonzo.Code.Data.Integer.Base
import qualified MAlonzo.Code.Data.Irrelevant
import qualified MAlonzo.Code.Data.Nat.Base
import qualified MAlonzo.Code.Data.Nat.Properties
import qualified MAlonzo.Code.Data.Sign.Base
import qualified MAlonzo.Code.Data.Sum.Base
import qualified MAlonzo.Code.Function.Base
import qualified MAlonzo.Code.Relation.Binary.Bundles
import qualified MAlonzo.Code.Relation.Binary.Consequences
import qualified MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd
import qualified MAlonzo.Code.Relation.Binary.Definitions
import qualified MAlonzo.Code.Relation.Binary.Morphism.Structures
import qualified MAlonzo.Code.Relation.Binary.PropositionalEquality.Core
import qualified MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties
import qualified MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple
import qualified MAlonzo.Code.Relation.Binary.Reasoning.Syntax
import qualified MAlonzo.Code.Relation.Binary.Structures
import qualified MAlonzo.Code.Relation.Nullary.Decidable.Core
import qualified MAlonzo.Code.Relation.Nullary.Negation.Core
import qualified MAlonzo.Code.Relation.Nullary.Reflects

-- Data.Integer.Properties._._Absorbs_
d__Absorbs__8 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d__Absorbs__8 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d__Absorbs__8 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._._DistributesOver_
d__DistributesOver__10 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d__DistributesOver__10 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d__DistributesOver__10 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._._DistributesOverʳ_
d__DistributesOver'691'__12 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d__DistributesOver'691'__12 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d__DistributesOver'691'__12 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._._DistributesOverˡ_
d__DistributesOver'737'__14 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d__DistributesOver'737'__14 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d__DistributesOver'737'__14 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._._IdempotentOn_
d__IdempotentOn__16 ::
  (Integer -> Integer -> Integer) -> Integer -> ()
d__IdempotentOn__16 :: (Integer -> Integer -> Integer) -> Integer -> ()
d__IdempotentOn__16 = (Integer -> Integer -> Integer) -> Integer -> ()
forall a. a
erased
-- Data.Integer.Properties._._MiddleFourExchange_
d__MiddleFourExchange__18 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d__MiddleFourExchange__18 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d__MiddleFourExchange__18 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Absorptive
d_Absorptive_20 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_Absorptive_20 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_Absorptive_20 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.AlmostCancellative
d_AlmostCancellative_22 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_AlmostCancellative_22 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_AlmostCancellative_22 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.AlmostLeftCancellative
d_AlmostLeftCancellative_24 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_AlmostLeftCancellative_24 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_AlmostLeftCancellative_24 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.AlmostRightCancellative
d_AlmostRightCancellative_26 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_AlmostRightCancellative_26 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_AlmostRightCancellative_26 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Alternative
d_Alternative_28 :: (Integer -> Integer -> Integer) -> ()
d_Alternative_28 :: (Integer -> Integer -> Integer) -> ()
d_Alternative_28 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Associative
d_Associative_30 :: (Integer -> Integer -> Integer) -> ()
d_Associative_30 :: (Integer -> Integer -> Integer) -> ()
d_Associative_30 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Cancellative
d_Cancellative_32 :: (Integer -> Integer -> Integer) -> ()
d_Cancellative_32 :: (Integer -> Integer -> Integer) -> ()
d_Cancellative_32 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Commutative
d_Commutative_34 :: (Integer -> Integer -> Integer) -> ()
d_Commutative_34 :: (Integer -> Integer -> Integer) -> ()
d_Commutative_34 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Congruent₁
d_Congruent'8321'_36 :: (Integer -> Integer) -> ()
d_Congruent'8321'_36 :: (Integer -> Integer) -> ()
d_Congruent'8321'_36 = (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Congruent₂
d_Congruent'8322'_38 :: (Integer -> Integer -> Integer) -> ()
d_Congruent'8322'_38 :: (Integer -> Integer -> Integer) -> ()
d_Congruent'8322'_38 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Conical
d_Conical_40 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Conical_40 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Conical_40 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Flexible
d_Flexible_42 :: (Integer -> Integer -> Integer) -> ()
d_Flexible_42 :: (Integer -> Integer -> Integer) -> ()
d_Flexible_42 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Idempotent
d_Idempotent_44 :: (Integer -> Integer -> Integer) -> ()
d_Idempotent_44 :: (Integer -> Integer -> Integer) -> ()
d_Idempotent_44 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.IdempotentFun
d_IdempotentFun_46 :: (Integer -> Integer) -> ()
d_IdempotentFun_46 :: (Integer -> Integer) -> ()
d_IdempotentFun_46 = (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Identical
d_Identical_48 :: (Integer -> Integer -> Integer) -> ()
d_Identical_48 :: (Integer -> Integer -> Integer) -> ()
d_Identical_48 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Identity
d_Identity_50 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Identity_50 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Identity_50 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Interchangable
d_Interchangable_52 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_Interchangable_52 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_Interchangable_52 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Inverse
d_Inverse_54 ::
  Integer ->
  (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_Inverse_54 :: Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_Inverse_54 = Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Invertible
d_Invertible_56 ::
  Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_Invertible_56 :: Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_Invertible_56 = Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
forall a. a
erased
-- Data.Integer.Properties._.Involutive
d_Involutive_58 :: (Integer -> Integer) -> ()
d_Involutive_58 :: (Integer -> Integer) -> ()
d_Involutive_58 = (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftAlternative
d_LeftAlternative_60 :: (Integer -> Integer -> Integer) -> ()
d_LeftAlternative_60 :: (Integer -> Integer -> Integer) -> ()
d_LeftAlternative_60 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftBol
d_LeftBol_62 :: (Integer -> Integer -> Integer) -> ()
d_LeftBol_62 :: (Integer -> Integer -> Integer) -> ()
d_LeftBol_62 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftCancellative
d_LeftCancellative_64 :: (Integer -> Integer -> Integer) -> ()
d_LeftCancellative_64 :: (Integer -> Integer -> Integer) -> ()
d_LeftCancellative_64 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftCongruent
d_LeftCongruent_66 :: (Integer -> Integer -> Integer) -> ()
d_LeftCongruent_66 :: (Integer -> Integer -> Integer) -> ()
d_LeftCongruent_66 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftConical
d_LeftConical_68 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftConical_68 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftConical_68 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftDivides
d_LeftDivides_70 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_LeftDivides_70 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_LeftDivides_70 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftDividesʳ
d_LeftDivides'691'_72 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_LeftDivides'691'_72 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_LeftDivides'691'_72 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftDividesˡ
d_LeftDivides'737'_74 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_LeftDivides'737'_74 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_LeftDivides'737'_74 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftIdentity
d_LeftIdentity_76 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftIdentity_76 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftIdentity_76 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftInverse
d_LeftInverse_78 ::
  Integer ->
  (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_LeftInverse_78 :: Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_LeftInverse_78 = Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftInvertible
d_LeftInvertible_80 ::
  Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_LeftInvertible_80 :: Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_LeftInvertible_80 = Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftSemimedial
d_LeftSemimedial_82 :: (Integer -> Integer -> Integer) -> ()
d_LeftSemimedial_82 :: (Integer -> Integer -> Integer) -> ()
d_LeftSemimedial_82 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.LeftZero
d_LeftZero_84 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftZero_84 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftZero_84 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Medial
d_Medial_86 :: (Integer -> Integer -> Integer) -> ()
d_Medial_86 :: (Integer -> Integer -> Integer) -> ()
d_Medial_86 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.MiddleBol
d_MiddleBol_88 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_MiddleBol_88 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> ()
d_MiddleBol_88 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> ()
forall a. a
erased
-- Data.Integer.Properties._.RightAlternative
d_RightAlternative_90 :: (Integer -> Integer -> Integer) -> ()
d_RightAlternative_90 :: (Integer -> Integer -> Integer) -> ()
d_RightAlternative_90 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightBol
d_RightBol_92 :: (Integer -> Integer -> Integer) -> ()
d_RightBol_92 :: (Integer -> Integer -> Integer) -> ()
d_RightBol_92 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightCancellative
d_RightCancellative_94 :: (Integer -> Integer -> Integer) -> ()
d_RightCancellative_94 :: (Integer -> Integer -> Integer) -> ()
d_RightCancellative_94 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightCongruent
d_RightCongruent_96 :: (Integer -> Integer -> Integer) -> ()
d_RightCongruent_96 :: (Integer -> Integer -> Integer) -> ()
d_RightCongruent_96 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightConical
d_RightConical_98 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_RightConical_98 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightConical_98 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightDivides
d_RightDivides_100 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_RightDivides_100 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_RightDivides_100 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightDividesʳ
d_RightDivides'691'_102 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_RightDivides'691'_102 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_RightDivides'691'_102 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightDividesˡ
d_RightDivides'737'_104 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_RightDivides'737'_104 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_RightDivides'737'_104 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightIdentity
d_RightIdentity_106 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_RightIdentity_106 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightIdentity_106 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightInverse
d_RightInverse_108 ::
  Integer ->
  (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_RightInverse_108 :: Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_RightInverse_108 = Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightInvertible
d_RightInvertible_110 ::
  Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_RightInvertible_110 :: Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_RightInvertible_110 = Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightSemimedial
d_RightSemimedial_112 :: (Integer -> Integer -> Integer) -> ()
d_RightSemimedial_112 :: (Integer -> Integer -> Integer) -> ()
d_RightSemimedial_112 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.RightZero
d_RightZero_114 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightZero_114 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightZero_114 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Selective
d_Selective_116 :: (Integer -> Integer -> Integer) -> ()
d_Selective_116 :: (Integer -> Integer -> Integer) -> ()
d_Selective_116 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.SelfInverse
d_SelfInverse_118 :: (Integer -> Integer) -> ()
d_SelfInverse_118 :: (Integer -> Integer) -> ()
d_SelfInverse_118 = (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.Semimedial
d_Semimedial_120 :: (Integer -> Integer -> Integer) -> ()
d_Semimedial_120 :: (Integer -> Integer -> Integer) -> ()
d_Semimedial_120 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.StarDestructive
d_StarDestructive_122 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarDestructive_122 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarDestructive_122 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.StarExpansive
d_StarExpansive_124 ::
  Integer ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarExpansive_124 :: Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
d_StarExpansive_124 = Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
forall a. a
erased
-- Data.Integer.Properties._.StarLeftDestructive
d_StarLeftDestructive_126 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarLeftDestructive_126 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarLeftDestructive_126 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.StarLeftExpansive
d_StarLeftExpansive_128 ::
  Integer ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarLeftExpansive_128 :: Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
d_StarLeftExpansive_128 = Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
forall a. a
erased
-- Data.Integer.Properties._.StarRightDestructive
d_StarRightDestructive_130 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarRightDestructive_130 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarRightDestructive_130 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.StarRightExpansive
d_StarRightExpansive_132 ::
  Integer ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarRightExpansive_132 :: Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
d_StarRightExpansive_132 = Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
forall a. a
erased
-- Data.Integer.Properties._.Zero
d_Zero_134 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Zero_134 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Zero_134 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup
d_IsAbelianGroup_138 :: p -> p -> p -> ()
d_IsAbelianGroup_138 p
a0 p
a1 p
a2 = ()
-- Data.Integer.Properties._.IsAlternativeMagma
d_IsAlternativeMagma_142 :: p -> ()
d_IsAlternativeMagma_142 p
a0 = ()
-- Data.Integer.Properties._.IsBand
d_IsBand_146 :: p -> ()
d_IsBand_146 p
a0 = ()
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring
d_IsCancellativeCommutativeSemiring_150 :: p -> p -> p -> p -> ()
d_IsCancellativeCommutativeSemiring_150 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsCommutativeBand
d_IsCommutativeBand_154 :: p -> ()
d_IsCommutativeBand_154 p
a0 = ()
-- Data.Integer.Properties._.IsCommutativeMagma
d_IsCommutativeMagma_158 :: p -> ()
d_IsCommutativeMagma_158 p
a0 = ()
-- Data.Integer.Properties._.IsCommutativeMonoid
d_IsCommutativeMonoid_162 :: p -> p -> ()
d_IsCommutativeMonoid_162 p
a0 p
a1 = ()
-- Data.Integer.Properties._.IsCommutativeRing
d_IsCommutativeRing_166 :: p -> p -> p -> p -> p -> ()
d_IsCommutativeRing_166 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Integer.Properties._.IsCommutativeSemigroup
d_IsCommutativeSemigroup_170 :: p -> ()
d_IsCommutativeSemigroup_170 p
a0 = ()
-- Data.Integer.Properties._.IsCommutativeSemiring
d_IsCommutativeSemiring_174 :: p -> p -> p -> p -> ()
d_IsCommutativeSemiring_174 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne
d_IsCommutativeSemiringWithoutOne_178 :: p -> p -> p -> ()
d_IsCommutativeSemiringWithoutOne_178 p
a0 p
a1 p
a2 = ()
-- Data.Integer.Properties._.IsFlexibleMagma
d_IsFlexibleMagma_182 :: p -> ()
d_IsFlexibleMagma_182 p
a0 = ()
-- Data.Integer.Properties._.IsGroup
d_IsGroup_186 :: p -> p -> p -> ()
d_IsGroup_186 p
a0 p
a1 p
a2 = ()
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid
d_IsIdempotentCommutativeMonoid_190 :: p -> p -> ()
d_IsIdempotentCommutativeMonoid_190 p
a0 p
a1 = ()
-- Data.Integer.Properties._.IsIdempotentMagma
d_IsIdempotentMagma_194 :: p -> ()
d_IsIdempotentMagma_194 p
a0 = ()
-- Data.Integer.Properties._.IsIdempotentMonoid
d_IsIdempotentMonoid_198 :: p -> p -> ()
d_IsIdempotentMonoid_198 p
a0 p
a1 = ()
-- Data.Integer.Properties._.IsIdempotentSemiring
d_IsIdempotentSemiring_202 :: p -> p -> p -> p -> ()
d_IsIdempotentSemiring_202 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsInvertibleMagma
d_IsInvertibleMagma_206 :: p -> p -> p -> ()
d_IsInvertibleMagma_206 p
a0 p
a1 p
a2 = ()
-- Data.Integer.Properties._.IsInvertibleUnitalMagma
d_IsInvertibleUnitalMagma_210 :: p -> p -> p -> ()
d_IsInvertibleUnitalMagma_210 p
a0 p
a1 p
a2 = ()
-- Data.Integer.Properties._.IsKleeneAlgebra
d_IsKleeneAlgebra_214 :: p -> p -> p -> p -> p -> ()
d_IsKleeneAlgebra_214 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Integer.Properties._.IsLeftBolLoop
d_IsLeftBolLoop_218 :: p -> p -> p -> p -> ()
d_IsLeftBolLoop_218 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsLoop
d_IsLoop_222 :: p -> p -> p -> p -> ()
d_IsLoop_222 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsMagma
d_IsMagma_226 :: p -> ()
d_IsMagma_226 p
a0 = ()
-- Data.Integer.Properties._.IsMedialMagma
d_IsMedialMagma_230 :: p -> ()
d_IsMedialMagma_230 p
a0 = ()
-- Data.Integer.Properties._.IsMiddleBolLoop
d_IsMiddleBolLoop_234 :: p -> p -> p -> p -> ()
d_IsMiddleBolLoop_234 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsMonoid
d_IsMonoid_238 :: p -> p -> ()
d_IsMonoid_238 p
a0 p
a1 = ()
-- Data.Integer.Properties._.IsMoufangLoop
d_IsMoufangLoop_242 :: p -> p -> p -> p -> ()
d_IsMoufangLoop_242 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsNearSemiring
d_IsNearSemiring_246 :: p -> p -> p -> ()
d_IsNearSemiring_246 p
a0 p
a1 p
a2 = ()
-- Data.Integer.Properties._.IsNearring
d_IsNearring_250 :: p -> p -> p -> p -> p -> ()
d_IsNearring_250 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Integer.Properties._.IsNonAssociativeRing
d_IsNonAssociativeRing_254 :: p -> p -> p -> p -> p -> ()
d_IsNonAssociativeRing_254 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Integer.Properties._.IsQuasigroup
d_IsQuasigroup_258 :: p -> p -> p -> ()
d_IsQuasigroup_258 p
a0 p
a1 p
a2 = ()
-- Data.Integer.Properties._.IsQuasiring
d_IsQuasiring_262 :: p -> p -> p -> p -> ()
d_IsQuasiring_262 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsRightBolLoop
d_IsRightBolLoop_266 :: p -> p -> p -> p -> ()
d_IsRightBolLoop_266 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsRing
d_IsRing_270 :: p -> p -> p -> p -> p -> ()
d_IsRing_270 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Integer.Properties._.IsRingWithoutOne
d_IsRingWithoutOne_274 :: p -> p -> p -> p -> ()
d_IsRingWithoutOne_274 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsSelectiveMagma
d_IsSelectiveMagma_278 :: p -> ()
d_IsSelectiveMagma_278 p
a0 = ()
-- Data.Integer.Properties._.IsSemigroup
d_IsSemigroup_282 :: p -> ()
d_IsSemigroup_282 p
a0 = ()
-- Data.Integer.Properties._.IsSemimedialMagma
d_IsSemimedialMagma_286 :: p -> ()
d_IsSemimedialMagma_286 p
a0 = ()
-- Data.Integer.Properties._.IsSemiring
d_IsSemiring_290 :: p -> p -> p -> p -> ()
d_IsSemiring_290 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsSemiringWithoutAnnihilatingZero
d_IsSemiringWithoutAnnihilatingZero_294 :: p -> p -> p -> p -> ()
d_IsSemiringWithoutAnnihilatingZero_294 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Integer.Properties._.IsSemiringWithoutOne
d_IsSemiringWithoutOne_298 :: p -> p -> p -> ()
d_IsSemiringWithoutOne_298 p
a0 p
a1 p
a2 = ()
-- Data.Integer.Properties._.IsSuccessorSet
d_IsSuccessorSet_302 :: p -> p -> ()
d_IsSuccessorSet_302 p
a0 p
a1 = ()
-- Data.Integer.Properties._.IsUnitalMagma
d_IsUnitalMagma_306 :: p -> p -> ()
d_IsUnitalMagma_306 p
a0 p
a1 = ()
-- Data.Integer.Properties._.IsAbelianGroup._//_
d__'47''47'__312 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer -> Integer -> Integer
d__'47''47'__312 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
d__'47''47'__312 Integer -> Integer -> Integer
v0 ~Integer
v1 Integer -> Integer
v2 ~T_IsAbelianGroup_1172
v3 = (Integer -> Integer -> Integer)
-> (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__312 Integer -> Integer -> Integer
v0 Integer -> Integer
v2
du__'47''47'__312 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__312 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__312 Integer -> Integer -> Integer
v0 Integer -> Integer
v1
  = ((Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any)
-> Any -> Any -> Integer -> Integer -> Integer
forall a b. a -> b
coe
      (Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any
MAlonzo.Code.Algebra.Structures.du__'47''47'__1136 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0)
      ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v1)
-- Data.Integer.Properties._.IsAbelianGroup.assoc
d_assoc_314 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_314 :: T_IsAbelianGroup_1172
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_314 = T_IsAbelianGroup_1172
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.comm
d_comm_316 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_316 :: T_IsAbelianGroup_1172 -> Integer -> Integer -> T__'8801'__12
d_comm_316 = T_IsAbelianGroup_1172 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.identity
d_identity_318 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_318 :: T_IsAbelianGroup_1172 -> T_Σ_14
d_identity_318 T_IsAbelianGroup_1172
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0)))
-- Data.Integer.Properties._.IsAbelianGroup.identityʳ
d_identity'691'_320 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_320 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
d_identity'691'_320 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.identityˡ
d_identity'737'_322 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_322 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
d_identity'737'_322 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.inverse
d_inverse_324 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_324 :: T_IsAbelianGroup_1172 -> T_Σ_14
d_inverse_324 T_IsAbelianGroup_1172
v0
  = (T_IsGroup_1074 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_1090
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Data.Integer.Properties._.IsAbelianGroup.inverseʳ
d_inverse'691'_326 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'691'_326 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
d_inverse'691'_326 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.inverseˡ
d_inverse'737'_328 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'737'_328 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
d_inverse'737'_328 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.isCommutativeMagma
d_isCommutativeMagma_330 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_330 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_330 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_330 T_IsAbelianGroup_1172
v3
du_isCommutativeMagma_330 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_330 :: T_IsAbelianGroup_1172 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_330 T_IsAbelianGroup_1172
v0
  = let v1 :: Any
v1
          = (T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764) -> Any -> Any
forall a b. a -> b
coe
              T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
              (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
         ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
            (Any -> Any
forall a b. a -> b
coe Any
v1)))
-- Data.Integer.Properties._.IsAbelianGroup.isCommutativeMonoid
d_isCommutativeMonoid_332 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_332 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_332 Integer -> Integer -> Integer
v0 Integer
v1 Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = (T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764)
-> T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244 T_IsAbelianGroup_1172
v3
-- Data.Integer.Properties._.IsAbelianGroup.isCommutativeSemigroup
d_isCommutativeSemigroup_334 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_334 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_334 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_334 T_IsAbelianGroup_1172
v3
du_isCommutativeSemigroup_334 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_334 :: T_IsAbelianGroup_1172 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_334 T_IsAbelianGroup_1172
v0
  = (T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
      ((T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764) -> Any -> Any
forall a b. a -> b
coe
         T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
         (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Data.Integer.Properties._.IsAbelianGroup.isEquivalence
d_isEquivalence_336 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_336 :: T_IsAbelianGroup_1172 -> T_IsEquivalence_28
d_isEquivalence_336 T_IsAbelianGroup_1172
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
               T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
               ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0)))))
-- Data.Integer.Properties._.IsAbelianGroup.isGroup
d_isGroup_338 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074
d_isGroup_338 :: T_IsAbelianGroup_1172 -> T_IsGroup_1074
d_isGroup_338 T_IsAbelianGroup_1172
v0
  = (T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> T_IsGroup_1074
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0)
-- Data.Integer.Properties._.IsAbelianGroup.isInvertibleMagma
d_isInvertibleMagma_340 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_340 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsInvertibleMagma_958
d_isInvertibleMagma_340 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_340 T_IsAbelianGroup_1172
v3
du_isInvertibleMagma_340 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
du_isInvertibleMagma_340 :: T_IsAbelianGroup_1172 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_340 T_IsAbelianGroup_1172
v0
  = (T_IsGroup_1074 -> T_IsInvertibleMagma_958)
-> Any -> T_IsInvertibleMagma_958
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1160
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Data.Integer.Properties._.IsAbelianGroup.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_342 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_342 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_342 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_342 T_IsAbelianGroup_1172
v3
du_isInvertibleUnitalMagma_342 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_342 :: T_IsAbelianGroup_1172 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_342 T_IsAbelianGroup_1172
v0
  = (T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012)
-> Any -> T_IsInvertibleUnitalMagma_1012
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012
MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1162
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Data.Integer.Properties._.IsAbelianGroup.isMagma
d_isMagma_344 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_344 :: T_IsAbelianGroup_1172 -> T_IsMagma_178
d_isMagma_344 T_IsAbelianGroup_1172
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
            T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
            ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))))
-- Data.Integer.Properties._.IsAbelianGroup.isMonoid
d_isMonoid_346 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_346 :: T_IsAbelianGroup_1172 -> T_IsMonoid_712
d_isMonoid_346 T_IsAbelianGroup_1172
v0
  = (T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Data.Integer.Properties._.IsAbelianGroup.isPartialEquivalence
d_isPartialEquivalence_348 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_348 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_348 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_348 T_IsAbelianGroup_1172
v3
du_isPartialEquivalence_348 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_348 :: T_IsAbelianGroup_1172 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_348 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_178
v4 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
                  T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                  ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                     T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v4))))))
-- Data.Integer.Properties._.IsAbelianGroup.isSemigroup
d_isSemigroup_350 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_350 :: T_IsAbelianGroup_1172 -> T_IsSemigroup_488
d_isSemigroup_350 T_IsAbelianGroup_1172
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0)))
-- Data.Integer.Properties._.IsAbelianGroup.isUnitalMagma
d_isUnitalMagma_352 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_352 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsUnitalMagma_666
d_isUnitalMagma_352 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3 = T_IsAbelianGroup_1172 -> T_IsUnitalMagma_666
du_isUnitalMagma_352 T_IsAbelianGroup_1172
v3
du_isUnitalMagma_352 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_352 :: T_IsAbelianGroup_1172 -> T_IsUnitalMagma_666
du_isUnitalMagma_352 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v1)))
-- Data.Integer.Properties._.IsAbelianGroup.refl
d_refl_354 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_354 :: T_IsAbelianGroup_1172 -> Integer -> T__'8801'__12
d_refl_354 = T_IsAbelianGroup_1172 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.reflexive
d_reflexive_356 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_356 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_356 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.setoid
d_setoid_358 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_358 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_Setoid_46
d_setoid_358 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3 = T_IsAbelianGroup_1172 -> T_Setoid_46
du_setoid_358 T_IsAbelianGroup_1172
v3
du_setoid_358 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_358 :: T_IsAbelianGroup_1172 -> T_Setoid_46
du_setoid_358 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
               T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v3)))))
-- Data.Integer.Properties._.IsAbelianGroup.sym
d_sym_360 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_360 :: T_IsAbelianGroup_1172
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_360 = T_IsAbelianGroup_1172
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.trans
d_trans_362 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_362 :: T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_362 = T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_364 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_unique'691''45''8315''185'_364 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'691''45''8315''185'_364 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_366 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_unique'737''45''8315''185'_366 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'737''45''8315''185'_366 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.⁻¹-cong
d_'8315''185''45'cong_368 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_368 :: T_IsAbelianGroup_1172
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_368 = T_IsAbelianGroup_1172
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.∙-cong
d_'8729''45'cong_370 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_370 :: T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_370 = T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.∙-congʳ
d_'8729''45'cong'691'_372 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_372 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_372 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAbelianGroup.∙-congˡ
d_'8729''45'cong'737'_374 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_374 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_374 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.alter
d_alter_378 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_alter_378 :: T_IsAlternativeMagma_290 -> T_Σ_14
d_alter_378 T_IsAlternativeMagma_290
v0
  = (T_IsAlternativeMagma_290 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsAlternativeMagma_290 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_alter_300 (T_IsAlternativeMagma_290 -> Any
forall a b. a -> b
coe T_IsAlternativeMagma_290
v0)
-- Data.Integer.Properties._.IsAlternativeMagma.alternativeʳ
d_alternative'691'_380 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_alternative'691'_380 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> Integer -> Integer -> T__'8801'__12
d_alternative'691'_380 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.alternativeˡ
d_alternative'737'_382 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_alternative'737'_382 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> Integer -> Integer -> T__'8801'__12
d_alternative'737'_382 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.isEquivalence
d_isEquivalence_384 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_384 :: T_IsAlternativeMagma_290 -> T_IsEquivalence_28
d_isEquivalence_384 T_IsAlternativeMagma_290
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsAlternativeMagma_290 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsAlternativeMagma_290 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_298 (T_IsAlternativeMagma_290 -> Any
forall a b. a -> b
coe T_IsAlternativeMagma_290
v0))
-- Data.Integer.Properties._.IsAlternativeMagma.isMagma
d_isMagma_386 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_386 :: T_IsAlternativeMagma_290 -> T_IsMagma_178
d_isMagma_386 T_IsAlternativeMagma_290
v0
  = (T_IsAlternativeMagma_290 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe T_IsAlternativeMagma_290 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_298 (T_IsAlternativeMagma_290 -> Any
forall a b. a -> b
coe T_IsAlternativeMagma_290
v0)
-- Data.Integer.Properties._.IsAlternativeMagma.isPartialEquivalence
d_isPartialEquivalence_388 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_388 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_388 ~Integer -> Integer -> Integer
v0 T_IsAlternativeMagma_290
v1 = T_IsAlternativeMagma_290 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_388 T_IsAlternativeMagma_290
v1
du_isPartialEquivalence_388 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_388 :: T_IsAlternativeMagma_290 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_388 T_IsAlternativeMagma_290
v0
  = let v1 :: T_IsMagma_178
v1
          = T_IsAlternativeMagma_290 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_298 (T_IsAlternativeMagma_290 -> T_IsAlternativeMagma_290
forall a b. a -> b
coe T_IsAlternativeMagma_290
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
         T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
         ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v1)))
-- Data.Integer.Properties._.IsAlternativeMagma.refl
d_refl_390 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_390 :: T_IsAlternativeMagma_290 -> Integer -> T__'8801'__12
d_refl_390 = T_IsAlternativeMagma_290 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.reflexive
d_reflexive_392 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_392 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_392 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.setoid
d_setoid_394 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_394 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> T_Setoid_46
d_setoid_394 ~Integer -> Integer -> Integer
v0 T_IsAlternativeMagma_290
v1 = T_IsAlternativeMagma_290 -> T_Setoid_46
du_setoid_394 T_IsAlternativeMagma_290
v1
du_setoid_394 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_394 :: T_IsAlternativeMagma_290 -> T_Setoid_46
du_setoid_394 T_IsAlternativeMagma_290
v0
  = (T_IsMagma_178 -> T_Setoid_46) -> Any -> T_Setoid_46
forall a b. a -> b
coe
      T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
      ((T_IsAlternativeMagma_290 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsAlternativeMagma_290 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_298 (T_IsAlternativeMagma_290 -> Any
forall a b. a -> b
coe T_IsAlternativeMagma_290
v0))
-- Data.Integer.Properties._.IsAlternativeMagma.sym
d_sym_396 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_396 :: T_IsAlternativeMagma_290
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_396 = T_IsAlternativeMagma_290
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.trans
d_trans_398 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_398 :: T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_398 = T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.∙-cong
d_'8729''45'cong_400 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_400 :: T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_400 = T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.∙-congʳ
d_'8729''45'cong'691'_402 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_402 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_402 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsAlternativeMagma.∙-congˡ
d_'8729''45'cong'737'_404 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_404 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_404 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.assoc
d_assoc_408 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_408 :: T_IsBand_526 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_408 = T_IsBand_526 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.idem
d_idem_410 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_410 :: T_IsBand_526 -> Integer -> T__'8801'__12
d_idem_410 = T_IsBand_526 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.isEquivalence
d_isEquivalence_412 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_412 :: T_IsBand_526 -> T_IsEquivalence_28
d_isEquivalence_412 T_IsBand_526
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsBand_526 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534 (T_IsBand_526 -> Any
forall a b. a -> b
coe T_IsBand_526
v0)))
-- Data.Integer.Properties._.IsBand.isMagma
d_isMagma_414 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_414 :: T_IsBand_526 -> T_IsMagma_178
d_isMagma_414 T_IsBand_526
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsBand_526 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534 (T_IsBand_526 -> Any
forall a b. a -> b
coe T_IsBand_526
v0))
-- Data.Integer.Properties._.IsBand.isPartialEquivalence
d_isPartialEquivalence_416 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_416 :: (Integer -> Integer -> Integer)
-> T_IsBand_526 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_416 ~Integer -> Integer -> Integer
v0 T_IsBand_526
v1 = T_IsBand_526 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_416 T_IsBand_526
v1
du_isPartialEquivalence_416 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_416 :: T_IsBand_526 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_416 T_IsBand_526
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534 (T_IsBand_526 -> T_IsBand_526
forall a b. a -> b
coe T_IsBand_526
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
            T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
            ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
               T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v2))))
-- Data.Integer.Properties._.IsBand.isSemigroup
d_isSemigroup_418 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_418 :: T_IsBand_526 -> T_IsSemigroup_488
d_isSemigroup_418 T_IsBand_526
v0
  = (T_IsBand_526 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534 (T_IsBand_526 -> Any
forall a b. a -> b
coe T_IsBand_526
v0)
-- Data.Integer.Properties._.IsBand.refl
d_refl_420 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_420 :: T_IsBand_526 -> Integer -> T__'8801'__12
d_refl_420 = T_IsBand_526 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.reflexive
d_reflexive_422 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_422 :: (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_422 = (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.setoid
d_setoid_424 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_424 :: (Integer -> Integer -> Integer) -> T_IsBand_526 -> T_Setoid_46
d_setoid_424 ~Integer -> Integer -> Integer
v0 T_IsBand_526
v1 = T_IsBand_526 -> T_Setoid_46
du_setoid_424 T_IsBand_526
v1
du_setoid_424 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_424 :: T_IsBand_526 -> T_Setoid_46
du_setoid_424 T_IsBand_526
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534 (T_IsBand_526 -> T_IsBand_526
forall a b. a -> b
coe T_IsBand_526
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
         T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v1)))
-- Data.Integer.Properties._.IsBand.sym
d_sym_426 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_426 :: T_IsBand_526
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_426 = T_IsBand_526
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.trans
d_trans_428 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_428 :: T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_428 = T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.∙-cong
d_'8729''45'cong_430 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_430 :: T_IsBand_526
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_430 = T_IsBand_526
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.∙-congʳ
d_'8729''45'cong'691'_432 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_432 :: (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_432 = (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsBand.∙-congˡ
d_'8729''45'cong'737'_434 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_434 :: (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_434 = (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-assoc
d_'42''45'assoc_438 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_438 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_438 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-cancelʳ-nonZero
d_'42''45'cancel'691''45'nonZero_440 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cancel'691''45'nonZero_440 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> (T__'8801'__12 -> T_Irrelevant_20)
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cancel'691''45'nonZero_440 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> (T__'8801'__12 -> T_Irrelevant_20)
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-cancelˡ-nonZero
d_'42''45'cancel'737''45'nonZero_442 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cancel'737''45'nonZero_442 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> (T__'8801'__12 -> T_Irrelevant_20)
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cancel'737''45'nonZero_442 = T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> (T__'8801'__12 -> T_Irrelevant_20)
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-comm
d_'42''45'comm_444 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_444 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_444 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-cong
d_'42''45'cong_446 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cong_446 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_446 = T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.∙-congʳ
d_'8729''45'cong'691'_448 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_448 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_448 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.∙-congˡ
d_'8729''45'cong'737'_450 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_450 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_450 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-identity
d_'42''45'identity_452 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_452 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Σ_14
d_'42''45'identity_452 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_1562
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
            ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
               (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.identityʳ
d_identity'691'_454 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_454 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_identity'691'_454 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.identityˡ
d_identity'737'_456 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_456 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_identity'737'_456 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isCommutativeMagma
d_isCommutativeMagma_458 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_458 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_458 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_458 T_IsCancellativeCommutativeSemiring_1872
v4
du_isCommutativeMagma_458 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_458 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_458 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> Any -> Any
forall a b. a -> b
coe
                 T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
                 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
            ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeSemigroup_1520
               (Any -> Any
forall a b. a -> b
coe Any
v2))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_460 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_460 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_460 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_460 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isCommutativeMonoid_460 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_460 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_460 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764)
-> Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeMonoid_1860
      ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
         (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_462 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_462 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_462 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_462 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isCommutativeSemigroup_462 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_462 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_462 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeSemigroup_1520
         ((T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
            (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1)))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-isMagma
d_'42''45'isMagma_464 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_'42''45'isMagma_464 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsMagma_178
d_'42''45'isMagma_464 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872 -> T_IsMagma_178
du_'42''45'isMagma_464 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isMagma_464 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_464 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsMagma_178
du_'42''45'isMagma_464 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsMagma_178
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMagma_178)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.du_'42''45'isMagma_1614
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v2))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-isMonoid
d_'42''45'isMonoid_466 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'42''45'isMonoid_466 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsMonoid_712
d_'42''45'isMonoid_466 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872 -> T_IsMonoid_712
du_'42''45'isMonoid_466 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isMonoid_466 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_466 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsMonoid_712
du_'42''45'isMonoid_466 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsMonoid_712
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMonoid_712)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.du_'42''45'isMonoid_1618
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v2))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.*-isSemigroup
d_'42''45'isSemigroup_468 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_'42''45'isSemigroup_468 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemigroup_488
d_'42''45'isSemigroup_468 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872 -> T_IsSemigroup_488
du_'42''45'isSemigroup_468 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isSemigroup_468 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_468 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsSemigroup_488
du_'42''45'isSemigroup_468 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsSemigroup_488)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.du_'42''45'isSemigroup_1616
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v2))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.assoc
d_assoc_470 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_470 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_470 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.comm
d_comm_472 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_472 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12
d_comm_472 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.∙-cong
d_'8729''45'cong_474 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_474 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_474 = T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.∙-congʳ
d_'8729''45'cong'691'_476 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_476 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_476 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.∙-congˡ
d_'8729''45'cong'737'_478 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_478 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_478 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.identity
d_identity_480 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_480 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Σ_14
d_identity_480 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
                  ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
                     (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.identityʳ
d_identity'691'_482 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_482 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_identity'691'_482 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.identityˡ
d_identity'737'_484 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_484 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_identity'737'_484 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isCommutativeMagma
d_isCommutativeMagma_486 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_486 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_486 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_486 T_IsCancellativeCommutativeSemiring_1872
v4
du_isCommutativeMagma_486 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_486 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_486 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsCommutativeMonoid_764
v4
                   = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                       (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
                  ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
                     (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v4))))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_488 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_488 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_488 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
            ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
               (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isCommutativeSemigroup
d_isCommutativeSemigroup_490 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_490 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_490 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_490 T_IsCancellativeCommutativeSemiring_1872
v4
du_isCommutativeSemigroup_490 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_490 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_490 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
               ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                  (T_IsSemiringWithoutAnnihilatingZero_1536 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3)))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isMagma
d_isMagma_492 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_492 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsMagma_178
d_isMagma_492 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
            ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
               ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                  ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
                     ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
                        T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
                        (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0)))))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isMonoid
d_isMonoid_494 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_494 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsMonoid_712
d_isMonoid_494 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsCommutativeMonoid_764 -> T_IsMonoid_712)
-> Any -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
      ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
               ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
                  (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0)))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isSemigroup
d_isSemigroup_496 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_496 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsSemigroup_488
d_isSemigroup_496 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
                  ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
                     (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isUnitalMagma
d_isUnitalMagma_498 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_498 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsUnitalMagma_666
d_isUnitalMagma_498 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4 = T_IsCancellativeCommutativeSemiring_1872 -> T_IsUnitalMagma_666
du_isUnitalMagma_498 T_IsCancellativeCommutativeSemiring_1872
v4
du_isUnitalMagma_498 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_498 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsUnitalMagma_666
du_isUnitalMagma_498 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsCommutativeMonoid_764
v4
                   = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                       (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
                  ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v4))))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.distrib
d_distrib_500 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_500 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Σ_14
d_distrib_500 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1564
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
            ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
               (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.distribʳ
d_distrib'691'_502 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_502 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_502 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.distribˡ
d_distrib'737'_504 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'737'_504 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_504 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isCommutativeSemiring
d_isCommutativeSemiring_506 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_506 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_506 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe
      T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
      (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0)
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isCommutativeSemiringWithoutOne
d_isCommutativeSemiringWithoutOne_508 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_508 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_508 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_508 T_IsCancellativeCommutativeSemiring_1872
v4
du_isCommutativeSemiringWithoutOne_508 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_508 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_508 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> Any -> T_IsCommutativeSemiringWithoutOne_1438
forall a b. a -> b
coe
      T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
      ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
         (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isEquivalence
d_isEquivalence_510 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_510 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsEquivalence_28
d_isEquivalence_510 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
               ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                  ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                     ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                        T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
                        ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
                           T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
                           (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))))))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isNearSemiring
d_isNearSemiring_512 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_512 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsNearSemiring_1260
d_isNearSemiring_512 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4 = T_IsCancellativeCommutativeSemiring_1872 -> T_IsNearSemiring_1260
du_isNearSemiring_512 T_IsCancellativeCommutativeSemiring_1872
v4
du_isNearSemiring_512 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_512 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsNearSemiring_1260
du_isNearSemiring_512 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsNearSemiring_1260
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_1428
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342) -> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
               (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v2))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isPartialEquivalence
d_isPartialEquivalence_514 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_514 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_514 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsPartialEquivalence_16
du_isPartialEquivalence_514 T_IsCancellativeCommutativeSemiring_1872
v4
du_isPartialEquivalence_514 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_514 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsPartialEquivalence_16
du_isPartialEquivalence_514 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsCommutativeMonoid_764
v4
                   = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                       (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsMonoid_712
v5
                      = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v4) in
                Any -> Any
forall a b. a -> b
coe
                  (let v6 :: T_IsSemigroup_488
v6
                         = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v5) in
                   Any -> Any
forall a b. a -> b
coe
                     (let v7 :: T_IsMagma_178
v7 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v6) in
                      Any -> Any
forall a b. a -> b
coe
                        ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
                           T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                           ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                              T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
                              (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v7)))))))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isSemiring
d_isSemiring_516 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_516 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsSemiring_1640
d_isSemiring_516 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640)
-> Any -> T_IsSemiring_1640
forall a b. a -> b
coe
      T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
      ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
         (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_518 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_518 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_518 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
      ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
         ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
            (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0)))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_520 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_520 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_520 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_520 T_IsCancellativeCommutativeSemiring_1872
v4
du_isSemiringWithoutOne_520 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_520 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_520 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
         ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1)))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.refl
d_refl_522 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_522 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> T__'8801'__12
d_refl_522 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.reflexive
d_reflexive_524 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_524 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_524 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.setoid
d_setoid_526 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_526 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_Setoid_46
d_setoid_526 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4 = T_IsCancellativeCommutativeSemiring_1872 -> T_Setoid_46
du_setoid_526 T_IsCancellativeCommutativeSemiring_1872
v4
du_setoid_526 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_526 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Setoid_46
du_setoid_526 T_IsCancellativeCommutativeSemiring_1872
v0
  = let v1 :: T_IsCommutativeSemiring_1750
v1
          = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
              (T_IsCancellativeCommutativeSemiring_1872
-> T_IsCancellativeCommutativeSemiring_1872
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsCommutativeMonoid_764
v4
                   = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                       (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsMonoid_712
v5
                      = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v4) in
                Any -> Any
forall a b. a -> b
coe
                  (let v6 :: T_IsSemigroup_488
v6
                         = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v5) in
                   Any -> Any
forall a b. a -> b
coe
                     ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
                        T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                        ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v6))))))))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.sym
d_sym_528 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_528 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_528 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.trans
d_trans_530 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_530 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_530 = T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.zero
d_zero_532 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_532 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Σ_14
d_zero_532 T_IsCancellativeCommutativeSemiring_1872
v0
  = (T_IsSemiring_1640 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1656
      ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764
         ((T_IsCancellativeCommutativeSemiring_1872
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1886
            (T_IsCancellativeCommutativeSemiring_1872 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1872
v0)))
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.zeroʳ
d_zero'691'_534 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'691'_534 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_zero'691'_534 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCancellativeCommutativeSemiring.zeroˡ
d_zero'737'_536 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'737'_536 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_zero'737'_536 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.assoc
d_assoc_540 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_540 :: T_IsCommutativeBand_612
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_540 = T_IsCommutativeBand_612
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.comm
d_comm_542 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_542 :: T_IsCommutativeBand_612 -> Integer -> Integer -> T__'8801'__12
d_comm_542 = T_IsCommutativeBand_612 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.idem
d_idem_544 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_544 :: T_IsCommutativeBand_612 -> Integer -> T__'8801'__12
d_idem_544 = T_IsCommutativeBand_612 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.isBand
d_isBand_546 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_546 :: T_IsCommutativeBand_612 -> T_IsBand_526
d_isBand_546 T_IsCommutativeBand_612
v0
  = (T_IsCommutativeBand_612 -> T_IsBand_526) -> Any -> T_IsBand_526
forall a b. a -> b
coe T_IsCommutativeBand_612 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.d_isBand_620 (T_IsCommutativeBand_612 -> Any
forall a b. a -> b
coe T_IsCommutativeBand_612
v0)
-- Data.Integer.Properties._.IsCommutativeBand.isCommutativeMagma
d_isCommutativeMagma_548 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_548 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612 -> T_IsCommutativeMagma_214
d_isCommutativeMagma_548 ~Integer -> Integer -> Integer
v0 T_IsCommutativeBand_612
v1 = T_IsCommutativeBand_612 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_548 T_IsCommutativeBand_612
v1
du_isCommutativeMagma_548 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_548 :: T_IsCommutativeBand_612 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_548 T_IsCommutativeBand_612
v0
  = (T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
      ((T_IsCommutativeBand_612 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeBand_612 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_654
         (T_IsCommutativeBand_612 -> Any
forall a b. a -> b
coe T_IsCommutativeBand_612
v0))
-- Data.Integer.Properties._.IsCommutativeBand.isCommutativeSemigroup
d_isCommutativeSemigroup_550 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_550 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612 -> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_550 Integer -> Integer -> Integer
v0 T_IsCommutativeBand_612
v1
  = (T_IsCommutativeBand_612 -> T_IsCommutativeSemigroup_568)
-> T_IsCommutativeBand_612 -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      T_IsCommutativeBand_612 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_654 T_IsCommutativeBand_612
v1
-- Data.Integer.Properties._.IsCommutativeBand.isEquivalence
d_isEquivalence_552 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_552 :: T_IsCommutativeBand_612 -> T_IsEquivalence_28
d_isEquivalence_552 T_IsCommutativeBand_612
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsBand_526 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534
            ((T_IsCommutativeBand_612 -> T_IsBand_526) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeBand_612 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.d_isBand_620 (T_IsCommutativeBand_612 -> Any
forall a b. a -> b
coe T_IsCommutativeBand_612
v0))))
-- Data.Integer.Properties._.IsCommutativeBand.isMagma
d_isMagma_554 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_554 :: T_IsCommutativeBand_612 -> T_IsMagma_178
d_isMagma_554 T_IsCommutativeBand_612
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsBand_526 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534
         ((T_IsCommutativeBand_612 -> T_IsBand_526) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeBand_612 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.d_isBand_620 (T_IsCommutativeBand_612 -> Any
forall a b. a -> b
coe T_IsCommutativeBand_612
v0)))
-- Data.Integer.Properties._.IsCommutativeBand.isPartialEquivalence
d_isPartialEquivalence_556 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_556 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_556 ~Integer -> Integer -> Integer
v0 T_IsCommutativeBand_612
v1 = T_IsCommutativeBand_612 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_556 T_IsCommutativeBand_612
v1
du_isPartialEquivalence_556 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_556 :: T_IsCommutativeBand_612 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_556 T_IsCommutativeBand_612
v0
  = let v1 :: T_IsBand_526
v1 = T_IsCommutativeBand_612 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.d_isBand_620 (T_IsCommutativeBand_612 -> T_IsCommutativeBand_612
forall a b. a -> b
coe T_IsCommutativeBand_612
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534 (T_IsBand_526 -> T_IsBand_526
forall a b. a -> b
coe T_IsBand_526
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Data.Integer.Properties._.IsCommutativeBand.isSemigroup
d_isSemigroup_558 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_558 :: T_IsCommutativeBand_612 -> T_IsSemigroup_488
d_isSemigroup_558 T_IsCommutativeBand_612
v0
  = (T_IsBand_526 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534
      ((T_IsCommutativeBand_612 -> T_IsBand_526) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeBand_612 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.d_isBand_620 (T_IsCommutativeBand_612 -> Any
forall a b. a -> b
coe T_IsCommutativeBand_612
v0))
-- Data.Integer.Properties._.IsCommutativeBand.refl
d_refl_560 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_560 :: T_IsCommutativeBand_612 -> Integer -> T__'8801'__12
d_refl_560 = T_IsCommutativeBand_612 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.reflexive
d_reflexive_562 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_562 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_562 = (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.setoid
d_setoid_564 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_564 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612 -> T_Setoid_46
d_setoid_564 ~Integer -> Integer -> Integer
v0 T_IsCommutativeBand_612
v1 = T_IsCommutativeBand_612 -> T_Setoid_46
du_setoid_564 T_IsCommutativeBand_612
v1
du_setoid_564 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_564 :: T_IsCommutativeBand_612 -> T_Setoid_46
du_setoid_564 T_IsCommutativeBand_612
v0
  = let v1 :: T_IsBand_526
v1 = T_IsCommutativeBand_612 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.d_isBand_620 (T_IsCommutativeBand_612 -> T_IsCommutativeBand_612
forall a b. a -> b
coe T_IsCommutativeBand_612
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsBand_526 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_534 (T_IsBand_526 -> T_IsBand_526
forall a b. a -> b
coe T_IsBand_526
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v2))))
-- Data.Integer.Properties._.IsCommutativeBand.sym
d_sym_566 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_566 :: T_IsCommutativeBand_612
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_566 = T_IsCommutativeBand_612
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.trans
d_trans_568 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_568 :: T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_568 = T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.∙-cong
d_'8729''45'cong_570 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_570 :: T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_570 = T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.∙-congʳ
d_'8729''45'cong'691'_572 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_572 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_572 = (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeBand.∙-congˡ
d_'8729''45'cong'737'_574 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_574 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_574 = (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMagma.comm
d_comm_578 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_578 :: T_IsCommutativeMagma_214 -> Integer -> Integer -> T__'8801'__12
d_comm_578 = T_IsCommutativeMagma_214 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMagma.isEquivalence
d_isEquivalence_580 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_580 :: T_IsCommutativeMagma_214 -> T_IsEquivalence_28
d_isEquivalence_580 T_IsCommutativeMagma_214
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsCommutativeMagma_214 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMagma_214 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_222 (T_IsCommutativeMagma_214 -> Any
forall a b. a -> b
coe T_IsCommutativeMagma_214
v0))
-- Data.Integer.Properties._.IsCommutativeMagma.isMagma
d_isMagma_582 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_582 :: T_IsCommutativeMagma_214 -> T_IsMagma_178
d_isMagma_582 T_IsCommutativeMagma_214
v0
  = (T_IsCommutativeMagma_214 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe T_IsCommutativeMagma_214 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_222 (T_IsCommutativeMagma_214 -> Any
forall a b. a -> b
coe T_IsCommutativeMagma_214
v0)
-- Data.Integer.Properties._.IsCommutativeMagma.isPartialEquivalence
d_isPartialEquivalence_584 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_584 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_584 ~Integer -> Integer -> Integer
v0 T_IsCommutativeMagma_214
v1 = T_IsCommutativeMagma_214 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_584 T_IsCommutativeMagma_214
v1
du_isPartialEquivalence_584 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_584 :: T_IsCommutativeMagma_214 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_584 T_IsCommutativeMagma_214
v0
  = let v1 :: T_IsMagma_178
v1
          = T_IsCommutativeMagma_214 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_222 (T_IsCommutativeMagma_214 -> T_IsCommutativeMagma_214
forall a b. a -> b
coe T_IsCommutativeMagma_214
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
         T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
         ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v1)))
-- Data.Integer.Properties._.IsCommutativeMagma.refl
d_refl_586 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_586 :: T_IsCommutativeMagma_214 -> Integer -> T__'8801'__12
d_refl_586 = T_IsCommutativeMagma_214 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMagma.reflexive
d_reflexive_588 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_588 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_588 = (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMagma.setoid
d_setoid_590 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_590 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214 -> T_Setoid_46
d_setoid_590 ~Integer -> Integer -> Integer
v0 T_IsCommutativeMagma_214
v1 = T_IsCommutativeMagma_214 -> T_Setoid_46
du_setoid_590 T_IsCommutativeMagma_214
v1
du_setoid_590 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_590 :: T_IsCommutativeMagma_214 -> T_Setoid_46
du_setoid_590 T_IsCommutativeMagma_214
v0
  = (T_IsMagma_178 -> T_Setoid_46) -> Any -> T_Setoid_46
forall a b. a -> b
coe
      T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
      ((T_IsCommutativeMagma_214 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMagma_214 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_222 (T_IsCommutativeMagma_214 -> Any
forall a b. a -> b
coe T_IsCommutativeMagma_214
v0))
-- Data.Integer.Properties._.IsCommutativeMagma.sym
d_sym_592 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_592 :: T_IsCommutativeMagma_214
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_592 = T_IsCommutativeMagma_214
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMagma.trans
d_trans_594 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_594 :: T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_594 = T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMagma.∙-cong
d_'8729''45'cong_596 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_596 :: T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_596 = T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMagma.∙-congʳ
d_'8729''45'cong'691'_598 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_598 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_598 = (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMagma.∙-congˡ
d_'8729''45'cong'737'_600 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_600 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_600 = (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.assoc
d_assoc_604 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_604 :: T_IsCommutativeMonoid_764
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_604 = T_IsCommutativeMonoid_764
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.comm
d_comm_606 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_606 :: T_IsCommutativeMonoid_764 -> Integer -> Integer -> T__'8801'__12
d_comm_606 = T_IsCommutativeMonoid_764 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.identity
d_identity_608 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_608 :: T_IsCommutativeMonoid_764 -> T_Σ_14
d_identity_608 T_IsCommutativeMonoid_764
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0))
-- Data.Integer.Properties._.IsCommutativeMonoid.identityʳ
d_identity'691'_610 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_610 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
d_identity'691'_610 = (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.identityˡ
d_identity'737'_612 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_612 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
d_identity'737'_612 = (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.isCommutativeMagma
d_isCommutativeMagma_614 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_614 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> T_IsCommutativeMagma_214
d_isCommutativeMagma_614 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsCommutativeMonoid_764
v2 = T_IsCommutativeMonoid_764 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_614 T_IsCommutativeMonoid_764
v2
du_isCommutativeMagma_614 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_614 :: T_IsCommutativeMonoid_764 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_614 T_IsCommutativeMonoid_764
v0
  = (T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
      ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
         (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0))
-- Data.Integer.Properties._.IsCommutativeMonoid.isCommutativeSemigroup
d_isCommutativeSemigroup_616 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_616 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_616 Integer -> Integer -> Integer
v0 Integer
v1 T_IsCommutativeMonoid_764
v2
  = (T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814 T_IsCommutativeMonoid_764
v2
-- Data.Integer.Properties._.IsCommutativeMonoid.isEquivalence
d_isEquivalence_618 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_618 :: T_IsCommutativeMonoid_764 -> T_IsEquivalence_28
d_isEquivalence_618 T_IsCommutativeMonoid_764
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0))))
-- Data.Integer.Properties._.IsCommutativeMonoid.isMagma
d_isMagma_620 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_620 :: T_IsCommutativeMonoid_764 -> T_IsMagma_178
d_isMagma_620 T_IsCommutativeMonoid_764
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0)))
-- Data.Integer.Properties._.IsCommutativeMonoid.isMonoid
d_isMonoid_622 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_622 :: T_IsCommutativeMonoid_764 -> T_IsMonoid_712
d_isMonoid_622 T_IsCommutativeMonoid_764
v0
  = (T_IsCommutativeMonoid_764 -> T_IsMonoid_712)
-> Any -> T_IsMonoid_712
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0)
-- Data.Integer.Properties._.IsCommutativeMonoid.isPartialEquivalence
d_isPartialEquivalence_624 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_624 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_624 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsCommutativeMonoid_764
v2
  = T_IsCommutativeMonoid_764 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_624 T_IsCommutativeMonoid_764
v2
du_isPartialEquivalence_624 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_624 :: T_IsCommutativeMonoid_764 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_624 T_IsCommutativeMonoid_764
v0
  = let v1 :: T_IsMonoid_712
v1
          = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Data.Integer.Properties._.IsCommutativeMonoid.isSemigroup
d_isSemigroup_626 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_626 :: T_IsCommutativeMonoid_764 -> T_IsSemigroup_488
d_isSemigroup_626 T_IsCommutativeMonoid_764
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0))
-- Data.Integer.Properties._.IsCommutativeMonoid.isUnitalMagma
d_isUnitalMagma_628 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_628 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> T_IsUnitalMagma_666
d_isUnitalMagma_628 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsCommutativeMonoid_764
v2 = T_IsCommutativeMonoid_764 -> T_IsUnitalMagma_666
du_isUnitalMagma_628 T_IsCommutativeMonoid_764
v2
du_isUnitalMagma_628 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_628 :: T_IsCommutativeMonoid_764 -> T_IsUnitalMagma_666
du_isUnitalMagma_628 T_IsCommutativeMonoid_764
v0
  = (T_IsMonoid_712 -> T_IsUnitalMagma_666)
-> Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0))
-- Data.Integer.Properties._.IsCommutativeMonoid.refl
d_refl_630 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_630 :: T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
d_refl_630 = T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.reflexive
d_reflexive_632 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_632 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_632 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.setoid
d_setoid_634 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_634 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> T_Setoid_46
d_setoid_634 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsCommutativeMonoid_764
v2 = T_IsCommutativeMonoid_764 -> T_Setoid_46
du_setoid_634 T_IsCommutativeMonoid_764
v2
du_setoid_634 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_634 :: T_IsCommutativeMonoid_764 -> T_Setoid_46
du_setoid_634 T_IsCommutativeMonoid_764
v0
  = let v1 :: T_IsMonoid_712
v1
          = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v2))))
-- Data.Integer.Properties._.IsCommutativeMonoid.sym
d_sym_636 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_636 :: T_IsCommutativeMonoid_764
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_636 = T_IsCommutativeMonoid_764
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.trans
d_trans_638 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_638 :: T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_638 = T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.∙-cong
d_'8729''45'cong_640 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_640 :: T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_640 = T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.∙-congʳ
d_'8729''45'cong'691'_642 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_642 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_642 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeMonoid.∙-congˡ
d_'8729''45'cong'737'_644 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_644 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_644 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing._//_
d__'47''47'__648 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> Integer -> Integer
d__'47''47'__648 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
d__'47''47'__648 Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 ~T_IsCommutativeRing_2888
v5 = (Integer -> Integer -> Integer)
-> (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__648 Integer -> Integer -> Integer
v0 Integer -> Integer
v2
du__'47''47'__648 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__648 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__648 Integer -> Integer -> Integer
v0 Integer -> Integer
v1
  = ((Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any)
-> Any -> Any -> Integer -> Integer -> Integer
forall a b. a -> b
coe
      (Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any
MAlonzo.Code.Algebra.Structures.du__'47''47'__1136 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0)
      ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v1)
-- Data.Integer.Properties._.IsCommutativeRing.*-assoc
d_'42''45'assoc_650 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_650 :: T_IsCommutativeRing_2888
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_650 = T_IsCommutativeRing_2888
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.*-comm
d_'42''45'comm_652 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_652 :: T_IsCommutativeRing_2888 -> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_652 = T_IsCommutativeRing_2888 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.*-cong
d_'42''45'cong_654 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cong_654 :: T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_654 = T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.∙-congʳ
d_'8729''45'cong'691'_656 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_656 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_656 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.∙-congˡ
d_'8729''45'cong'737'_658 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_658 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_658 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.*-identity
d_'42''45'identity_660 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_660 :: T_IsCommutativeRing_2888 -> T_Σ_14
d_'42''45'identity_660 T_IsCommutativeRing_2888
v0
  = (T_IsRing_2740 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsRing_2740 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_2768
      ((T_IsCommutativeRing_2888 -> T_IsRing_2740) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0))
-- Data.Integer.Properties._.IsCommutativeRing.identityʳ
d_identity'691'_662 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_662 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_identity'691'_662 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.identityˡ
d_identity'737'_664 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_664 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_identity'737'_664 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.isCommutativeMagma
d_isCommutativeMagma_666 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_666 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_666 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_666 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isCommutativeMagma_666 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_666 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_666 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = let v5 :: Any
v5
          = ((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsCommutativeRing_2888
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
              (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiring_3018
              ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0) ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3) (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v6 :: Any
v6
             = (T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> Any -> Any
forall a b. a -> b
coe
                 T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
                 (Any -> Any
forall a b. a -> b
coe Any
v5) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
            ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeSemigroup_1520
               (Any -> Any
forall a b. a -> b
coe Any
v6))))
-- Data.Integer.Properties._.IsCommutativeRing.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_668 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_668 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_668 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_668 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isCommutativeMonoid_668 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_668 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_668 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = (T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764)
-> Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeMonoid_1860
      (((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsCommutativeRing_2888
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiring_3018
         ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0) ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3) (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4))
-- Data.Integer.Properties._.IsCommutativeRing.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_670 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_670 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_670 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_670 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isCommutativeSemigroup_670 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_670 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_670 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = let v5 :: Any
v5
          = ((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsCommutativeRing_2888
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
              (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiring_3018
              ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0) ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3) (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4) in
    Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeSemigroup_1520
         ((T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
            (Any -> Any
forall a b. a -> b
coe Any
v5)))
-- Data.Integer.Properties._.IsCommutativeRing.*-isMagma
d_'42''45'isMagma_672 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_'42''45'isMagma_672 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMagma_178
d_'42''45'isMagma_672 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMagma_178
du_'42''45'isMagma_672 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isMagma_672 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_672 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMagma_178
du_'42''45'isMagma_672 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = let v5 :: T_IsRing_2740
v5
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4) in
    Any -> T_IsMagma_178
forall a b. a -> b
coe
      (let v6 :: Any
v6
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v5) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsNearSemiring_1260 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
            T_IsNearSemiring_1260 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.du_'42''45'isMagma_1324
            (((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsRingWithoutOne_2368
 -> T_IsNearSemiring_1260)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
               (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsRingWithoutOne_2368
-> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_2470 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0)
               ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3) (Any -> Any
forall a b. a -> b
coe Any
v6))))
-- Data.Integer.Properties._.IsCommutativeRing.*-isMonoid
d_'42''45'isMonoid_674 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'42''45'isMonoid_674 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMonoid_712
d_'42''45'isMonoid_674 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMonoid_712
du_'42''45'isMonoid_674 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isMonoid_674 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_674 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMonoid_712
du_'42''45'isMonoid_674 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = ((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsRing_2740
 -> T_IsMonoid_712)
-> Any -> Any -> Any -> Any -> Any -> T_IsMonoid_712
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsRing_2740
-> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.du_'42''45'isMonoid_2860 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0)
      ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3)
      ((T_IsCommutativeRing_2888 -> T_IsRing_2740) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4))
-- Data.Integer.Properties._.IsCommutativeRing.*-isSemigroup
d_'42''45'isSemigroup_676 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_'42''45'isSemigroup_676 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemigroup_488
d_'42''45'isSemigroup_676 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemigroup_488
du_'42''45'isSemigroup_676 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isSemigroup_676 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_676 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemigroup_488
du_'42''45'isSemigroup_676 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = let v5 :: T_IsRing_2740
v5
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4) in
    Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      (let v6 :: Any
v6
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v5) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsNearSemiring_1260 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsNearSemiring_1260 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.du_'42''45'isSemigroup_1326
            (((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsRingWithoutOne_2368
 -> T_IsNearSemiring_1260)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
               (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsRingWithoutOne_2368
-> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_2470 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0)
               ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3) (Any -> Any
forall a b. a -> b
coe Any
v6))))
-- Data.Integer.Properties._.IsCommutativeRing.assoc
d_assoc_678 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_678 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_assoc_678 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.comm
d_comm_680 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_680 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
d_comm_680 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.∙-cong
d_'8729''45'cong_682 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_682 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_682 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.∙-congʳ
d_'8729''45'cong'691'_684 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_684 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_684 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.∙-congˡ
d_'8729''45'cong'737'_686 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_686 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_686 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.identity
d_identity_688 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_688 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
d_identity_688 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5 = T_IsCommutativeRing_2888 -> T_Σ_14
du_identity_688 T_IsCommutativeRing_2888
v5
du_identity_688 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_identity_688 :: T_IsCommutativeRing_2888 -> T_Σ_14
du_identity_688 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_Σ_14
forall a b. a -> b
coe
      ((T_IsMonoid_712 -> T_Σ_14) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
            T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
            ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe
               T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
               ((T_IsRing_2740 -> T_IsAbelianGroup_1172) -> Any -> Any
forall a b. a -> b
coe
                  T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
                  (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1)))))
-- Data.Integer.Properties._.IsCommutativeRing.identityʳ
d_identity'691'_690 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_690 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_identity'691'_690 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.identityˡ
d_identity'737'_692 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_692 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_identity'737'_692 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.+-isAbelianGroup
d_'43''45'isAbelianGroup_694 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_694 :: T_IsCommutativeRing_2888 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_694 T_IsCommutativeRing_2888
v0
  = (T_IsRing_2740 -> T_IsAbelianGroup_1172)
-> Any -> T_IsAbelianGroup_1172
forall a b. a -> b
coe
      T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
      ((T_IsCommutativeRing_2888 -> T_IsRing_2740) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0))
-- Data.Integer.Properties._.IsCommutativeRing.isCommutativeMagma
d_isCommutativeMagma_696 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_696 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_696 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_696 T_IsCommutativeRing_2888
v5
du_isCommutativeMagma_696 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_696 :: T_IsCommutativeRing_2888 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_696 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsAbelianGroup_1172
v3
                = T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2386
                    (Any -> T_IsRingWithoutOne_2368
forall a b. a -> b
coe Any
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: Any
v4
                   = (T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764) -> Any -> Any
forall a b. a -> b
coe
                       T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
                       (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
                  ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
                     (Any -> Any
forall a b. a -> b
coe Any
v4))))))
-- Data.Integer.Properties._.IsCommutativeRing.isCommutativeMonoid
d_isCommutativeMonoid_698 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_698 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_698 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_698 T_IsCommutativeRing_2888
v5
du_isCommutativeMonoid_698 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_isCommutativeMonoid_698 :: T_IsCommutativeRing_2888 -> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_698 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764) -> Any -> Any
forall a b. a -> b
coe
            T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
            ((T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172) -> Any -> Any
forall a b. a -> b
coe
               T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2386
               (Any -> Any
forall a b. a -> b
coe Any
v2))))
-- Data.Integer.Properties._.IsCommutativeRing.isCommutativeSemigroup
d_isCommutativeSemigroup_700 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_700 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_700 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_700 T_IsCommutativeRing_2888
v5
du_isCommutativeSemigroup_700 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_700 :: T_IsCommutativeRing_2888 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_700 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsAbelianGroup_1172
v3
                = T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2386
                    (Any -> T_IsRingWithoutOne_2368
forall a b. a -> b
coe Any
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
               ((T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764) -> Any -> Any
forall a b. a -> b
coe
                  T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
                  (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3)))))
-- Data.Integer.Properties._.IsCommutativeRing.isGroup
d_isGroup_702 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074
d_isGroup_702 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsGroup_1074
d_isGroup_702 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5 = T_IsCommutativeRing_2888 -> T_IsGroup_1074
du_isGroup_702 T_IsCommutativeRing_2888
v5
du_isGroup_702 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074
du_isGroup_702 :: T_IsCommutativeRing_2888 -> T_IsGroup_1074
du_isGroup_702 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsGroup_1074
forall a b. a -> b
coe
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe
         T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
         ((T_IsRing_2740 -> T_IsAbelianGroup_1172) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
            (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1)))
-- Data.Integer.Properties._.IsCommutativeRing.isInvertibleMagma
d_isInvertibleMagma_704 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_704 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsInvertibleMagma_958
d_isInvertibleMagma_704 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_704 T_IsCommutativeRing_2888
v5
du_isInvertibleMagma_704 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
du_isInvertibleMagma_704 :: T_IsCommutativeRing_2888 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_704 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsInvertibleMagma_958
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsAbelianGroup_1172
v3
                = T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2386
                    (Any -> T_IsRingWithoutOne_2368
forall a b. a -> b
coe Any
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsGroup_1074 -> T_IsInvertibleMagma_958) -> Any -> Any
forall a b. a -> b
coe
               T_IsGroup_1074 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1160
               ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3)))))
-- Data.Integer.Properties._.IsCommutativeRing.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_706 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_706 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_706 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_706 T_IsCommutativeRing_2888
v5
du_isInvertibleUnitalMagma_706 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_706 :: T_IsCommutativeRing_2888 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_706 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsInvertibleUnitalMagma_1012
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsAbelianGroup_1172
v3
                = T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2386
                    (Any -> T_IsRingWithoutOne_2368
forall a b. a -> b
coe Any
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012) -> Any -> Any
forall a b. a -> b
coe
               T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012
MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1162
               ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3)))))
-- Data.Integer.Properties._.IsCommutativeRing.isMagma
d_isMagma_708 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_708 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMagma_178
d_isMagma_708 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5 = T_IsCommutativeRing_2888 -> T_IsMagma_178
du_isMagma_708 T_IsCommutativeRing_2888
v5
du_isMagma_708 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_isMagma_708 :: T_IsCommutativeRing_2888 -> T_IsMagma_178
du_isMagma_708 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsMagma_178
forall a b. a -> b
coe
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
               T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
               ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe
                  T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                  ((T_IsRing_2740 -> T_IsAbelianGroup_1172) -> Any -> Any
forall a b. a -> b
coe
                     T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
                     (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1))))))
-- Data.Integer.Properties._.IsCommutativeRing.isMonoid
d_isMonoid_710 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_710 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMonoid_712
d_isMonoid_710 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5 = T_IsCommutativeRing_2888 -> T_IsMonoid_712
du_isMonoid_710 T_IsCommutativeRing_2888
v5
du_isMonoid_710 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_isMonoid_710 :: T_IsCommutativeRing_2888 -> T_IsMonoid_712
du_isMonoid_710 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsMonoid_712
forall a b. a -> b
coe
      ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe
            T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
            ((T_IsRing_2740 -> T_IsAbelianGroup_1172) -> Any -> Any
forall a b. a -> b
coe
               T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
               (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1))))
-- Data.Integer.Properties._.IsCommutativeRing.isSemigroup
d_isSemigroup_712 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_712 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemigroup_488
d_isSemigroup_712 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5 = T_IsCommutativeRing_2888 -> T_IsSemigroup_488
du_isSemigroup_712 T_IsCommutativeRing_2888
v5
du_isSemigroup_712 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_isSemigroup_712 :: T_IsCommutativeRing_2888 -> T_IsSemigroup_488
du_isSemigroup_712 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
            T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
            ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe
               T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
               ((T_IsRing_2740 -> T_IsAbelianGroup_1172) -> Any -> Any
forall a b. a -> b
coe
                  T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
                  (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1)))))
-- Data.Integer.Properties._.IsCommutativeRing.isUnitalMagma
d_isUnitalMagma_714 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_714 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsUnitalMagma_666
d_isUnitalMagma_714 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsUnitalMagma_666
du_isUnitalMagma_714 T_IsCommutativeRing_2888
v5
du_isUnitalMagma_714 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_714 :: T_IsCommutativeRing_2888 -> T_IsUnitalMagma_666
du_isUnitalMagma_714 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsAbelianGroup_1172
v3
                = T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2386
                    (Any -> T_IsRingWithoutOne_2368
forall a b. a -> b
coe Any
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsGroup_1074
v4
                   = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
                  ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v4))))))
-- Data.Integer.Properties._.IsCommutativeRing.⁻¹-cong
d_'8315''185''45'cong_716 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_716 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8315''185''45'cong_716 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.inverse
d_inverse_718 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_718 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
d_inverse_718 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5 = T_IsCommutativeRing_2888 -> T_Σ_14
du_inverse_718 T_IsCommutativeRing_2888
v5
du_inverse_718 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_inverse_718 :: T_IsCommutativeRing_2888 -> T_Σ_14
du_inverse_718 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_Σ_14
forall a b. a -> b
coe
      ((T_IsGroup_1074 -> T_Σ_14) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_1074 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_1090
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe
            T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
            ((T_IsRing_2740 -> T_IsAbelianGroup_1172) -> Any -> Any
forall a b. a -> b
coe
               T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
               (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1))))
-- Data.Integer.Properties._.IsCommutativeRing.inverseʳ
d_inverse'691'_720 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'691'_720 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_inverse'691'_720 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.inverseˡ
d_inverse'737'_722 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'737'_722 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_inverse'737'_722 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.distrib
d_distrib_724 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_724 :: T_IsCommutativeRing_2888 -> T_Σ_14
d_distrib_724 T_IsCommutativeRing_2888
v0
  = (T_IsRing_2740 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsRing_2740 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_2770
      ((T_IsCommutativeRing_2888 -> T_IsRing_2740) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0))
-- Data.Integer.Properties._.IsCommutativeRing.distribʳ
d_distrib'691'_726 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_726 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_726 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.distribˡ
d_distrib'737'_728 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'737'_728 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_728 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.isCommutativeSemiring
d_isCommutativeSemiring_730 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_730 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_730 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 Integer
v4 T_IsCommutativeRing_2888
v5
  = ((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsCommutativeRing_2888
 -> T_IsCommutativeSemiring_1750)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiring_3018 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1
      Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
-- Data.Integer.Properties._.IsCommutativeRing.isCommutativeSemiringWithoutOne
d_isCommutativeSemiringWithoutOne_732 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_732 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_732 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_732 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isCommutativeSemiringWithoutOne_732 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_732 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_732 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = (T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> Any -> T_IsCommutativeSemiringWithoutOne_1438
forall a b. a -> b
coe
      T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
      (((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsCommutativeRing_2888
 -> T_IsCommutativeSemiring_1750)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiring_3018
         ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0) ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3) (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4))
-- Data.Integer.Properties._.IsCommutativeRing.isEquivalence
d_isEquivalence_734 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_734 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsEquivalence_28
d_isEquivalence_734 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsEquivalence_28
du_isEquivalence_734 T_IsCommutativeRing_2888
v5
du_isEquivalence_734 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
du_isEquivalence_734 :: T_IsCommutativeRing_2888 -> T_IsEquivalence_28
du_isEquivalence_734 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
         T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
                  T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
                  ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> Any -> Any
forall a b. a -> b
coe
                     T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                     ((T_IsRing_2740 -> T_IsAbelianGroup_1172) -> Any -> Any
forall a b. a -> b
coe
                        T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
                        (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1)))))))
-- Data.Integer.Properties._.IsCommutativeRing.isNearSemiring
d_isNearSemiring_736 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_736 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsNearSemiring_1260
d_isNearSemiring_736 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsNearSemiring_1260
du_isNearSemiring_736 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isNearSemiring_736 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_736 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsNearSemiring_1260
du_isNearSemiring_736 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = let v5 :: T_IsRing_2740
v5
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4) in
    Any -> T_IsNearSemiring_1260
forall a b. a -> b
coe
      (((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsRingWithoutOne_2368
 -> T_IsNearSemiring_1260)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsRingWithoutOne_2368
-> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_2470 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0)
         ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3)
         ((T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v5)))
-- Data.Integer.Properties._.IsCommutativeRing.isPartialEquivalence
d_isPartialEquivalence_738 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_738 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_738 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_738 T_IsCommutativeRing_2888
v5
du_isPartialEquivalence_738 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_738 :: T_IsCommutativeRing_2888 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_738 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsAbelianGroup_1172
v3
                = T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2386
                    (Any -> T_IsRingWithoutOne_2368
forall a b. a -> b
coe Any
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsGroup_1074
v4
                   = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsMonoid_712
v5
                      = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v4) in
                Any -> Any
forall a b. a -> b
coe
                  (let v6 :: T_IsSemigroup_488
v6
                         = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v5) in
                   Any -> Any
forall a b. a -> b
coe
                     (let v7 :: T_IsMagma_178
v7 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v6) in
                      Any -> Any
forall a b. a -> b
coe
                        ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
                           T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                           ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                              T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
                              (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v7)))))))))
-- Data.Integer.Properties._.IsCommutativeRing.isRing
d_isRing_740 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740
d_isRing_740 :: T_IsCommutativeRing_2888 -> T_IsRing_2740
d_isRing_740 T_IsCommutativeRing_2888
v0
  = (T_IsCommutativeRing_2888 -> T_IsRing_2740) -> Any -> T_IsRing_2740
forall a b. a -> b
coe T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0)
-- Data.Integer.Properties._.IsCommutativeRing.isRingWithoutOne
d_isRingWithoutOne_742 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsRingWithoutOne_2368
d_isRingWithoutOne_742 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsRingWithoutOne_2368
d_isRingWithoutOne_742 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888 -> T_IsRingWithoutOne_2368
du_isRingWithoutOne_742 T_IsCommutativeRing_2888
v5
du_isRingWithoutOne_742 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsRingWithoutOne_2368
du_isRingWithoutOne_742 :: T_IsCommutativeRing_2888 -> T_IsRingWithoutOne_2368
du_isRingWithoutOne_742 T_IsCommutativeRing_2888
v0
  = (T_IsRing_2740 -> T_IsRingWithoutOne_2368)
-> Any -> T_IsRingWithoutOne_2368
forall a b. a -> b
coe
      T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
      ((T_IsCommutativeRing_2888 -> T_IsRing_2740) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0))
-- Data.Integer.Properties._.IsCommutativeRing.isSemiring
d_isSemiring_744 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_744 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiring_1640
d_isSemiring_744 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiring_1640
du_isSemiring_744 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isSemiring_744 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
du_isSemiring_744 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiring_1640
du_isSemiring_744 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = ((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsRing_2740
 -> T_IsSemiring_1640)
-> Any -> Any -> Any -> Any -> Any -> T_IsSemiring_1640
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsRing_2740
-> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.du_isSemiring_2870 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0)
      ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3)
      ((T_IsCommutativeRing_2888 -> T_IsRing_2740) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4))
-- Data.Integer.Properties._.IsCommutativeRing.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_746 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_746 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_746 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = T_IsCommutativeRing_2888
-> T_IsSemiringWithoutAnnihilatingZero_1536
du_isSemiringWithoutAnnihilatingZero_746 T_IsCommutativeRing_2888
v5
du_isSemiringWithoutAnnihilatingZero_746 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
du_isSemiringWithoutAnnihilatingZero_746 :: T_IsCommutativeRing_2888
-> T_IsSemiringWithoutAnnihilatingZero_1536
du_isSemiringWithoutAnnihilatingZero_746 T_IsCommutativeRing_2888
v0
  = (T_IsRing_2740 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe
      T_IsRing_2740 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutAnnihilatingZero_2868
      ((T_IsCommutativeRing_2888 -> T_IsRing_2740) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0))
-- Data.Integer.Properties._.IsCommutativeRing.isSemiringWithoutOne
d_isSemiringWithoutOne_748 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_748 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_748 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5
  = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_748 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isSemiringWithoutOne_748 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_748 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_748 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = let v5 :: T_IsRing_2740
v5
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4) in
    Any -> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
         (((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsRing_2740
 -> T_IsSemiring_1640)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
            (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsRing_2740
-> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.du_isSemiring_2870 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0)
            ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1) ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3) (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v5)))
-- Data.Integer.Properties._.IsCommutativeRing.refl
d_refl_750 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_750 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_refl_750 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.reflexive
d_reflexive_752 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_752 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_752 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.setoid
d_setoid_754 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_754 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_Setoid_46
d_setoid_754 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5 = T_IsCommutativeRing_2888 -> T_Setoid_46
du_setoid_754 T_IsCommutativeRing_2888
v5
du_setoid_754 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_754 :: T_IsCommutativeRing_2888 -> T_Setoid_46
du_setoid_754 T_IsCommutativeRing_2888
v0
  = let v1 :: T_IsRing_2740
v1
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
                 T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772
                 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsAbelianGroup_1172
v3
                = T_IsRingWithoutOne_2368 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2386
                    (Any -> T_IsRingWithoutOne_2368
forall a b. a -> b
coe Any
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsGroup_1074
v4
                   = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsMonoid_712
v5
                      = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v4) in
                Any -> Any
forall a b. a -> b
coe
                  (let v6 :: T_IsSemigroup_488
v6
                         = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v5) in
                   Any -> Any
forall a b. a -> b
coe
                     ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
                        T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                        ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v6))))))))
-- Data.Integer.Properties._.IsCommutativeRing.sym
d_sym_756 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_756 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_sym_756 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.trans
d_trans_758 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_758 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_758 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_760 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_unique'691''45''8315''185'_760 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'691''45''8315''185'_760 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_762 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_unique'737''45''8315''185'_762 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'737''45''8315''185'_762 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.zero
d_zero_764 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_764 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
d_zero_764 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 ~Integer
v4 T_IsCommutativeRing_2888
v5 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
du_zero_764 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_zero_764 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_zero_764 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
du_zero_764 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v4
  = let v5 :: T_IsRing_2740
v5
          = T_IsCommutativeRing_2888 -> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.d_isRing_2904 (T_IsCommutativeRing_2888 -> T_IsCommutativeRing_2888
forall a b. a -> b
coe T_IsCommutativeRing_2888
v4) in
    Any -> T_Σ_14
forall a b. a -> b
coe
      (((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> (Any -> Any)
 -> Any
 -> T_IsRingWithoutOne_2368
 -> T_Σ_14)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Any)
-> Any
-> T_IsRingWithoutOne_2368
-> T_Σ_14
MAlonzo.Code.Algebra.Structures.du_zero_2468 ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v0) ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
v1)
         ((Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3)
         ((T_IsRing_2740 -> T_IsRingWithoutOne_2368) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_2740 -> T_IsRingWithoutOne_2368
MAlonzo.Code.Algebra.Structures.du_isRingWithoutOne_2772 (T_IsRing_2740 -> Any
forall a b. a -> b
coe T_IsRing_2740
v5)))
-- Data.Integer.Properties._.IsCommutativeRing.zeroʳ
d_zero'691'_766 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'691'_766 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_zero'691'_766 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeRing.zeroˡ
d_zero'737'_768 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'737'_768 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_zero'737'_768 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.assoc
d_assoc_772 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_772 :: T_IsCommutativeSemigroup_568
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_772 = T_IsCommutativeSemigroup_568
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.comm
d_comm_774 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_774 :: T_IsCommutativeSemigroup_568 -> Integer -> Integer -> T__'8801'__12
d_comm_774 = T_IsCommutativeSemigroup_568 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.isCommutativeMagma
d_isCommutativeMagma_776 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_776 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
d_isCommutativeMagma_776 Integer -> Integer -> Integer
v0 T_IsCommutativeSemigroup_568
v1
  = (T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
forall a b. a -> b
coe T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606 T_IsCommutativeSemigroup_568
v1
-- Data.Integer.Properties._.IsCommutativeSemigroup.isEquivalence
d_isEquivalence_778 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_778 :: T_IsCommutativeSemigroup_568 -> T_IsEquivalence_28
d_isEquivalence_778 T_IsCommutativeSemigroup_568
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_576 (T_IsCommutativeSemigroup_568 -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_568
v0)))
-- Data.Integer.Properties._.IsCommutativeSemigroup.isMagma
d_isMagma_780 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_780 :: T_IsCommutativeSemigroup_568 -> T_IsMagma_178
d_isMagma_780 T_IsCommutativeSemigroup_568
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_576 (T_IsCommutativeSemigroup_568 -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_568
v0))
-- Data.Integer.Properties._.IsCommutativeSemigroup.isPartialEquivalence
d_isPartialEquivalence_782 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_782 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_782 ~Integer -> Integer -> Integer
v0 T_IsCommutativeSemigroup_568
v1 = T_IsCommutativeSemigroup_568 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_782 T_IsCommutativeSemigroup_568
v1
du_isPartialEquivalence_782 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_782 :: T_IsCommutativeSemigroup_568 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_782 T_IsCommutativeSemigroup_568
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_576 (T_IsCommutativeSemigroup_568 -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe T_IsCommutativeSemigroup_568
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
            T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
            ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
               T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v2))))
-- Data.Integer.Properties._.IsCommutativeSemigroup.isSemigroup
d_isSemigroup_784 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_784 :: T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488
d_isSemigroup_784 T_IsCommutativeSemigroup_568
v0
  = (T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488)
-> Any -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_576 (T_IsCommutativeSemigroup_568 -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_568
v0)
-- Data.Integer.Properties._.IsCommutativeSemigroup.refl
d_refl_786 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_786 :: T_IsCommutativeSemigroup_568 -> Integer -> T__'8801'__12
d_refl_786 = T_IsCommutativeSemigroup_568 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.reflexive
d_reflexive_788 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_788 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_788 = (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.setoid
d_setoid_790 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_790 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568 -> T_Setoid_46
d_setoid_790 ~Integer -> Integer -> Integer
v0 T_IsCommutativeSemigroup_568
v1 = T_IsCommutativeSemigroup_568 -> T_Setoid_46
du_setoid_790 T_IsCommutativeSemigroup_568
v1
du_setoid_790 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_790 :: T_IsCommutativeSemigroup_568 -> T_Setoid_46
du_setoid_790 T_IsCommutativeSemigroup_568
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_576 (T_IsCommutativeSemigroup_568 -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe T_IsCommutativeSemigroup_568
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
         T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v1)))
-- Data.Integer.Properties._.IsCommutativeSemigroup.sym
d_sym_792 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_792 :: T_IsCommutativeSemigroup_568
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_792 = T_IsCommutativeSemigroup_568
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.trans
d_trans_794 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_794 :: T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_794 = T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.∙-cong
d_'8729''45'cong_796 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_796 :: T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_796 = T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.∙-congʳ
d_'8729''45'cong'691'_798 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_798 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_798 = (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemigroup.∙-congˡ
d_'8729''45'cong'737'_800 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_800 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_800 = (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.*-assoc
d_'42''45'assoc_804 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_804 :: T_IsCommutativeSemiring_1750
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_804 = T_IsCommutativeSemiring_1750
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.*-comm
d_'42''45'comm_806 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_806 :: T_IsCommutativeSemiring_1750 -> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_806 = T_IsCommutativeSemiring_1750 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.*-cong
d_'42''45'cong_808 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cong_808 :: T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_808 = T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.∙-congʳ
d_'8729''45'cong'691'_810 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_810 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_810 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.∙-congˡ
d_'8729''45'cong'737'_812 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_812 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_812 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.*-identity
d_'42''45'identity_814 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_814 :: T_IsCommutativeSemiring_1750 -> T_Σ_14
d_'42''45'identity_814 T_IsCommutativeSemiring_1750
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_1562
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)))
-- Data.Integer.Properties._.IsCommutativeSemiring.identityʳ
d_identity'691'_816 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_816 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_identity'691'_816 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.identityˡ
d_identity'737'_818 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_818 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_identity'737'_818 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.isCommutativeMagma
d_isCommutativeMagma_820 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_820 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_820 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_820 T_IsCommutativeSemiring_1750
v4
du_isCommutativeMagma_820 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_820 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_820 T_IsCommutativeSemiring_1750
v0
  = let v1 :: Any
v1
          = (T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> Any -> Any
forall a b. a -> b
coe
              T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
              (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
         ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeSemigroup_1520
            (Any -> Any
forall a b. a -> b
coe Any
v1)))
-- Data.Integer.Properties._.IsCommutativeSemiring.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_822 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_822 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_822 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer
v2 Integer
v3 T_IsCommutativeSemiring_1750
v4
  = (T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764)
-> T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeMonoid_1860
      T_IsCommutativeSemiring_1750
v4
-- Data.Integer.Properties._.IsCommutativeSemiring.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_824 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_824 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_824 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_824 T_IsCommutativeSemiring_1750
v4
du_'42''45'isCommutativeSemigroup_824 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_824 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_824 T_IsCommutativeSemiring_1750
v0
  = (T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsCommutativeSemigroup_568)
-> Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeSemigroup_1520
      ((T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
         (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0))
-- Data.Integer.Properties._.IsCommutativeSemiring.*-isMagma
d_'42''45'isMagma_826 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_'42''45'isMagma_826 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsMagma_178
d_'42''45'isMagma_826 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsMagma_178
du_'42''45'isMagma_826 T_IsCommutativeSemiring_1750
v4
du_'42''45'isMagma_826 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_826 :: T_IsCommutativeSemiring_1750 -> T_IsMagma_178
du_'42''45'isMagma_826 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsMagma_178
forall a b. a -> b
coe
      ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMagma_178)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.du_'42''45'isMagma_1614
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v1)))
-- Data.Integer.Properties._.IsCommutativeSemiring.*-isMonoid
d_'42''45'isMonoid_828 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'42''45'isMonoid_828 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsMonoid_712
d_'42''45'isMonoid_828 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsMonoid_712
du_'42''45'isMonoid_828 T_IsCommutativeSemiring_1750
v4
du_'42''45'isMonoid_828 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_828 :: T_IsCommutativeSemiring_1750 -> T_IsMonoid_712
du_'42''45'isMonoid_828 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsMonoid_712
forall a b. a -> b
coe
      ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMonoid_712)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.du_'42''45'isMonoid_1618
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v1)))
-- Data.Integer.Properties._.IsCommutativeSemiring.*-isSemigroup
d_'42''45'isSemigroup_830 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_'42''45'isSemigroup_830 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsSemigroup_488
d_'42''45'isSemigroup_830 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsSemigroup_488
du_'42''45'isSemigroup_830 T_IsCommutativeSemiring_1750
v4
du_'42''45'isSemigroup_830 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_830 :: T_IsCommutativeSemiring_1750 -> T_IsSemigroup_488
du_'42''45'isSemigroup_830 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsSemigroup_488)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.du_'42''45'isSemigroup_1616
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v1)))
-- Data.Integer.Properties._.IsCommutativeSemiring.assoc
d_assoc_832 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_832 :: T_IsCommutativeSemiring_1750
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_832 = T_IsCommutativeSemiring_1750
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.comm
d_comm_834 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_834 :: T_IsCommutativeSemiring_1750 -> Integer -> Integer -> T__'8801'__12
d_comm_834 = T_IsCommutativeSemiring_1750 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.∙-cong
d_'8729''45'cong_836 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_836 :: T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_836 = T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.∙-congʳ
d_'8729''45'cong'691'_838 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_838 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_838 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.∙-congˡ
d_'8729''45'cong'737'_840 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_840 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_840 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.identity
d_identity_842 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_842 :: T_IsCommutativeSemiring_1750 -> T_Σ_14
d_identity_842 T_IsCommutativeSemiring_1750
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)))))
-- Data.Integer.Properties._.IsCommutativeSemiring.identityʳ
d_identity'691'_844 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_844 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_identity'691'_844 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.identityˡ
d_identity'737'_846 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_846 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_identity'737'_846 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.isCommutativeMagma
d_isCommutativeMagma_848 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_848 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_848 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_848 T_IsCommutativeSemiring_1750
v4
du_isCommutativeMagma_848 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_848 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_848 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsCommutativeMonoid_764
v3
                = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                    (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
               ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
                  (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v3)))))
-- Data.Integer.Properties._.IsCommutativeSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_850 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_850 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_850 T_IsCommutativeSemiring_1750
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)))
-- Data.Integer.Properties._.IsCommutativeSemiring.isCommutativeSemigroup
d_isCommutativeSemigroup_852 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_852 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_852 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_852 T_IsCommutativeSemiring_1750
v4
du_isCommutativeSemigroup_852 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_852 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_852 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
            ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
               (T_IsSemiringWithoutAnnihilatingZero_1536 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2))))
-- Data.Integer.Properties._.IsCommutativeSemiring.isMagma
d_isMagma_854 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_854 :: T_IsCommutativeSemiring_1750 -> T_IsMagma_178
d_isMagma_854 T_IsCommutativeSemiring_1750
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
            ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
               ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                  ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0))))))
-- Data.Integer.Properties._.IsCommutativeSemiring.isMonoid
d_isMonoid_856 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_856 :: T_IsCommutativeSemiring_1750 -> T_IsMonoid_712
d_isMonoid_856 T_IsCommutativeSemiring_1750
v0
  = (T_IsCommutativeMonoid_764 -> T_IsMonoid_712)
-> Any -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
      ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0))))
-- Data.Integer.Properties._.IsCommutativeSemiring.isSemigroup
d_isSemigroup_858 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_858 :: T_IsCommutativeSemiring_1750 -> T_IsSemigroup_488
d_isSemigroup_858 T_IsCommutativeSemiring_1750
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)))))
-- Data.Integer.Properties._.IsCommutativeSemiring.isUnitalMagma
d_isUnitalMagma_860 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_860 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsUnitalMagma_666
d_isUnitalMagma_860 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4 = T_IsCommutativeSemiring_1750 -> T_IsUnitalMagma_666
du_isUnitalMagma_860 T_IsCommutativeSemiring_1750
v4
du_isUnitalMagma_860 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_860 :: T_IsCommutativeSemiring_1750 -> T_IsUnitalMagma_666
du_isUnitalMagma_860 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsCommutativeMonoid_764
v3
                = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                    (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> Any -> Any
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
               ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v3)))))
-- Data.Integer.Properties._.IsCommutativeSemiring.distrib
d_distrib_862 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_862 :: T_IsCommutativeSemiring_1750 -> T_Σ_14
d_distrib_862 T_IsCommutativeSemiring_1750
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1564
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)))
-- Data.Integer.Properties._.IsCommutativeSemiring.distribʳ
d_distrib'691'_864 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_864 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_864 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.distribˡ
d_distrib'737'_866 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'737'_866 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_866 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.isCommutativeSemiringWithoutOne
d_isCommutativeSemiringWithoutOne_868 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_868 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_868 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer
v2 Integer
v3 T_IsCommutativeSemiring_1750
v4
  = (T_IsCommutativeSemiring_1750
 -> T_IsCommutativeSemiringWithoutOne_1438)
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
forall a b. a -> b
coe
      T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemiringWithoutOne_1852
      T_IsCommutativeSemiring_1750
v4
-- Data.Integer.Properties._.IsCommutativeSemiring.isEquivalence
d_isEquivalence_870 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_870 :: T_IsCommutativeSemiring_1750 -> T_IsEquivalence_28
d_isEquivalence_870 T_IsCommutativeSemiring_1750
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
               ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                  ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                     ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                        T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)))))))
-- Data.Integer.Properties._.IsCommutativeSemiring.isNearSemiring
d_isNearSemiring_872 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_872 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsNearSemiring_1260
d_isNearSemiring_872 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4 = T_IsCommutativeSemiring_1750 -> T_IsNearSemiring_1260
du_isNearSemiring_872 T_IsCommutativeSemiring_1750
v4
du_isNearSemiring_872 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_872 :: T_IsCommutativeSemiring_1750 -> T_IsNearSemiring_1260
du_isNearSemiring_872 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsNearSemiring_1260
forall a b. a -> b
coe
      ((T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_1428
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
            (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v1)))
-- Data.Integer.Properties._.IsCommutativeSemiring.isPartialEquivalence
d_isPartialEquivalence_874 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_874 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_874 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_874 T_IsCommutativeSemiring_1750
v4
du_isPartialEquivalence_874 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_874 :: T_IsCommutativeSemiring_1750 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_874 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsCommutativeMonoid_764
v3
                = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                    (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsMonoid_712
v4
                   = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsSemigroup_488
v5
                      = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v4) in
                Any -> Any
forall a b. a -> b
coe
                  (let v6 :: T_IsMagma_178
v6 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v5) in
                   Any -> Any
forall a b. a -> b
coe
                     ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
                        T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                        ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                           T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v6))))))))
-- Data.Integer.Properties._.IsCommutativeSemiring.isSemiring
d_isSemiring_876 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_876 :: T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
d_isSemiring_876 T_IsCommutativeSemiring_1750
v0
  = (T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640)
-> Any -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)
-- Data.Integer.Properties._.IsCommutativeSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_878 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_878 :: T_IsCommutativeSemiring_1750
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_878 T_IsCommutativeSemiring_1750
v0
  = (T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
      ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0))
-- Data.Integer.Properties._.IsCommutativeSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_880 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_880 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_880 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_880 T_IsCommutativeSemiring_1750
v4
du_isSemiringWithoutOne_880 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_880 :: T_IsCommutativeSemiring_1750 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_880 T_IsCommutativeSemiring_1750
v0
  = (T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342)
-> Any -> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
      ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0))
-- Data.Integer.Properties._.IsCommutativeSemiring.refl
d_refl_882 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_882 :: T_IsCommutativeSemiring_1750 -> Integer -> T__'8801'__12
d_refl_882 = T_IsCommutativeSemiring_1750 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.reflexive
d_reflexive_884 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_884 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_884 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.setoid
d_setoid_886 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_886 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_Setoid_46
d_setoid_886 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4 = T_IsCommutativeSemiring_1750 -> T_Setoid_46
du_setoid_886 T_IsCommutativeSemiring_1750
v4
du_setoid_886 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_886 :: T_IsCommutativeSemiring_1750 -> T_Setoid_46
du_setoid_886 T_IsCommutativeSemiring_1750
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsCommutativeMonoid_764
v3
                = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                    (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsMonoid_712
v4
                   = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsSemigroup_488
v5
                      = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v4) in
                Any -> Any
forall a b. a -> b
coe
                  ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
                     T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                     ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v5)))))))
-- Data.Integer.Properties._.IsCommutativeSemiring.sym
d_sym_888 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_888 :: T_IsCommutativeSemiring_1750
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_888 = T_IsCommutativeSemiring_1750
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.trans
d_trans_890 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_890 :: T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_890 = T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.zero
d_zero_892 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_892 :: T_IsCommutativeSemiring_1750 -> T_Σ_14
d_zero_892 T_IsCommutativeSemiring_1750
v0
  = (T_IsSemiring_1640 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1656
      ((T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0))
-- Data.Integer.Properties._.IsCommutativeSemiring.zeroʳ
d_zero'691'_894 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'691'_894 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_zero'691'_894 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiring.zeroˡ
d_zero'737'_896 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'737'_896 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_zero'737'_896 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.*-assoc
d_'42''45'assoc_900 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_900 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_900 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.*-comm
d_'42''45'comm_902 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_902 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_902 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.*-cong
d_'42''45'cong_904 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cong_904 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_904 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.∙-congʳ
d_'8729''45'cong'691'_906 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_906 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_906 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.∙-congˡ
d_'8729''45'cong'737'_908 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_908 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_908 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.isCommutativeMagma
d_isCommutativeMagma_910 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_910 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_910 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_910 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isCommutativeMagma_910 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_910 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_910 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeSemigroup_1520
         (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_912 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_912 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_912 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = (T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsCommutativeSemigroup_568)
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_'42''45'isCommutativeSemigroup_1520
      T_IsCommutativeSemiringWithoutOne_1438
v3
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.*-isMagma
d_'42''45'isMagma_914 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_'42''45'isMagma_914 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsMagma_178
d_'42''45'isMagma_914 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3 = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsMagma_178
du_'42''45'isMagma_914 T_IsCommutativeSemiringWithoutOne_1438
v3
du_'42''45'isMagma_914 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_914 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsMagma_178
du_'42''45'isMagma_914 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsSemiringWithoutOne_1342 -> T_IsMagma_178)
-> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_1342 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.du_'42''45'isMagma_1410
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
         (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.*-isSemigroup
d_'42''45'isSemigroup_916 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_'42''45'isSemigroup_916 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemigroup_488
d_'42''45'isSemigroup_916 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsSemigroup_488
du_'42''45'isSemigroup_916 T_IsCommutativeSemiringWithoutOne_1438
v3
du_'42''45'isSemigroup_916 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_916 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsSemigroup_488
du_'42''45'isSemigroup_916 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsSemiringWithoutOne_1342 -> T_IsSemigroup_488)
-> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_1342 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.du_'42''45'isSemigroup_1412
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
         (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.assoc
d_assoc_918 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_918 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_918 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.comm
d_comm_920 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_920 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12
d_comm_920 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.∙-cong
d_'8729''45'cong_922 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_922 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_922 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.∙-congʳ
d_'8729''45'cong'691'_924 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_924 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_924 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.∙-congˡ
d_'8729''45'cong'737'_926 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_926 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_926 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.identity
d_identity_928 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_928 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_Σ_14
d_identity_928 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
            ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
               (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.identityʳ
d_identity'691'_930 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_930 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
d_identity'691'_930 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.identityˡ
d_identity'737'_932 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_932 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
d_identity'737'_932 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.isCommutativeMagma
d_isCommutativeMagma_934 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_934 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_934 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_934 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isCommutativeMagma_934 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_934 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_934 T_IsCommutativeSemiringWithoutOne_1438
v0
  = let v1 :: T_IsSemiringWithoutOne_1342
v1
          = T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
              (T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemiringWithoutOne_1438
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v2 :: T_IsCommutativeMonoid_764
v2
             = T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
                 (T_IsSemiringWithoutOne_1342 -> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
            ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
               (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v2))))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_936 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_936 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_936 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764)
-> Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
         (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.isCommutativeSemigroup
d_isCommutativeSemigroup_938 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_938 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_938 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_938 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isCommutativeSemigroup_938 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_938 :: T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_938 T_IsCommutativeSemiringWithoutOne_1438
v0
  = let v1 :: T_IsSemiringWithoutOne_1342
v1
          = T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
              (T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemiringWithoutOne_1438
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0) in
    Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
         ((T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
            (T_IsSemiringWithoutOne_1342 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v1)))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.isMonoid
d_isMonoid_940 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_940 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsMonoid_712
d_isMonoid_940 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsCommutativeMonoid_764 -> T_IsMonoid_712)
-> Any -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
      ((T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
         ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
            (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0)))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.distrib
d_distrib_942 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_942 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_Σ_14
d_distrib_942 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsSemiringWithoutOne_1342 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_1342 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1366
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
         (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.distribʳ
d_distrib'691'_944 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_944 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_944 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.distribˡ
d_distrib'737'_946 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'737'_946 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_946 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.isEquivalence
d_isEquivalence_948 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_948 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsEquivalence_28
d_isEquivalence_948 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
               ((T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
                  ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
                     (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))))))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.isNearSemiring
d_isNearSemiring_950 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_950 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsNearSemiring_1260
d_isNearSemiring_950 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3 = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsNearSemiring_1260
du_isNearSemiring_950 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isNearSemiring_950 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_950 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsNearSemiring_1260
du_isNearSemiring_950 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260)
-> Any -> T_IsNearSemiring_1260
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_1428
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
         (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.isPartialEquivalence
d_isPartialEquivalence_952 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_952 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_952 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_952 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isPartialEquivalence_952 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_952 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_952 T_IsCommutativeSemiringWithoutOne_1438
v0
  = let v1 :: T_IsSemiringWithoutOne_1342
v1
          = T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
              (T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemiringWithoutOne_1438
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
         T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
         ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
            T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
            ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
               T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
               ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                  ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                     ((T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                        T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
                        (T_IsSemiringWithoutOne_1342 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v1)))))))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.isSemiringWithoutOne
d_isSemiringWithoutOne_954 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_954 :: T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_954 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe
      T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
      (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0)
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.refl
d_refl_956 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_956 :: T_IsCommutativeSemiringWithoutOne_1438 -> Integer -> T__'8801'__12
d_refl_956 = T_IsCommutativeSemiringWithoutOne_1438 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.reflexive
d_reflexive_958 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_958 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_958 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.setoid
d_setoid_960 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_960 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_Setoid_46
d_setoid_960 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3 = T_IsCommutativeSemiringWithoutOne_1438 -> T_Setoid_46
du_setoid_960 T_IsCommutativeSemiringWithoutOne_1438
v3
du_setoid_960 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_960 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_Setoid_46
du_setoid_960 T_IsCommutativeSemiringWithoutOne_1438
v0
  = let v1 :: T_IsSemiringWithoutOne_1342
v1
          = T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
              (T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemiringWithoutOne_1438
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsCommutativeMonoid_764
v2
             = T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
                 (T_IsSemiringWithoutOne_1342 -> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_712
v3
                = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_488
v4
                   = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                  ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v4))))))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.sym
d_sym_962 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_962 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_962 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.trans
d_trans_964 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_964 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_964 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.zero
d_zero_966 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_966 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_Σ_14
d_zero_966 T_IsCommutativeSemiringWithoutOne_1438
v0
  = (T_IsSemiringWithoutOne_1342 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_1342 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1368
      ((T_IsCommutativeSemiringWithoutOne_1438
 -> T_IsSemiringWithoutOne_1342)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1450
         (T_IsCommutativeSemiringWithoutOne_1438 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1438
v0))
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.zeroʳ
d_zero'691'_968 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'691'_968 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
d_zero'691'_968 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsCommutativeSemiringWithoutOne.zeroˡ
d_zero'737'_970 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'737'_970 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
d_zero'737'_970 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsFlexibleMagma.flex
d_flex_974 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_flex_974 :: T_IsFlexibleMagma_332 -> Integer -> Integer -> T__'8801'__12
d_flex_974 = T_IsFlexibleMagma_332 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsFlexibleMagma.isEquivalence
d_isEquivalence_976 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_976 :: T_IsFlexibleMagma_332 -> T_IsEquivalence_28
d_isEquivalence_976 T_IsFlexibleMagma_332
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsFlexibleMagma_332 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsFlexibleMagma_332 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_340 (T_IsFlexibleMagma_332 -> Any
forall a b. a -> b
coe T_IsFlexibleMagma_332
v0))
-- Data.Integer.Properties._.IsFlexibleMagma.isMagma
d_isMagma_978 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_978 :: T_IsFlexibleMagma_332 -> T_IsMagma_178
d_isMagma_978 T_IsFlexibleMagma_332
v0
  = (T_IsFlexibleMagma_332 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe T_IsFlexibleMagma_332 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_340 (T_IsFlexibleMagma_332 -> Any
forall a b. a -> b
coe T_IsFlexibleMagma_332
v0)
-- Data.Integer.Properties._.IsFlexibleMagma.isPartialEquivalence
d_isPartialEquivalence_980 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_980 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_980 ~Integer -> Integer -> Integer
v0 T_IsFlexibleMagma_332
v1 = T_IsFlexibleMagma_332 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_980 T_IsFlexibleMagma_332
v1
du_isPartialEquivalence_980 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_980 :: T_IsFlexibleMagma_332 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_980 T_IsFlexibleMagma_332
v0
  = let v1 :: T_IsMagma_178
v1
          = T_IsFlexibleMagma_332 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_340 (T_IsFlexibleMagma_332 -> T_IsFlexibleMagma_332
forall a b. a -> b
coe T_IsFlexibleMagma_332
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
         T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
         ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v1)))
-- Data.Integer.Properties._.IsFlexibleMagma.refl
d_refl_982 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_982 :: T_IsFlexibleMagma_332 -> Integer -> T__'8801'__12
d_refl_982 = T_IsFlexibleMagma_332 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsFlexibleMagma.reflexive
d_reflexive_984 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_984 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_984 = (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsFlexibleMagma.setoid
d_setoid_986 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_986 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332 -> T_Setoid_46
d_setoid_986 ~Integer -> Integer -> Integer
v0 T_IsFlexibleMagma_332
v1 = T_IsFlexibleMagma_332 -> T_Setoid_46
du_setoid_986 T_IsFlexibleMagma_332
v1
du_setoid_986 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_986 :: T_IsFlexibleMagma_332 -> T_Setoid_46
du_setoid_986 T_IsFlexibleMagma_332
v0
  = (T_IsMagma_178 -> T_Setoid_46) -> Any -> T_Setoid_46
forall a b. a -> b
coe
      T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
      ((T_IsFlexibleMagma_332 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsFlexibleMagma_332 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_340 (T_IsFlexibleMagma_332 -> Any
forall a b. a -> b
coe T_IsFlexibleMagma_332
v0))
-- Data.Integer.Properties._.IsFlexibleMagma.sym
d_sym_988 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_988 :: T_IsFlexibleMagma_332
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_988 = T_IsFlexibleMagma_332
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsFlexibleMagma.trans
d_trans_990 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_990 :: T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_990 = T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsFlexibleMagma.∙-cong
d_'8729''45'cong_992 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_992 :: T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_992 = T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsFlexibleMagma.∙-congʳ
d_'8729''45'cong'691'_994 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_994 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_994 = (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsFlexibleMagma.∙-congˡ
d_'8729''45'cong'737'_996 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_996 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_996 = (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup._-_
d__'45'__1000 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> Integer -> Integer
d__'45'__1000 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
d__'45'__1000 Integer -> Integer -> Integer
v0 Integer
v1 Integer -> Integer
v2 T_IsGroup_1074
v3
  = ((Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> Integer
forall a b. a -> b
coe (Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any
MAlonzo.Code.Algebra.Structures.du__'45'__1142 Integer -> Integer -> Integer
v0 Integer -> Integer
v2
-- Data.Integer.Properties._.IsGroup._//_
d__'47''47'__1002 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> Integer -> Integer
d__'47''47'__1002 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
d__'47''47'__1002 Integer -> Integer -> Integer
v0 Integer
v1 Integer -> Integer
v2 T_IsGroup_1074
v3 Integer
v4 Integer
v5
  = ((Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> Integer
forall a b. a -> b
coe
      (Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any
MAlonzo.Code.Algebra.Structures.du__'47''47'__1136 Integer -> Integer -> Integer
v0 Integer -> Integer
v2 Integer
v4 Integer
v5
-- Data.Integer.Properties._.IsGroup._\\_
d__'92''92'__1004 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> Integer -> Integer
d__'92''92'__1004 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
d__'92''92'__1004 Integer -> Integer -> Integer
v0 Integer
v1 Integer -> Integer
v2 T_IsGroup_1074
v3 Integer
v4 Integer
v5
  = ((Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> Integer
forall a b. a -> b
coe
      (Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any
MAlonzo.Code.Algebra.Structures.du__'92''92'__1130 Integer -> Integer -> Integer
v0 Integer -> Integer
v2 Integer
v4 Integer
v5
-- Data.Integer.Properties._.IsGroup.assoc
d_assoc_1006 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1006 :: T_IsGroup_1074 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1006 = T_IsGroup_1074 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.identity
d_identity_1008 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1008 :: T_IsGroup_1074 -> T_Σ_14
d_identity_1008 T_IsGroup_1074
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v0))
-- Data.Integer.Properties._.IsGroup.identityʳ
d_identity'691'_1010 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1010 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
d_identity'691'_1010 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.identityˡ
d_identity'737'_1012 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1012 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
d_identity'737'_1012 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.inverse
d_inverse_1014 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_1014 :: T_IsGroup_1074 -> T_Σ_14
d_inverse_1014 T_IsGroup_1074
v0
  = (T_IsGroup_1074 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsGroup_1074 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_1090 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v0)
-- Data.Integer.Properties._.IsGroup.inverseʳ
d_inverse'691'_1016 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'691'_1016 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
d_inverse'691'_1016 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.inverseˡ
d_inverse'737'_1018 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'737'_1018 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
d_inverse'737'_1018 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.isEquivalence
d_isEquivalence_1020 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1020 :: T_IsGroup_1074 -> T_IsEquivalence_28
d_isEquivalence_1020 T_IsGroup_1074
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v0))))
-- Data.Integer.Properties._.IsGroup.isInvertibleMagma
d_isInvertibleMagma_1022 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_1022 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> T_IsInvertibleMagma_958
d_isInvertibleMagma_1022 Integer -> Integer -> Integer
v0 Integer
v1 Integer -> Integer
v2 T_IsGroup_1074
v3
  = (T_IsGroup_1074 -> T_IsInvertibleMagma_958)
-> T_IsGroup_1074 -> T_IsInvertibleMagma_958
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1160 T_IsGroup_1074
v3
-- Data.Integer.Properties._.IsGroup.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_1024 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_1024 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_1024 Integer -> Integer -> Integer
v0 Integer
v1 Integer -> Integer
v2 T_IsGroup_1074
v3
  = (T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012)
-> T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012
MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1162 T_IsGroup_1074
v3
-- Data.Integer.Properties._.IsGroup.isMagma
d_isMagma_1026 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1026 :: T_IsGroup_1074 -> T_IsMagma_178
d_isMagma_1026 T_IsGroup_1074
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v0)))
-- Data.Integer.Properties._.IsGroup.isMonoid
d_isMonoid_1028 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1028 :: T_IsGroup_1074 -> T_IsMonoid_712
d_isMonoid_1028 T_IsGroup_1074
v0
  = (T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> T_IsMonoid_712
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v0)
-- Data.Integer.Properties._.IsGroup.isPartialEquivalence
d_isPartialEquivalence_1030 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1030 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1030 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsGroup_1074
v3
  = T_IsGroup_1074 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1030 T_IsGroup_1074
v3
du_isPartialEquivalence_1030 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1030 :: T_IsGroup_1074 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1030 T_IsGroup_1074
v0
  = let v1 :: T_IsMonoid_712
v1
          = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Data.Integer.Properties._.IsGroup.isSemigroup
d_isSemigroup_1032 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1032 :: T_IsGroup_1074 -> T_IsSemigroup_488
d_isSemigroup_1032 T_IsGroup_1074
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v0))
-- Data.Integer.Properties._.IsGroup.isUnitalMagma
d_isUnitalMagma_1034 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1034 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> T_IsUnitalMagma_666
d_isUnitalMagma_1034 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsGroup_1074
v3 = T_IsGroup_1074 -> T_IsUnitalMagma_666
du_isUnitalMagma_1034 T_IsGroup_1074
v3
du_isUnitalMagma_1034 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1034 :: T_IsGroup_1074 -> T_IsUnitalMagma_666
du_isUnitalMagma_1034 T_IsGroup_1074
v0
  = (T_IsMonoid_712 -> T_IsUnitalMagma_666)
-> Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
      ((T_IsGroup_1074 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> Any
forall a b. a -> b
coe T_IsGroup_1074
v0))
-- Data.Integer.Properties._.IsGroup.refl
d_refl_1036 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1036 :: T_IsGroup_1074 -> Integer -> T__'8801'__12
d_refl_1036 = T_IsGroup_1074 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.reflexive
d_reflexive_1038 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1038 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1038 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.setoid
d_setoid_1040 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1040 :: (Integer -> Integer -> Integer)
-> Integer -> (Integer -> Integer) -> T_IsGroup_1074 -> T_Setoid_46
d_setoid_1040 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsGroup_1074
v3 = T_IsGroup_1074 -> T_Setoid_46
du_setoid_1040 T_IsGroup_1074
v3
du_setoid_1040 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1040 :: T_IsGroup_1074 -> T_Setoid_46
du_setoid_1040 T_IsGroup_1074
v0
  = let v1 :: T_IsMonoid_712
v1
          = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v2))))
-- Data.Integer.Properties._.IsGroup.sym
d_sym_1042 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1042 :: T_IsGroup_1074
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1042 = T_IsGroup_1074
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.trans
d_trans_1044 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1044 :: T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1044 = T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_1046 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_unique'691''45''8315''185'_1046 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'691''45''8315''185'_1046 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_1048 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_unique'737''45''8315''185'_1048 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'737''45''8315''185'_1048 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.⁻¹-cong
d_'8315''185''45'cong_1050 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_1050 :: T_IsGroup_1074
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_1050 = T_IsGroup_1074
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.∙-cong
d_'8729''45'cong_1052 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1052 :: T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1052 = T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.∙-congʳ
d_'8729''45'cong'691'_1054 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1054 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1054 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsGroup.∙-congˡ
d_'8729''45'cong'737'_1056 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1056 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1056 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.assoc
d_assoc_1060 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1060 :: T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1060 = T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.comm
d_comm_1062 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1062 :: T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> T__'8801'__12
d_comm_1062 = T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.idem
d_idem_1064 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_1064 :: T_IsIdempotentCommutativeMonoid_884 -> Integer -> T__'8801'__12
d_idem_1064 = T_IsIdempotentCommutativeMonoid_884 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.identity
d_identity_1066 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1066 :: T_IsIdempotentCommutativeMonoid_884 -> T_Σ_14
d_identity_1066 T_IsIdempotentCommutativeMonoid_884
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894
            (T_IsIdempotentCommutativeMonoid_884 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0)))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.identityʳ
d_identity'691'_1068 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1068 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> T__'8801'__12
d_identity'691'_1068 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.identityˡ
d_identity'737'_1070 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1070 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> T__'8801'__12
d_identity'737'_1070 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isBand
d_isBand_1072 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_1072 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentCommutativeMonoid_884 -> T_IsBand_526
d_isBand_1072 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2 = T_IsIdempotentCommutativeMonoid_884 -> T_IsBand_526
du_isBand_1072 T_IsIdempotentCommutativeMonoid_884
v2
du_isBand_1072 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
du_isBand_1072 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsBand_526
du_isBand_1072 T_IsIdempotentCommutativeMonoid_884
v0
  = (T_IsIdempotentMonoid_826 -> T_IsBand_526) -> Any -> T_IsBand_526
forall a b. a -> b
coe
      T_IsIdempotentMonoid_826 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.du_isBand_876
      ((T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826
MAlonzo.Code.Algebra.Structures.du_isIdempotentMonoid_942 (T_IsIdempotentCommutativeMonoid_884 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isCommutativeBand
d_isCommutativeBand_1074 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
d_isCommutativeBand_1074 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsCommutativeBand_612
d_isCommutativeBand_1074 Integer -> Integer -> Integer
v0 Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2
  = (T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeBand_612)
-> T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeBand_612
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeBand_612
MAlonzo.Code.Algebra.Structures.du_isCommutativeBand_948 T_IsIdempotentCommutativeMonoid_884
v2
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isCommutativeMagma
d_isCommutativeMagma_1076 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_1076 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_1076 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2
  = T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1076 T_IsIdempotentCommutativeMonoid_884
v2
du_isCommutativeMagma_1076 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_1076 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1076 T_IsIdempotentCommutativeMonoid_884
v0
  = let v1 :: T_IsCommutativeMonoid_764
v1
          = T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894
              (T_IsIdempotentCommutativeMonoid_884
-> T_IsIdempotentCommutativeMonoid_884
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
         ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
            (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v1)))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isCommutativeMonoid
d_isCommutativeMonoid_1078 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_1078 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_1078 T_IsIdempotentCommutativeMonoid_884
v0
  = (T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764)
-> Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894 (T_IsIdempotentCommutativeMonoid_884 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0)
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isCommutativeSemigroup
d_isCommutativeSemigroup_1080 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1080 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1080 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2
  = T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1080 T_IsIdempotentCommutativeMonoid_884
v2
du_isCommutativeSemigroup_1080 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1080 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1080 T_IsIdempotentCommutativeMonoid_884
v0
  = (T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
      ((T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894 (T_IsIdempotentCommutativeMonoid_884 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isEquivalence
d_isEquivalence_1082 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1082 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsEquivalence_28
d_isEquivalence_1082 T_IsIdempotentCommutativeMonoid_884
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
               ((T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894
                  (T_IsIdempotentCommutativeMonoid_884 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0)))))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isIdempotentMonoid
d_isIdempotentMonoid_1084 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1084 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1084 Integer -> Integer -> Integer
v0 Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2
  = (T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826)
-> T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826
MAlonzo.Code.Algebra.Structures.du_isIdempotentMonoid_942 T_IsIdempotentCommutativeMonoid_884
v2
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isMagma
d_isMagma_1086 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1086 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsMagma_178
d_isMagma_1086 T_IsIdempotentCommutativeMonoid_884
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
            ((T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
               T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894
               (T_IsIdempotentCommutativeMonoid_884 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0))))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isMonoid
d_isMonoid_1088 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1088 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsMonoid_712
d_isMonoid_1088 T_IsIdempotentCommutativeMonoid_884
v0
  = (T_IsCommutativeMonoid_764 -> T_IsMonoid_712)
-> Any -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
      ((T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894 (T_IsIdempotentCommutativeMonoid_884 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isPartialEquivalence
d_isPartialEquivalence_1090 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1090 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1090 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2
  = T_IsIdempotentCommutativeMonoid_884 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1090 T_IsIdempotentCommutativeMonoid_884
v2
du_isPartialEquivalence_1090 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1090 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1090 T_IsIdempotentCommutativeMonoid_884
v0
  = let v1 :: T_IsCommutativeMonoid_764
v1
          = T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894
              (T_IsIdempotentCommutativeMonoid_884
-> T_IsIdempotentCommutativeMonoid_884
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_178
v4 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
                  T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                  ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                     T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v4))))))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isSemigroup
d_isSemigroup_1092 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1092 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsSemigroup_488
d_isSemigroup_1092 T_IsIdempotentCommutativeMonoid_884
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894
            (T_IsIdempotentCommutativeMonoid_884 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0)))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.isUnitalMagma
d_isUnitalMagma_1094 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1094 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsUnitalMagma_666
d_isUnitalMagma_1094 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2 = T_IsIdempotentCommutativeMonoid_884 -> T_IsUnitalMagma_666
du_isUnitalMagma_1094 T_IsIdempotentCommutativeMonoid_884
v2
du_isUnitalMagma_1094 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1094 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsUnitalMagma_666
du_isUnitalMagma_1094 T_IsIdempotentCommutativeMonoid_884
v0
  = let v1 :: T_IsCommutativeMonoid_764
v1
          = T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894
              (T_IsIdempotentCommutativeMonoid_884
-> T_IsIdempotentCommutativeMonoid_884
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0) in
    Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
         ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v1)))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.refl
d_refl_1096 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1096 :: T_IsIdempotentCommutativeMonoid_884 -> Integer -> T__'8801'__12
d_refl_1096 = T_IsIdempotentCommutativeMonoid_884 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.reflexive
d_reflexive_1098 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1098 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1098 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.setoid
d_setoid_1100 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1100 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentCommutativeMonoid_884 -> T_Setoid_46
d_setoid_1100 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2 = T_IsIdempotentCommutativeMonoid_884 -> T_Setoid_46
du_setoid_1100 T_IsIdempotentCommutativeMonoid_884
v2
du_setoid_1100 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1100 :: T_IsIdempotentCommutativeMonoid_884 -> T_Setoid_46
du_setoid_1100 T_IsIdempotentCommutativeMonoid_884
v0
  = let v1 :: T_IsCommutativeMonoid_764
v1
          = T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_894
              (T_IsIdempotentCommutativeMonoid_884
-> T_IsIdempotentCommutativeMonoid_884
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_884
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
               T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v3)))))
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.sym
d_sym_1102 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1102 :: T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1102 = T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.trans
d_trans_1104 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1104 :: T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1104 = T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.∙-cong
d_'8729''45'cong_1106 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1106 :: T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1106 = T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.∙-congʳ
d_'8729''45'cong'691'_1108 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1108 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1108 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentCommutativeMonoid.∙-congˡ
d_'8729''45'cong'737'_1110 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1110 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1110 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMagma.idem
d_idem_1114 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_1114 :: T_IsIdempotentMagma_252 -> Integer -> T__'8801'__12
d_idem_1114 = T_IsIdempotentMagma_252 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMagma.isEquivalence
d_isEquivalence_1116 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1116 :: T_IsIdempotentMagma_252 -> T_IsEquivalence_28
d_isEquivalence_1116 T_IsIdempotentMagma_252
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsIdempotentMagma_252 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentMagma_252 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_260 (T_IsIdempotentMagma_252 -> Any
forall a b. a -> b
coe T_IsIdempotentMagma_252
v0))
-- Data.Integer.Properties._.IsIdempotentMagma.isMagma
d_isMagma_1118 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1118 :: T_IsIdempotentMagma_252 -> T_IsMagma_178
d_isMagma_1118 T_IsIdempotentMagma_252
v0
  = (T_IsIdempotentMagma_252 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe T_IsIdempotentMagma_252 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_260 (T_IsIdempotentMagma_252 -> Any
forall a b. a -> b
coe T_IsIdempotentMagma_252
v0)
-- Data.Integer.Properties._.IsIdempotentMagma.isPartialEquivalence
d_isPartialEquivalence_1120 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1120 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1120 ~Integer -> Integer -> Integer
v0 T_IsIdempotentMagma_252
v1
  = T_IsIdempotentMagma_252 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1120 T_IsIdempotentMagma_252
v1
du_isPartialEquivalence_1120 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1120 :: T_IsIdempotentMagma_252 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1120 T_IsIdempotentMagma_252
v0
  = let v1 :: T_IsMagma_178
v1
          = T_IsIdempotentMagma_252 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_260 (T_IsIdempotentMagma_252 -> T_IsIdempotentMagma_252
forall a b. a -> b
coe T_IsIdempotentMagma_252
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
         T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
         ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v1)))
-- Data.Integer.Properties._.IsIdempotentMagma.refl
d_refl_1122 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1122 :: T_IsIdempotentMagma_252 -> Integer -> T__'8801'__12
d_refl_1122 = T_IsIdempotentMagma_252 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMagma.reflexive
d_reflexive_1124 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1124 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1124 = (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMagma.setoid
d_setoid_1126 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1126 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252 -> T_Setoid_46
d_setoid_1126 ~Integer -> Integer -> Integer
v0 T_IsIdempotentMagma_252
v1 = T_IsIdempotentMagma_252 -> T_Setoid_46
du_setoid_1126 T_IsIdempotentMagma_252
v1
du_setoid_1126 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1126 :: T_IsIdempotentMagma_252 -> T_Setoid_46
du_setoid_1126 T_IsIdempotentMagma_252
v0
  = (T_IsMagma_178 -> T_Setoid_46) -> Any -> T_Setoid_46
forall a b. a -> b
coe
      T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
      ((T_IsIdempotentMagma_252 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentMagma_252 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_260 (T_IsIdempotentMagma_252 -> Any
forall a b. a -> b
coe T_IsIdempotentMagma_252
v0))
-- Data.Integer.Properties._.IsIdempotentMagma.sym
d_sym_1128 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1128 :: T_IsIdempotentMagma_252
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1128 = T_IsIdempotentMagma_252
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMagma.trans
d_trans_1130 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1130 :: T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1130 = T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMagma.∙-cong
d_'8729''45'cong_1132 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1132 :: T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1132 = T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMagma.∙-congʳ
d_'8729''45'cong'691'_1134 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1134 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1134 = (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMagma.∙-congˡ
d_'8729''45'cong'737'_1136 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1136 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1136 = (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.assoc
d_assoc_1140 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1140 :: T_IsIdempotentMonoid_826
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1140 = T_IsIdempotentMonoid_826
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.idem
d_idem_1142 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_1142 :: T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
d_idem_1142 = T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.identity
d_identity_1144 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1144 :: T_IsIdempotentMonoid_826 -> T_Σ_14
d_identity_1144 T_IsIdempotentMonoid_826
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsIdempotentMonoid_826 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_836 (T_IsIdempotentMonoid_826 -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826
v0))
-- Data.Integer.Properties._.IsIdempotentMonoid.identityʳ
d_identity'691'_1146 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1146 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
d_identity'691'_1146 = (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.identityˡ
d_identity'737'_1148 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1148 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
d_identity'737'_1148 = (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.isBand
d_isBand_1150 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_1150 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> T_IsBand_526
d_isBand_1150 Integer -> Integer -> Integer
v0 Integer
v1 T_IsIdempotentMonoid_826
v2
  = (T_IsIdempotentMonoid_826 -> T_IsBand_526)
-> T_IsIdempotentMonoid_826 -> T_IsBand_526
forall a b. a -> b
coe T_IsIdempotentMonoid_826 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.du_isBand_876 T_IsIdempotentMonoid_826
v2
-- Data.Integer.Properties._.IsIdempotentMonoid.isEquivalence
d_isEquivalence_1152 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1152 :: T_IsIdempotentMonoid_826 -> T_IsEquivalence_28
d_isEquivalence_1152 T_IsIdempotentMonoid_826
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsIdempotentMonoid_826 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_836 (T_IsIdempotentMonoid_826 -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826
v0))))
-- Data.Integer.Properties._.IsIdempotentMonoid.isMagma
d_isMagma_1154 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1154 :: T_IsIdempotentMonoid_826 -> T_IsMagma_178
d_isMagma_1154 T_IsIdempotentMonoid_826
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsIdempotentMonoid_826 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_836 (T_IsIdempotentMonoid_826 -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826
v0)))
-- Data.Integer.Properties._.IsIdempotentMonoid.isMonoid
d_isMonoid_1156 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1156 :: T_IsIdempotentMonoid_826 -> T_IsMonoid_712
d_isMonoid_1156 T_IsIdempotentMonoid_826
v0
  = (T_IsIdempotentMonoid_826 -> T_IsMonoid_712)
-> Any -> T_IsMonoid_712
forall a b. a -> b
coe T_IsIdempotentMonoid_826 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_836 (T_IsIdempotentMonoid_826 -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826
v0)
-- Data.Integer.Properties._.IsIdempotentMonoid.isPartialEquivalence
d_isPartialEquivalence_1158 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1158 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1158 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentMonoid_826
v2
  = T_IsIdempotentMonoid_826 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1158 T_IsIdempotentMonoid_826
v2
du_isPartialEquivalence_1158 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1158 :: T_IsIdempotentMonoid_826 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1158 T_IsIdempotentMonoid_826
v0
  = let v1 :: T_IsMonoid_712
v1
          = T_IsIdempotentMonoid_826 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_836 (T_IsIdempotentMonoid_826 -> T_IsIdempotentMonoid_826
forall a b. a -> b
coe T_IsIdempotentMonoid_826
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Data.Integer.Properties._.IsIdempotentMonoid.isSemigroup
d_isSemigroup_1160 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1160 :: T_IsIdempotentMonoid_826 -> T_IsSemigroup_488
d_isSemigroup_1160 T_IsIdempotentMonoid_826
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsIdempotentMonoid_826 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_836 (T_IsIdempotentMonoid_826 -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826
v0))
-- Data.Integer.Properties._.IsIdempotentMonoid.isUnitalMagma
d_isUnitalMagma_1162 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1162 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> T_IsUnitalMagma_666
d_isUnitalMagma_1162 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentMonoid_826
v2 = T_IsIdempotentMonoid_826 -> T_IsUnitalMagma_666
du_isUnitalMagma_1162 T_IsIdempotentMonoid_826
v2
du_isUnitalMagma_1162 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1162 :: T_IsIdempotentMonoid_826 -> T_IsUnitalMagma_666
du_isUnitalMagma_1162 T_IsIdempotentMonoid_826
v0
  = (T_IsMonoid_712 -> T_IsUnitalMagma_666)
-> Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
      ((T_IsIdempotentMonoid_826 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_836 (T_IsIdempotentMonoid_826 -> Any
forall a b. a -> b
coe T_IsIdempotentMonoid_826
v0))
-- Data.Integer.Properties._.IsIdempotentMonoid.refl
d_refl_1164 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1164 :: T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
d_refl_1164 = T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.reflexive
d_reflexive_1166 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1166 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1166 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.setoid
d_setoid_1168 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1168 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> T_Setoid_46
d_setoid_1168 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentMonoid_826
v2 = T_IsIdempotentMonoid_826 -> T_Setoid_46
du_setoid_1168 T_IsIdempotentMonoid_826
v2
du_setoid_1168 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1168 :: T_IsIdempotentMonoid_826 -> T_Setoid_46
du_setoid_1168 T_IsIdempotentMonoid_826
v0
  = let v1 :: T_IsMonoid_712
v1
          = T_IsIdempotentMonoid_826 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_836 (T_IsIdempotentMonoid_826 -> T_IsIdempotentMonoid_826
forall a b. a -> b
coe T_IsIdempotentMonoid_826
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v2))))
-- Data.Integer.Properties._.IsIdempotentMonoid.sym
d_sym_1170 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1170 :: T_IsIdempotentMonoid_826
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1170 = T_IsIdempotentMonoid_826
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.trans
d_trans_1172 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1172 :: T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1172 = T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.∙-cong
d_'8729''45'cong_1174 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1174 :: T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1174 = T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.∙-congʳ
d_'8729''45'cong'691'_1176 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1176 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1176 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentMonoid.∙-congˡ
d_'8729''45'cong'737'_1178 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1178 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1178 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.*-assoc
d_'42''45'assoc_1182 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_1182 :: T_IsIdempotentSemiring_1998
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_1182 = T_IsIdempotentSemiring_1998
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.*-cong
d_'42''45'cong_1184 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cong_1184 :: T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_1184 = T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.∙-congʳ
d_'8729''45'cong'691'_1186 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1186 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1186 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.∙-congˡ
d_'8729''45'cong'737'_1188 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1188 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1188 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.*-identity
d_'42''45'identity_1190 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_1190 :: T_IsIdempotentSemiring_1998 -> T_Σ_14
d_'42''45'identity_1190 T_IsIdempotentSemiring_1998
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_1562
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0)))
-- Data.Integer.Properties._.IsIdempotentSemiring.identityʳ
d_identity'691'_1192 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1192 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_identity'691'_1192 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.identityˡ
d_identity'737'_1194 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1194 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_identity'737'_1194 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.*-isMagma
d_'42''45'isMagma_1196 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_'42''45'isMagma_1196 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsMagma_178
d_'42''45'isMagma_1196 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsMagma_178
du_'42''45'isMagma_1196 T_IsIdempotentSemiring_1998
v4
du_'42''45'isMagma_1196 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_1196 :: T_IsIdempotentSemiring_1998 -> T_IsMagma_178
du_'42''45'isMagma_1196 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsMagma_178
forall a b. a -> b
coe
      ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMagma_178)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.du_'42''45'isMagma_1614
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v1)))
-- Data.Integer.Properties._.IsIdempotentSemiring.*-isMonoid
d_'42''45'isMonoid_1198 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'42''45'isMonoid_1198 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsMonoid_712
d_'42''45'isMonoid_1198 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsMonoid_712
du_'42''45'isMonoid_1198 T_IsIdempotentSemiring_1998
v4
du_'42''45'isMonoid_1198 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_1198 :: T_IsIdempotentSemiring_1998 -> T_IsMonoid_712
du_'42''45'isMonoid_1198 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsMonoid_712
forall a b. a -> b
coe
      ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMonoid_712)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.du_'42''45'isMonoid_1618
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v1)))
-- Data.Integer.Properties._.IsIdempotentSemiring.*-isSemigroup
d_'42''45'isSemigroup_1200 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_'42''45'isSemigroup_1200 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsSemigroup_488
d_'42''45'isSemigroup_1200 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsSemigroup_488
du_'42''45'isSemigroup_1200 T_IsIdempotentSemiring_1998
v4
du_'42''45'isSemigroup_1200 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_1200 :: T_IsIdempotentSemiring_1998 -> T_IsSemigroup_488
du_'42''45'isSemigroup_1200 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsSemigroup_488)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.du_'42''45'isSemigroup_1616
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v1)))
-- Data.Integer.Properties._.IsIdempotentSemiring.assoc
d_assoc_1202 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1202 :: T_IsIdempotentSemiring_1998
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1202 = T_IsIdempotentSemiring_1998
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.comm
d_comm_1204 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1204 :: T_IsIdempotentSemiring_1998 -> Integer -> Integer -> T__'8801'__12
d_comm_1204 = T_IsIdempotentSemiring_1998 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.∙-cong
d_'8729''45'cong_1206 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1206 :: T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1206 = T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.∙-congʳ
d_'8729''45'cong'691'_1208 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1208 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1208 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.∙-congˡ
d_'8729''45'cong'737'_1210 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1210 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1210 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.+-idem
d_'43''45'idem_1212 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'idem_1212 :: T_IsIdempotentSemiring_1998 -> Integer -> T__'8801'__12
d_'43''45'idem_1212 = T_IsIdempotentSemiring_1998 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.identity
d_identity_1214 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1214 :: T_IsIdempotentSemiring_1998 -> T_Σ_14
d_identity_1214 T_IsIdempotentSemiring_1998
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0)))))
-- Data.Integer.Properties._.IsIdempotentSemiring.identityʳ
d_identity'691'_1216 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1216 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_identity'691'_1216 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.identityˡ
d_identity'737'_1218 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1218 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_identity'737'_1218 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.isBand
d_isBand_1220 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_1220 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsBand_526
d_isBand_1220 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4 = T_IsIdempotentSemiring_1998 -> T_IsBand_526
du_isBand_1220 T_IsIdempotentSemiring_1998
v4
du_isBand_1220 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
du_isBand_1220 :: T_IsIdempotentSemiring_1998 -> T_IsBand_526
du_isBand_1220 T_IsIdempotentSemiring_1998
v0
  = let v1 :: Any
v1
          = (T_IsIdempotentSemiring_1998
 -> T_IsIdempotentCommutativeMonoid_884)
-> Any -> Any
forall a b. a -> b
coe
              T_IsIdempotentSemiring_1998 -> T_IsIdempotentCommutativeMonoid_884
MAlonzo.Code.Algebra.Structures.du_'43''45'isIdempotentCommutativeMonoid_2100
              (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsBand_526
forall a b. a -> b
coe
      ((T_IsIdempotentMonoid_826 -> T_IsBand_526) -> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentMonoid_826 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.du_isBand_876
         ((T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826
MAlonzo.Code.Algebra.Structures.du_isIdempotentMonoid_942
            (Any -> Any
forall a b. a -> b
coe Any
v1)))
-- Data.Integer.Properties._.IsIdempotentSemiring.isCommutativeBand
d_isCommutativeBand_1222 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
d_isCommutativeBand_1222 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsCommutativeBand_612
d_isCommutativeBand_1222 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsCommutativeBand_612
du_isCommutativeBand_1222 T_IsIdempotentSemiring_1998
v4
du_isCommutativeBand_1222 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
du_isCommutativeBand_1222 :: T_IsIdempotentSemiring_1998 -> T_IsCommutativeBand_612
du_isCommutativeBand_1222 T_IsIdempotentSemiring_1998
v0
  = (T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeBand_612)
-> Any -> T_IsCommutativeBand_612
forall a b. a -> b
coe
      T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeBand_612
MAlonzo.Code.Algebra.Structures.du_isCommutativeBand_948
      ((T_IsIdempotentSemiring_1998
 -> T_IsIdempotentCommutativeMonoid_884)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentSemiring_1998 -> T_IsIdempotentCommutativeMonoid_884
MAlonzo.Code.Algebra.Structures.du_'43''45'isIdempotentCommutativeMonoid_2100
         (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0))
-- Data.Integer.Properties._.IsIdempotentSemiring.isCommutativeMagma
d_isCommutativeMagma_1224 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_1224 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_1224 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1224 T_IsIdempotentSemiring_1998
v4
du_isCommutativeMagma_1224 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_1224 :: T_IsIdempotentSemiring_1998 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1224 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsCommutativeMonoid_764
v3
                = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                    (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
               ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
                  (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v3)))))
-- Data.Integer.Properties._.IsIdempotentSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_1226 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_1226 :: T_IsIdempotentSemiring_1998 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_1226 T_IsIdempotentSemiring_1998
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0)))
-- Data.Integer.Properties._.IsIdempotentSemiring.isCommutativeSemigroup
d_isCommutativeSemigroup_1228 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1228 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1228 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1228 T_IsIdempotentSemiring_1998
v4
du_isCommutativeSemigroup_1228 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1228 :: T_IsIdempotentSemiring_1998 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1228 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
            ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
               (T_IsSemiringWithoutAnnihilatingZero_1536 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2))))
-- Data.Integer.Properties._.IsIdempotentSemiring.+-isIdempotentCommutativeMonoid
d_'43''45'isIdempotentCommutativeMonoid_1230 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884
d_'43''45'isIdempotentCommutativeMonoid_1230 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsIdempotentCommutativeMonoid_884
d_'43''45'isIdempotentCommutativeMonoid_1230 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer
v2 Integer
v3 T_IsIdempotentSemiring_1998
v4
  = (T_IsIdempotentSemiring_1998
 -> T_IsIdempotentCommutativeMonoid_884)
-> T_IsIdempotentSemiring_1998
-> T_IsIdempotentCommutativeMonoid_884
forall a b. a -> b
coe
      T_IsIdempotentSemiring_1998 -> T_IsIdempotentCommutativeMonoid_884
MAlonzo.Code.Algebra.Structures.du_'43''45'isIdempotentCommutativeMonoid_2100
      T_IsIdempotentSemiring_1998
v4
-- Data.Integer.Properties._.IsIdempotentSemiring.isIdempotentMonoid
d_isIdempotentMonoid_1232 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1232 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1232 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1232 T_IsIdempotentSemiring_1998
v4
du_isIdempotentMonoid_1232 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1232 :: T_IsIdempotentSemiring_1998 -> T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1232 T_IsIdempotentSemiring_1998
v0
  = (T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826)
-> Any -> T_IsIdempotentMonoid_826
forall a b. a -> b
coe
      T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826
MAlonzo.Code.Algebra.Structures.du_isIdempotentMonoid_942
      ((T_IsIdempotentSemiring_1998
 -> T_IsIdempotentCommutativeMonoid_884)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentSemiring_1998 -> T_IsIdempotentCommutativeMonoid_884
MAlonzo.Code.Algebra.Structures.du_'43''45'isIdempotentCommutativeMonoid_2100
         (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0))
-- Data.Integer.Properties._.IsIdempotentSemiring.isMagma
d_isMagma_1234 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1234 :: T_IsIdempotentSemiring_1998 -> T_IsMagma_178
d_isMagma_1234 T_IsIdempotentSemiring_1998
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
            ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
               ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                  ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                     T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0))))))
-- Data.Integer.Properties._.IsIdempotentSemiring.isMonoid
d_isMonoid_1236 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1236 :: T_IsIdempotentSemiring_1998 -> T_IsMonoid_712
d_isMonoid_1236 T_IsIdempotentSemiring_1998
v0
  = (T_IsCommutativeMonoid_764 -> T_IsMonoid_712)
-> Any -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
      ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0))))
-- Data.Integer.Properties._.IsIdempotentSemiring.isSemigroup
d_isSemigroup_1238 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1238 :: T_IsIdempotentSemiring_1998 -> T_IsSemigroup_488
d_isSemigroup_1238 T_IsIdempotentSemiring_1998
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0)))))
-- Data.Integer.Properties._.IsIdempotentSemiring.isUnitalMagma
d_isUnitalMagma_1240 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1240 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsUnitalMagma_666
d_isUnitalMagma_1240 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4 = T_IsIdempotentSemiring_1998 -> T_IsUnitalMagma_666
du_isUnitalMagma_1240 T_IsIdempotentSemiring_1998
v4
du_isUnitalMagma_1240 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1240 :: T_IsIdempotentSemiring_1998 -> T_IsUnitalMagma_666
du_isUnitalMagma_1240 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsCommutativeMonoid_764
v3
                = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                    (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> Any -> Any
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
               ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v3)))))
-- Data.Integer.Properties._.IsIdempotentSemiring.distrib
d_distrib_1242 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_1242 :: T_IsIdempotentSemiring_1998 -> T_Σ_14
d_distrib_1242 T_IsIdempotentSemiring_1998
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1564
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0)))
-- Data.Integer.Properties._.IsIdempotentSemiring.distribʳ
d_distrib'691'_1244 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_1244 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_1244 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.distribˡ
d_distrib'737'_1246 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'737'_1246 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_1246 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.isEquivalence
d_isEquivalence_1248 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1248 :: T_IsIdempotentSemiring_1998 -> T_IsEquivalence_28
d_isEquivalence_1248 T_IsIdempotentSemiring_1998
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
               ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                  ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                     ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                        T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0)))))))
-- Data.Integer.Properties._.IsIdempotentSemiring.isNearSemiring
d_isNearSemiring_1250 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_1250 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsNearSemiring_1260
d_isNearSemiring_1250 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsNearSemiring_1260
du_isNearSemiring_1250 T_IsIdempotentSemiring_1998
v4
du_isNearSemiring_1250 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_1250 :: T_IsIdempotentSemiring_1998 -> T_IsNearSemiring_1260
du_isNearSemiring_1250 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsNearSemiring_1260
forall a b. a -> b
coe
      ((T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_1428
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
            (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v1)))
-- Data.Integer.Properties._.IsIdempotentSemiring.isPartialEquivalence
d_isPartialEquivalence_1252 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1252 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1252 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1252 T_IsIdempotentSemiring_1998
v4
du_isPartialEquivalence_1252 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1252 :: T_IsIdempotentSemiring_1998 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1252 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsCommutativeMonoid_764
v3
                = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                    (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsMonoid_712
v4
                   = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsSemigroup_488
v5
                      = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v4) in
                Any -> Any
forall a b. a -> b
coe
                  (let v6 :: T_IsMagma_178
v6 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v5) in
                   Any -> Any
forall a b. a -> b
coe
                     ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
                        T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                        ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                           T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v6))))))))
-- Data.Integer.Properties._.IsIdempotentSemiring.isSemiring
d_isSemiring_1254 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_1254 :: T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
d_isSemiring_1254 T_IsIdempotentSemiring_1998
v0
  = (T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640)
-> Any -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0)
-- Data.Integer.Properties._.IsIdempotentSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_1256 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_1256 :: T_IsIdempotentSemiring_1998
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_1256 T_IsIdempotentSemiring_1998
v0
  = (T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
      ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0))
-- Data.Integer.Properties._.IsIdempotentSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_1258 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_1258 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_1258 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1258 T_IsIdempotentSemiring_1998
v4
du_isSemiringWithoutOne_1258 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1258 :: T_IsIdempotentSemiring_1998 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1258 T_IsIdempotentSemiring_1998
v0
  = (T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342)
-> Any -> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
      ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0))
-- Data.Integer.Properties._.IsIdempotentSemiring.refl
d_refl_1260 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1260 :: T_IsIdempotentSemiring_1998 -> Integer -> T__'8801'__12
d_refl_1260 = T_IsIdempotentSemiring_1998 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.reflexive
d_reflexive_1262 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1262 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1262 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.setoid
d_setoid_1264 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1264 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_Setoid_46
d_setoid_1264 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4 = T_IsIdempotentSemiring_1998 -> T_Setoid_46
du_setoid_1264 T_IsIdempotentSemiring_1998
v4
du_setoid_1264 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1264 :: T_IsIdempotentSemiring_1998 -> T_Setoid_46
du_setoid_1264 T_IsIdempotentSemiring_1998
v0
  = let v1 :: T_IsSemiring_1640
v1
          = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemiringWithoutAnnihilatingZero_1536
v2
             = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                 (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsCommutativeMonoid_764
v3
                = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                    (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsMonoid_712
v4
                   = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsSemigroup_488
v5
                      = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v4) in
                Any -> Any
forall a b. a -> b
coe
                  ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
                     T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                     ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v5)))))))
-- Data.Integer.Properties._.IsIdempotentSemiring.sym
d_sym_1266 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1266 :: T_IsIdempotentSemiring_1998
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1266 = T_IsIdempotentSemiring_1998
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.trans
d_trans_1268 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1268 :: T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1268 = T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.zero
d_zero_1270 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_1270 :: T_IsIdempotentSemiring_1998 -> T_Σ_14
d_zero_1270 T_IsIdempotentSemiring_1998
v0
  = (T_IsSemiring_1640 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1656
      ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v0))
-- Data.Integer.Properties._.IsIdempotentSemiring.zeroʳ
d_zero'691'_1272 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'691'_1272 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_zero'691'_1272 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsIdempotentSemiring.zeroˡ
d_zero'737'_1274 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'737'_1274 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_zero'737'_1274 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.inverse
d_inverse_1278 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_1278 :: T_IsInvertibleMagma_958 -> T_Σ_14
d_inverse_1278 T_IsInvertibleMagma_958
v0
  = (T_IsInvertibleMagma_958 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsInvertibleMagma_958 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_974 (T_IsInvertibleMagma_958 -> Any
forall a b. a -> b
coe T_IsInvertibleMagma_958
v0)
-- Data.Integer.Properties._.IsInvertibleMagma.inverseʳ
d_inverse'691'_1280 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'691'_1280 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> T__'8801'__12
d_inverse'691'_1280 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.inverseˡ
d_inverse'737'_1282 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'737'_1282 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> T__'8801'__12
d_inverse'737'_1282 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.isEquivalence
d_isEquivalence_1284 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1284 :: T_IsInvertibleMagma_958 -> T_IsEquivalence_28
d_isEquivalence_1284 T_IsInvertibleMagma_958
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsInvertibleMagma_958 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsInvertibleMagma_958 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_972 (T_IsInvertibleMagma_958 -> Any
forall a b. a -> b
coe T_IsInvertibleMagma_958
v0))
-- Data.Integer.Properties._.IsInvertibleMagma.isMagma
d_isMagma_1286 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1286 :: T_IsInvertibleMagma_958 -> T_IsMagma_178
d_isMagma_1286 T_IsInvertibleMagma_958
v0
  = (T_IsInvertibleMagma_958 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe T_IsInvertibleMagma_958 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_972 (T_IsInvertibleMagma_958 -> Any
forall a b. a -> b
coe T_IsInvertibleMagma_958
v0)
-- Data.Integer.Properties._.IsInvertibleMagma.isPartialEquivalence
d_isPartialEquivalence_1288 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1288 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1288 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsInvertibleMagma_958
v3
  = T_IsInvertibleMagma_958 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1288 T_IsInvertibleMagma_958
v3
du_isPartialEquivalence_1288 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1288 :: T_IsInvertibleMagma_958 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1288 T_IsInvertibleMagma_958
v0
  = let v1 :: T_IsMagma_178
v1
          = T_IsInvertibleMagma_958 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_972 (T_IsInvertibleMagma_958 -> T_IsInvertibleMagma_958
forall a b. a -> b
coe T_IsInvertibleMagma_958
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
         T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
         ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v1)))
-- Data.Integer.Properties._.IsInvertibleMagma.refl
d_refl_1290 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1290 :: T_IsInvertibleMagma_958 -> Integer -> T__'8801'__12
d_refl_1290 = T_IsInvertibleMagma_958 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.reflexive
d_reflexive_1292 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1292 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1292 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.setoid
d_setoid_1294 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1294 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> T_Setoid_46
d_setoid_1294 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsInvertibleMagma_958
v3 = T_IsInvertibleMagma_958 -> T_Setoid_46
du_setoid_1294 T_IsInvertibleMagma_958
v3
du_setoid_1294 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1294 :: T_IsInvertibleMagma_958 -> T_Setoid_46
du_setoid_1294 T_IsInvertibleMagma_958
v0
  = (T_IsMagma_178 -> T_Setoid_46) -> Any -> T_Setoid_46
forall a b. a -> b
coe
      T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
      ((T_IsInvertibleMagma_958 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsInvertibleMagma_958 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_972 (T_IsInvertibleMagma_958 -> Any
forall a b. a -> b
coe T_IsInvertibleMagma_958
v0))
-- Data.Integer.Properties._.IsInvertibleMagma.sym
d_sym_1296 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1296 :: T_IsInvertibleMagma_958
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1296 = T_IsInvertibleMagma_958
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.trans
d_trans_1298 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1298 :: T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1298 = T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.⁻¹-cong
d_'8315''185''45'cong_1300 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_1300 :: T_IsInvertibleMagma_958
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_1300 = T_IsInvertibleMagma_958
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.∙-cong
d_'8729''45'cong_1302 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1302 :: T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1302 = T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.∙-congʳ
d_'8729''45'cong'691'_1304 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1304 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1304 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleMagma.∙-congˡ
d_'8729''45'cong'737'_1306 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1306 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1306 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.identity
d_identity_1310 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1310 :: T_IsInvertibleUnitalMagma_1012 -> T_Σ_14
d_identity_1310 T_IsInvertibleUnitalMagma_1012
v0
  = (T_IsInvertibleUnitalMagma_1012 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_1026 (T_IsInvertibleUnitalMagma_1012 -> Any
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012
v0)
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.identityʳ
d_identity'691'_1312 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1312 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
d_identity'691'_1312 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.identityˡ
d_identity'737'_1314 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1314 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
d_identity'737'_1314 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.inverse
d_inverse_1316 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_1316 :: T_IsInvertibleUnitalMagma_1012 -> T_Σ_14
d_inverse_1316 T_IsInvertibleUnitalMagma_1012
v0
  = (T_IsInvertibleMagma_958 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsInvertibleMagma_958 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_974
      ((T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958)
-> Any -> Any
forall a b. a -> b
coe
         T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.d_isInvertibleMagma_1024 (T_IsInvertibleUnitalMagma_1012 -> Any
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012
v0))
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.inverseʳ
d_inverse'691'_1318 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'691'_1318 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
d_inverse'691'_1318 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.inverseˡ
d_inverse'737'_1320 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_inverse'737'_1320 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
d_inverse'737'_1320 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.isEquivalence
d_isEquivalence_1322 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1322 :: T_IsInvertibleUnitalMagma_1012 -> T_IsEquivalence_28
d_isEquivalence_1322 T_IsInvertibleUnitalMagma_1012
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsInvertibleMagma_958 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsInvertibleMagma_958 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_972
         ((T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958)
-> Any -> Any
forall a b. a -> b
coe
            T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.d_isInvertibleMagma_1024 (T_IsInvertibleUnitalMagma_1012 -> Any
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012
v0)))
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.isInvertibleMagma
d_isInvertibleMagma_1324 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_1324 :: T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958
d_isInvertibleMagma_1324 T_IsInvertibleUnitalMagma_1012
v0
  = (T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958)
-> Any -> T_IsInvertibleMagma_958
forall a b. a -> b
coe
      T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.d_isInvertibleMagma_1024 (T_IsInvertibleUnitalMagma_1012 -> Any
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012
v0)
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.isMagma
d_isMagma_1326 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1326 :: T_IsInvertibleUnitalMagma_1012 -> T_IsMagma_178
d_isMagma_1326 T_IsInvertibleUnitalMagma_1012
v0
  = (T_IsInvertibleMagma_958 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsInvertibleMagma_958 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_972
      ((T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958)
-> Any -> Any
forall a b. a -> b
coe
         T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.d_isInvertibleMagma_1024 (T_IsInvertibleUnitalMagma_1012 -> Any
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012
v0))
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.isPartialEquivalence
d_isPartialEquivalence_1328 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1328 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1328 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsInvertibleUnitalMagma_1012
v3
  = T_IsInvertibleUnitalMagma_1012 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1328 T_IsInvertibleUnitalMagma_1012
v3
du_isPartialEquivalence_1328 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1328 :: T_IsInvertibleUnitalMagma_1012 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1328 T_IsInvertibleUnitalMagma_1012
v0
  = let v1 :: T_IsInvertibleMagma_958
v1
          = T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.d_isInvertibleMagma_1024
              (T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleUnitalMagma_1012
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2 = T_IsInvertibleMagma_958 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_972 (T_IsInvertibleMagma_958 -> T_IsInvertibleMagma_958
forall a b. a -> b
coe T_IsInvertibleMagma_958
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
            T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
            ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
               T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v2))))
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.isUnitalMagma
d_isUnitalMagma_1330 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1330 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> T_IsUnitalMagma_666
d_isUnitalMagma_1330 Integer -> Integer -> Integer
v0 Integer
v1 Integer -> Integer
v2 T_IsInvertibleUnitalMagma_1012
v3
  = (T_IsInvertibleUnitalMagma_1012 -> T_IsUnitalMagma_666)
-> T_IsInvertibleUnitalMagma_1012 -> T_IsUnitalMagma_666
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_1064 T_IsInvertibleUnitalMagma_1012
v3
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.refl
d_refl_1332 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1332 :: T_IsInvertibleUnitalMagma_1012 -> Integer -> T__'8801'__12
d_refl_1332 = T_IsInvertibleUnitalMagma_1012 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.reflexive
d_reflexive_1334 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1334 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1334 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.setoid
d_setoid_1336 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1336 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> T_Setoid_46
d_setoid_1336 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsInvertibleUnitalMagma_1012
v3 = T_IsInvertibleUnitalMagma_1012 -> T_Setoid_46
du_setoid_1336 T_IsInvertibleUnitalMagma_1012
v3
du_setoid_1336 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1336 :: T_IsInvertibleUnitalMagma_1012 -> T_Setoid_46
du_setoid_1336 T_IsInvertibleUnitalMagma_1012
v0
  = let v1 :: T_IsInvertibleMagma_958
v1
          = T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.d_isInvertibleMagma_1024
              (T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleUnitalMagma_1012
forall a b. a -> b
coe T_IsInvertibleUnitalMagma_1012
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
         T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
         ((T_IsInvertibleMagma_958 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsInvertibleMagma_958 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_972 (T_IsInvertibleMagma_958 -> Any
forall a b. a -> b
coe T_IsInvertibleMagma_958
v1)))
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.sym
d_sym_1338 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1338 :: T_IsInvertibleUnitalMagma_1012
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1338 = T_IsInvertibleUnitalMagma_1012
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.trans
d_trans_1340 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1340 :: T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1340 = T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.⁻¹-cong
d_'8315''185''45'cong_1342 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_1342 :: T_IsInvertibleUnitalMagma_1012
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_1342 = T_IsInvertibleUnitalMagma_1012
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.∙-cong
d_'8729''45'cong_1344 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1344 :: T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1344 = T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.∙-congʳ
d_'8729''45'cong'691'_1346 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1346 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1346 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsInvertibleUnitalMagma.∙-congˡ
d_'8729''45'cong'737'_1348 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1348 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1348 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.*-assoc
d_'42''45'assoc_1352 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_1352 :: T_IsKleeneAlgebra_2122
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_1352 = T_IsKleeneAlgebra_2122
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.*-cong
d_'42''45'cong_1354 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cong_1354 :: T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_1354 = T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.∙-congʳ
d_'8729''45'cong'691'_1356 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1356 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1356 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.∙-congˡ
d_'8729''45'cong'737'_1358 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1358 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1358 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.*-identity
d_'42''45'identity_1360 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_1360 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_'42''45'identity_1360 T_IsKleeneAlgebra_2122
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_1562
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
            ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
               T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
               (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0))))
-- Data.Integer.Properties._.IsKleeneAlgebra.identityʳ
d_identity'691'_1362 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1362 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_identity'691'_1362 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.identityˡ
d_identity'737'_1364 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1364 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_identity'737'_1364 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.*-isMagma
d_'42''45'isMagma_1366 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_'42''45'isMagma_1366 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsMagma_178
d_'42''45'isMagma_1366 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsMagma_178
du_'42''45'isMagma_1366 T_IsKleeneAlgebra_2122
v5
du_'42''45'isMagma_1366 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_1366 :: T_IsKleeneAlgebra_2122 -> T_IsMagma_178
du_'42''45'isMagma_1366 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsMagma_178
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMagma_178)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.du_'42''45'isMagma_1614
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v2))))
-- Data.Integer.Properties._.IsKleeneAlgebra.*-isMonoid
d_'42''45'isMonoid_1368 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'42''45'isMonoid_1368 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsMonoid_712
d_'42''45'isMonoid_1368 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsMonoid_712
du_'42''45'isMonoid_1368 T_IsKleeneAlgebra_2122
v5
du_'42''45'isMonoid_1368 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_1368 :: T_IsKleeneAlgebra_2122 -> T_IsMonoid_712
du_'42''45'isMonoid_1368 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsMonoid_712
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMonoid_712)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.du_'42''45'isMonoid_1618
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v2))))
-- Data.Integer.Properties._.IsKleeneAlgebra.*-isSemigroup
d_'42''45'isSemigroup_1370 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_'42''45'isSemigroup_1370 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsSemigroup_488
d_'42''45'isSemigroup_1370 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsSemigroup_488
du_'42''45'isSemigroup_1370 T_IsKleeneAlgebra_2122
v5
du_'42''45'isSemigroup_1370 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_1370 :: T_IsKleeneAlgebra_2122 -> T_IsSemigroup_488
du_'42''45'isSemigroup_1370 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsSemigroup_488)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.du_'42''45'isSemigroup_1616
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v2))))
-- Data.Integer.Properties._.IsKleeneAlgebra.assoc
d_assoc_1372 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1372 :: T_IsKleeneAlgebra_2122
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1372 = T_IsKleeneAlgebra_2122
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.comm
d_comm_1374 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1374 :: T_IsKleeneAlgebra_2122 -> Integer -> Integer -> T__'8801'__12
d_comm_1374 = T_IsKleeneAlgebra_2122 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.∙-cong
d_'8729''45'cong_1376 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1376 :: T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1376 = T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.∙-congʳ
d_'8729''45'cong'691'_1378 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1378 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1378 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.∙-congˡ
d_'8729''45'cong'737'_1380 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1380 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1380 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.+-idem
d_'43''45'idem_1382 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'idem_1382 :: T_IsKleeneAlgebra_2122 -> Integer -> T__'8801'__12
d_'43''45'idem_1382 = T_IsKleeneAlgebra_2122 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.identity
d_identity_1384 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1384 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_identity_1384 T_IsKleeneAlgebra_2122
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                  T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
                  ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
                     (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0))))))
-- Data.Integer.Properties._.IsKleeneAlgebra.identityʳ
d_identity'691'_1386 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1386 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_identity'691'_1386 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.identityˡ
d_identity'737'_1388 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1388 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_identity'737'_1388 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.isBand
d_isBand_1390 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_1390 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsBand_526
d_isBand_1390 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5 = T_IsKleeneAlgebra_2122 -> T_IsBand_526
du_isBand_1390 T_IsKleeneAlgebra_2122
v5
du_isBand_1390 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
du_isBand_1390 :: T_IsKleeneAlgebra_2122 -> T_IsBand_526
du_isBand_1390 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsBand_526
forall a b. a -> b
coe
      (let v2 :: Any
v2
             = (T_IsIdempotentSemiring_1998
 -> T_IsIdempotentCommutativeMonoid_884)
-> Any -> Any
forall a b. a -> b
coe
                 T_IsIdempotentSemiring_1998 -> T_IsIdempotentCommutativeMonoid_884
MAlonzo.Code.Algebra.Structures.du_'43''45'isIdempotentCommutativeMonoid_2100
                 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsIdempotentMonoid_826 -> T_IsBand_526) -> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentMonoid_826 -> T_IsBand_526
MAlonzo.Code.Algebra.Structures.du_isBand_876
            ((T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826)
-> Any -> Any
forall a b. a -> b
coe
               T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826
MAlonzo.Code.Algebra.Structures.du_isIdempotentMonoid_942
               (Any -> Any
forall a b. a -> b
coe Any
v2))))
-- Data.Integer.Properties._.IsKleeneAlgebra.isCommutativeBand
d_isCommutativeBand_1392 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
d_isCommutativeBand_1392 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsCommutativeBand_612
d_isCommutativeBand_1392 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsCommutativeBand_612
du_isCommutativeBand_1392 T_IsKleeneAlgebra_2122
v5
du_isCommutativeBand_1392 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
du_isCommutativeBand_1392 :: T_IsKleeneAlgebra_2122 -> T_IsCommutativeBand_612
du_isCommutativeBand_1392 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsCommutativeBand_612
forall a b. a -> b
coe
      ((T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeBand_612)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeBand_612
MAlonzo.Code.Algebra.Structures.du_isCommutativeBand_948
         ((T_IsIdempotentSemiring_1998
 -> T_IsIdempotentCommutativeMonoid_884)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentSemiring_1998 -> T_IsIdempotentCommutativeMonoid_884
MAlonzo.Code.Algebra.Structures.du_'43''45'isIdempotentCommutativeMonoid_2100
            (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1)))
-- Data.Integer.Properties._.IsKleeneAlgebra.isCommutativeMagma
d_isCommutativeMagma_1394 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_1394 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_1394 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1394 T_IsKleeneAlgebra_2122
v5
du_isCommutativeMagma_1394 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_1394 :: T_IsKleeneAlgebra_2122 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1394 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsCommutativeMonoid_764
v4
                   = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                       (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
                  ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
                     (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v4))))))
-- Data.Integer.Properties._.IsKleeneAlgebra.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_1396 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_1396 :: T_IsKleeneAlgebra_2122 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_1396 T_IsKleeneAlgebra_2122
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
            ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
               T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
               (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0))))
-- Data.Integer.Properties._.IsKleeneAlgebra.isCommutativeSemigroup
d_isCommutativeSemigroup_1398 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1398 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1398 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1398 T_IsKleeneAlgebra_2122
v5
du_isCommutativeSemigroup_1398 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1398 :: T_IsKleeneAlgebra_2122 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1398 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
               ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                  (T_IsSemiringWithoutAnnihilatingZero_1536 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3)))))
-- Data.Integer.Properties._.IsKleeneAlgebra.+-isIdempotentCommutativeMonoid
d_'43''45'isIdempotentCommutativeMonoid_1400 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884
d_'43''45'isIdempotentCommutativeMonoid_1400 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsIdempotentCommutativeMonoid_884
d_'43''45'isIdempotentCommutativeMonoid_1400 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsIdempotentCommutativeMonoid_884
du_'43''45'isIdempotentCommutativeMonoid_1400 T_IsKleeneAlgebra_2122
v5
du_'43''45'isIdempotentCommutativeMonoid_1400 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884
du_'43''45'isIdempotentCommutativeMonoid_1400 :: T_IsKleeneAlgebra_2122 -> T_IsIdempotentCommutativeMonoid_884
du_'43''45'isIdempotentCommutativeMonoid_1400 T_IsKleeneAlgebra_2122
v0
  = (T_IsIdempotentSemiring_1998
 -> T_IsIdempotentCommutativeMonoid_884)
-> Any -> T_IsIdempotentCommutativeMonoid_884
forall a b. a -> b
coe
      T_IsIdempotentSemiring_1998 -> T_IsIdempotentCommutativeMonoid_884
MAlonzo.Code.Algebra.Structures.du_'43''45'isIdempotentCommutativeMonoid_2100
      ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
         T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
         (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0))
-- Data.Integer.Properties._.IsKleeneAlgebra.isIdempotentMonoid
d_isIdempotentMonoid_1402 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1402 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1402 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1402 T_IsKleeneAlgebra_2122
v5
du_isIdempotentMonoid_1402 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1402 :: T_IsKleeneAlgebra_2122 -> T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1402 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsIdempotentMonoid_826
forall a b. a -> b
coe
      ((T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentCommutativeMonoid_884 -> T_IsIdempotentMonoid_826
MAlonzo.Code.Algebra.Structures.du_isIdempotentMonoid_942
         ((T_IsIdempotentSemiring_1998
 -> T_IsIdempotentCommutativeMonoid_884)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentSemiring_1998 -> T_IsIdempotentCommutativeMonoid_884
MAlonzo.Code.Algebra.Structures.du_'43''45'isIdempotentCommutativeMonoid_2100
            (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1)))
-- Data.Integer.Properties._.IsKleeneAlgebra.isMagma
d_isMagma_1404 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1404 :: T_IsKleeneAlgebra_2122 -> T_IsMagma_178
d_isMagma_1404 T_IsKleeneAlgebra_2122
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
            ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
               ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                  ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                     T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
                     ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
                        T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
                        (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0)))))))
-- Data.Integer.Properties._.IsKleeneAlgebra.isMonoid
d_isMonoid_1406 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1406 :: T_IsKleeneAlgebra_2122 -> T_IsMonoid_712
d_isMonoid_1406 T_IsKleeneAlgebra_2122
v0
  = (T_IsCommutativeMonoid_764 -> T_IsMonoid_712)
-> Any -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
      ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
         ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
            ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
               T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
               ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
                  (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0)))))
-- Data.Integer.Properties._.IsKleeneAlgebra.isSemigroup
d_isSemigroup_1408 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1408 :: T_IsKleeneAlgebra_2122 -> T_IsSemigroup_488
d_isSemigroup_1408 T_IsKleeneAlgebra_2122
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
         ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
               ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                  T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
                  ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
                     (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0))))))
-- Data.Integer.Properties._.IsKleeneAlgebra.isUnitalMagma
d_isUnitalMagma_1410 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1410 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsUnitalMagma_666
d_isUnitalMagma_1410 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsUnitalMagma_666
du_isUnitalMagma_1410 T_IsKleeneAlgebra_2122
v5
du_isUnitalMagma_1410 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1410 :: T_IsKleeneAlgebra_2122 -> T_IsUnitalMagma_666
du_isUnitalMagma_1410 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsCommutativeMonoid_764
v4
                   = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                       (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
                  ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v4))))))
-- Data.Integer.Properties._.IsKleeneAlgebra.distrib
d_distrib_1412 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_1412 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_distrib_1412 T_IsKleeneAlgebra_2122
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1564
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
         ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
            ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
               T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
               (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0))))
-- Data.Integer.Properties._.IsKleeneAlgebra.distribʳ
d_distrib'691'_1414 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_1414 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_1414 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.distribˡ
d_distrib'737'_1416 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'737'_1416 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_1416 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.isEquivalence
d_isEquivalence_1418 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1418 :: T_IsKleeneAlgebra_2122 -> T_IsEquivalence_28
d_isEquivalence_1418 T_IsKleeneAlgebra_2122
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
               ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                  ((T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                     ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
                        T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
                        ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
                           T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
                           (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0))))))))
-- Data.Integer.Properties._.IsKleeneAlgebra.isIdempotentSemiring
d_isIdempotentSemiring_1420 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998
d_isIdempotentSemiring_1420 :: T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
d_isIdempotentSemiring_1420 T_IsKleeneAlgebra_2122
v0
  = (T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe
      T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
      (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0)
-- Data.Integer.Properties._.IsKleeneAlgebra.isNearSemiring
d_isNearSemiring_1422 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_1422 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsNearSemiring_1260
d_isNearSemiring_1422 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsNearSemiring_1260
du_isNearSemiring_1422 T_IsKleeneAlgebra_2122
v5
du_isNearSemiring_1422 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_1422 :: T_IsKleeneAlgebra_2122 -> T_IsNearSemiring_1260
du_isNearSemiring_1422 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsNearSemiring_1260
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_1342 -> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_1428
            ((T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342) -> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
               (T_IsSemiring_1640 -> Any
forall a b. a -> b
coe T_IsSemiring_1640
v2))))
-- Data.Integer.Properties._.IsKleeneAlgebra.isPartialEquivalence
d_isPartialEquivalence_1424 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1424 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1424 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1424 T_IsKleeneAlgebra_2122
v5
du_isPartialEquivalence_1424 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1424 :: T_IsKleeneAlgebra_2122 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1424 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsCommutativeMonoid_764
v4
                   = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                       (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsMonoid_712
v5
                      = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v4) in
                Any -> Any
forall a b. a -> b
coe
                  (let v6 :: T_IsSemigroup_488
v6
                         = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v5) in
                   Any -> Any
forall a b. a -> b
coe
                     (let v7 :: T_IsMagma_178
v7 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v6) in
                      Any -> Any
forall a b. a -> b
coe
                        ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
                           T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                           ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                              T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
                              (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v7)))))))))
-- Data.Integer.Properties._.IsKleeneAlgebra.isSemiring
d_isSemiring_1426 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_1426 :: T_IsKleeneAlgebra_2122 -> T_IsSemiring_1640
d_isSemiring_1426 T_IsKleeneAlgebra_2122
v0
  = (T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640)
-> Any -> T_IsSemiring_1640
forall a b. a -> b
coe
      T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
      ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
         T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
         (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0))
-- Data.Integer.Properties._.IsKleeneAlgebra.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_1428 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_1428 :: T_IsKleeneAlgebra_2122 -> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_1428 T_IsKleeneAlgebra_2122
v0
  = (T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> Any -> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
      ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
         ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
            T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
            (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0)))
-- Data.Integer.Properties._.IsKleeneAlgebra.isSemiringWithoutOne
d_isSemiringWithoutOne_1430 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_1430 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_1430 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5
  = T_IsKleeneAlgebra_2122 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1430 T_IsKleeneAlgebra_2122
v5
du_isSemiringWithoutOne_1430 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1430 :: T_IsKleeneAlgebra_2122 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1430 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe
      ((T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1640 -> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1730
         ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> Any
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1)))
-- Data.Integer.Properties._.IsKleeneAlgebra.refl
d_refl_1432 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1432 :: T_IsKleeneAlgebra_2122 -> Integer -> T__'8801'__12
d_refl_1432 = T_IsKleeneAlgebra_2122 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.reflexive
d_reflexive_1434 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1434 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1434 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.setoid
d_setoid_1436 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1436 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_Setoid_46
d_setoid_1436 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsKleeneAlgebra_2122
v5 = T_IsKleeneAlgebra_2122 -> T_Setoid_46
du_setoid_1436 T_IsKleeneAlgebra_2122
v5
du_setoid_1436 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1436 :: T_IsKleeneAlgebra_2122 -> T_Setoid_46
du_setoid_1436 T_IsKleeneAlgebra_2122
v0
  = let v1 :: T_IsIdempotentSemiring_1998
v1
          = T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
              (T_IsKleeneAlgebra_2122 -> T_IsKleeneAlgebra_2122
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1640
v2
             = T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012 (T_IsIdempotentSemiring_1998 -> T_IsIdempotentSemiring_1998
forall a b. a -> b
coe T_IsIdempotentSemiring_1998
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsSemiringWithoutAnnihilatingZero_1536
v3
                = T_IsSemiring_1640 -> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1654
                    (T_IsSemiring_1640 -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsSemiring_1640
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsCommutativeMonoid_764
v4
                   = T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
                       (T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v3) in
             Any -> Any
forall a b. a -> b
coe
               (let v5 :: T_IsMonoid_712
v5
                      = T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v4) in
                Any -> Any
forall a b. a -> b
coe
                  (let v6 :: T_IsSemigroup_488
v6
                         = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v5) in
                   Any -> Any
forall a b. a -> b
coe
                     ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
                        T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                        ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v6))))))))
-- Data.Integer.Properties._.IsKleeneAlgebra.starDestructive
d_starDestructive_1438 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_starDestructive_1438 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_starDestructive_1438 T_IsKleeneAlgebra_2122
v0
  = (T_IsKleeneAlgebra_2122 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsKleeneAlgebra_2122 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_starDestructive_2144 (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0)
-- Data.Integer.Properties._.IsKleeneAlgebra.starDestructiveʳ
d_starDestructive'691'_1440 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_starDestructive'691'_1440 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_starDestructive'691'_1440 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.starDestructiveˡ
d_starDestructive'737'_1442 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_starDestructive'737'_1442 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_starDestructive'737'_1442 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.starExpansive
d_starExpansive_1444 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_starExpansive_1444 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_starExpansive_1444 T_IsKleeneAlgebra_2122
v0
  = (T_IsKleeneAlgebra_2122 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsKleeneAlgebra_2122 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_starExpansive_2142 (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0)
-- Data.Integer.Properties._.IsKleeneAlgebra.starExpansiveʳ
d_starExpansive'691'_1446 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_starExpansive'691'_1446 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_starExpansive'691'_1446 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.starExpansiveˡ
d_starExpansive'737'_1448 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_starExpansive'737'_1448 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_starExpansive'737'_1448 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.sym
d_sym_1450 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1450 :: T_IsKleeneAlgebra_2122
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1450 = T_IsKleeneAlgebra_2122
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.trans
d_trans_1452 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1452 :: T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1452 = T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.zero
d_zero_1454 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_1454 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_zero_1454 T_IsKleeneAlgebra_2122
v0
  = (T_IsSemiring_1640 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiring_1640 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1656
      ((T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640) -> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_2012
         ((T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998)
-> Any -> Any
forall a b. a -> b
coe
            T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
MAlonzo.Code.Algebra.Structures.d_isIdempotentSemiring_2140
            (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0)))
-- Data.Integer.Properties._.IsKleeneAlgebra.zeroʳ
d_zero'691'_1456 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'691'_1456 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_zero'691'_1456 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsKleeneAlgebra.zeroˡ
d_zero'737'_1458 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'737'_1458 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_zero'737'_1458 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.//-cong
d_'47''47''45'cong_1462 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong_1462 :: T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong_1462 = T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.//-congʳ
d_'47''47''45'cong'691'_1464 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong'691'_1464 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong'691'_1464 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.//-congˡ
d_'47''47''45'cong'737'_1466 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong'737'_1466 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong'737'_1466 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.\\-cong
d_'92''92''45'cong_1468 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong_1468 :: T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong_1468 = T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.\\-congʳ
d_'92''92''45'cong'691'_1470 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong'691'_1470 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong'691'_1470 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.\\-congˡ
d_'92''92''45'cong'737'_1472 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong'737'_1472 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong'737'_1472 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.identity
d_identity_1474 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1474 :: T_IsLeftBolLoop_3202 -> T_Σ_14
d_identity_1474 T_IsLeftBolLoop_3202
v0
  = (T_IsLoop_3122 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsLoop_3122 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_3138
      ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0))
-- Data.Integer.Properties._.IsLeftBolLoop.identityʳ
d_identity'691'_1476 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1476 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> T__'8801'__12
d_identity'691'_1476 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.identityˡ
d_identity'737'_1478 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1478 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> T__'8801'__12
d_identity'737'_1478 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.isEquivalence
d_isEquivalence_1480 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1480 :: T_IsLeftBolLoop_3202 -> T_IsEquivalence_28
d_isEquivalence_1480 T_IsLeftBolLoop_3202
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056
         ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
            T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
            ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0))))
-- Data.Integer.Properties._.IsLeftBolLoop.isLoop
d_isLoop_1482 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122
d_isLoop_1482 :: T_IsLeftBolLoop_3202 -> T_IsLoop_3122
d_isLoop_1482 T_IsLeftBolLoop_3202
v0
  = (T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> T_IsLoop_3122
forall a b. a -> b
coe T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0)
-- Data.Integer.Properties._.IsLeftBolLoop.isMagma
d_isMagma_1484 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1484 :: T_IsLeftBolLoop_3202 -> T_IsMagma_178
d_isMagma_1484 T_IsLeftBolLoop_3202
v0
  = (T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
         T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
         ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0)))
-- Data.Integer.Properties._.IsLeftBolLoop.isPartialEquivalence
d_isPartialEquivalence_1486 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1486 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1486 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer -> Integer
v2 ~Integer
v3 T_IsLeftBolLoop_3202
v4
  = T_IsLeftBolLoop_3202 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1486 T_IsLeftBolLoop_3202
v4
du_isPartialEquivalence_1486 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1486 :: T_IsLeftBolLoop_3202 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1486 T_IsLeftBolLoop_3202
v0
  = let v1 :: T_IsLoop_3122
v1
          = T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> T_IsLeftBolLoop_3202
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsQuasigroup_3038
v2
             = T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> T_IsLoop_3122
forall a b. a -> b
coe T_IsLoop_3122
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3
                = T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056 (T_IsQuasigroup_3038 -> T_IsQuasigroup_3038
forall a b. a -> b
coe T_IsQuasigroup_3038
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Data.Integer.Properties._.IsLeftBolLoop.isQuasigroup
d_isQuasigroup_1488 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Algebra.Structures.T_IsQuasigroup_3038
d_isQuasigroup_1488 :: T_IsLeftBolLoop_3202 -> T_IsQuasigroup_3038
d_isQuasigroup_1488 T_IsLeftBolLoop_3202
v0
  = (T_IsLoop_3122 -> T_IsQuasigroup_3038)
-> Any -> T_IsQuasigroup_3038
forall a b. a -> b
coe
      T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
      ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0))
-- Data.Integer.Properties._.IsLeftBolLoop.leftBol
d_leftBol_1490 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftBol_1490 :: T_IsLeftBolLoop_3202
-> Integer -> Integer -> Integer -> T__'8801'__12
d_leftBol_1490 = T_IsLeftBolLoop_3202
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.leftDivides
d_leftDivides_1492 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_leftDivides_1492 :: T_IsLeftBolLoop_3202 -> T_Σ_14
d_leftDivides_1492 T_IsLeftBolLoop_3202
v0
  = (T_IsQuasigroup_3038 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_leftDivides_3062
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
         T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
         ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0)))
-- Data.Integer.Properties._.IsLeftBolLoop.leftDividesʳ
d_leftDivides'691'_1494 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftDivides'691'_1494 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'691'_1494 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.leftDividesˡ
d_leftDivides'737'_1496 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftDivides'737'_1496 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'737'_1496 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.refl
d_refl_1498 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1498 :: T_IsLeftBolLoop_3202 -> Integer -> T__'8801'__12
d_refl_1498 = T_IsLeftBolLoop_3202 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.reflexive
d_reflexive_1500 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1500 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1500 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.rightDivides
d_rightDivides_1502 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_rightDivides_1502 :: T_IsLeftBolLoop_3202 -> T_Σ_14
d_rightDivides_1502 T_IsLeftBolLoop_3202
v0
  = (T_IsQuasigroup_3038 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_rightDivides_3064
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
         T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
         ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> Any
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0)))
-- Data.Integer.Properties._.IsLeftBolLoop.rightDividesʳ
d_rightDivides'691'_1504 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_rightDivides'691'_1504 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'691'_1504 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.rightDividesˡ
d_rightDivides'737'_1506 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_rightDivides'737'_1506 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'737'_1506 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.setoid
d_setoid_1508 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1508 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> T_Setoid_46
d_setoid_1508 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer -> Integer
v2 ~Integer
v3 T_IsLeftBolLoop_3202
v4 = T_IsLeftBolLoop_3202 -> T_Setoid_46
du_setoid_1508 T_IsLeftBolLoop_3202
v4
du_setoid_1508 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1508 :: T_IsLeftBolLoop_3202 -> T_Setoid_46
du_setoid_1508 T_IsLeftBolLoop_3202
v0
  = let v1 :: T_IsLoop_3122
v1
          = T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> T_IsLeftBolLoop_3202
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsQuasigroup_3038
v2
             = T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> T_IsLoop_3122
forall a b. a -> b
coe T_IsLoop_3122
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056 (T_IsQuasigroup_3038 -> Any
forall a b. a -> b
coe T_IsQuasigroup_3038
v2))))
-- Data.Integer.Properties._.IsLeftBolLoop.sym
d_sym_1510 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1510 :: T_IsLeftBolLoop_3202
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1510 = T_IsLeftBolLoop_3202
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.trans
d_trans_1512 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1512 :: T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1512 = T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.∙-cong
d_'8729''45'cong_1514 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1514 :: T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1514 = T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.∙-congʳ
d_'8729''45'cong'691'_1516 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1516 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1516 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLeftBolLoop.∙-congˡ
d_'8729''45'cong'737'_1518 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1518 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1518 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.//-cong
d_'47''47''45'cong_1522 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong_1522 :: T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong_1522 = T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.//-congʳ
d_'47''47''45'cong'691'_1524 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong'691'_1524 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong'691'_1524 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.//-congˡ
d_'47''47''45'cong'737'_1526 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong'737'_1526 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong'737'_1526 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.\\-cong
d_'92''92''45'cong_1528 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong_1528 :: T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong_1528 = T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.\\-congʳ
d_'92''92''45'cong'691'_1530 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong'691'_1530 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong'691'_1530 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.\\-congˡ
d_'92''92''45'cong'737'_1532 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong'737'_1532 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong'737'_1532 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.identity
d_identity_1534 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1534 :: T_IsLoop_3122 -> T_Σ_14
d_identity_1534 T_IsLoop_3122
v0
  = (T_IsLoop_3122 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsLoop_3122 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_3138 (T_IsLoop_3122 -> Any
forall a b. a -> b
coe T_IsLoop_3122
v0)
-- Data.Integer.Properties._.IsLoop.identityʳ
d_identity'691'_1536 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1536 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> T__'8801'__12
d_identity'691'_1536 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.identityˡ
d_identity'737'_1538 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1538 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> T__'8801'__12
d_identity'737'_1538 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.isEquivalence
d_isEquivalence_1540 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1540 :: T_IsLoop_3122 -> T_IsEquivalence_28
d_isEquivalence_1540 T_IsLoop_3122
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056
         ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> Any
forall a b. a -> b
coe T_IsLoop_3122
v0)))
-- Data.Integer.Properties._.IsLoop.isMagma
d_isMagma_1542 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1542 :: T_IsLoop_3122 -> T_IsMagma_178
d_isMagma_1542 T_IsLoop_3122
v0
  = (T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> Any
forall a b. a -> b
coe T_IsLoop_3122
v0))
-- Data.Integer.Properties._.IsLoop.isPartialEquivalence
d_isPartialEquivalence_1544 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1544 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1544 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer -> Integer
v2 ~Integer
v3 T_IsLoop_3122
v4
  = T_IsLoop_3122 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1544 T_IsLoop_3122
v4
du_isPartialEquivalence_1544 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1544 :: T_IsLoop_3122 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1544 T_IsLoop_3122
v0
  = let v1 :: T_IsQuasigroup_3038
v1
          = T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> T_IsLoop_3122
forall a b. a -> b
coe T_IsLoop_3122
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2
             = T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056 (T_IsQuasigroup_3038 -> T_IsQuasigroup_3038
forall a b. a -> b
coe T_IsQuasigroup_3038
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
            T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
            ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
               T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v2))))
-- Data.Integer.Properties._.IsLoop.isQuasigroup
d_isQuasigroup_1546 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Algebra.Structures.T_IsQuasigroup_3038
d_isQuasigroup_1546 :: T_IsLoop_3122 -> T_IsQuasigroup_3038
d_isQuasigroup_1546 T_IsLoop_3122
v0
  = (T_IsLoop_3122 -> T_IsQuasigroup_3038)
-> Any -> T_IsQuasigroup_3038
forall a b. a -> b
coe T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> Any
forall a b. a -> b
coe T_IsLoop_3122
v0)
-- Data.Integer.Properties._.IsLoop.leftDivides
d_leftDivides_1548 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_leftDivides_1548 :: T_IsLoop_3122 -> T_Σ_14
d_leftDivides_1548 T_IsLoop_3122
v0
  = (T_IsQuasigroup_3038 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_leftDivides_3062
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> Any
forall a b. a -> b
coe T_IsLoop_3122
v0))
-- Data.Integer.Properties._.IsLoop.leftDividesʳ
d_leftDivides'691'_1550 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftDivides'691'_1550 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'691'_1550 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.leftDividesˡ
d_leftDivides'737'_1552 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftDivides'737'_1552 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'737'_1552 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.refl
d_refl_1554 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1554 :: T_IsLoop_3122 -> Integer -> T__'8801'__12
d_refl_1554 = T_IsLoop_3122 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.reflexive
d_reflexive_1556 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1556 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1556 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.rightDivides
d_rightDivides_1558 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_rightDivides_1558 :: T_IsLoop_3122 -> T_Σ_14
d_rightDivides_1558 T_IsLoop_3122
v0
  = (T_IsQuasigroup_3038 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_rightDivides_3064
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> Any
forall a b. a -> b
coe T_IsLoop_3122
v0))
-- Data.Integer.Properties._.IsLoop.rightDividesʳ
d_rightDivides'691'_1560 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_rightDivides'691'_1560 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'691'_1560 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.rightDividesˡ
d_rightDivides'737'_1562 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_rightDivides'737'_1562 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'737'_1562 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.setoid
d_setoid_1564 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1564 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> T_Setoid_46
d_setoid_1564 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer -> Integer
v2 ~Integer
v3 T_IsLoop_3122
v4 = T_IsLoop_3122 -> T_Setoid_46
du_setoid_1564 T_IsLoop_3122
v4
du_setoid_1564 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1564 :: T_IsLoop_3122 -> T_Setoid_46
du_setoid_1564 T_IsLoop_3122
v0
  = let v1 :: T_IsQuasigroup_3038
v1
          = T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> T_IsLoop_3122
forall a b. a -> b
coe T_IsLoop_3122
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
         T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
         ((T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056 (T_IsQuasigroup_3038 -> Any
forall a b. a -> b
coe T_IsQuasigroup_3038
v1)))
-- Data.Integer.Properties._.IsLoop.sym
d_sym_1566 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1566 :: T_IsLoop_3122
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1566 = T_IsLoop_3122
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.trans
d_trans_1568 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1568 :: T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1568 = T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.∙-cong
d_'8729''45'cong_1570 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1570 :: T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1570 = T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.∙-congʳ
d_'8729''45'cong'691'_1572 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1572 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1572 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsLoop.∙-congˡ
d_'8729''45'cong'737'_1574 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1574 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1574 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMagma.isEquivalence
d_isEquivalence_1578 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1578 :: T_IsMagma_178 -> T_IsEquivalence_28
d_isEquivalence_1578 T_IsMagma_178
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v0)
-- Data.Integer.Properties._.IsMagma.isPartialEquivalence
d_isPartialEquivalence_1580 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1580 :: (Integer -> Integer -> Integer)
-> T_IsMagma_178 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1580 ~Integer -> Integer -> Integer
v0 T_IsMagma_178
v1
  = T_IsMagma_178 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1580 T_IsMagma_178
v1
du_isPartialEquivalence_1580 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1580 :: T_IsMagma_178 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1580 T_IsMagma_178
v0
  = (T_IsEquivalence_28 -> T_IsPartialEquivalence_16)
-> Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
      ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v0))
-- Data.Integer.Properties._.IsMagma.refl
d_refl_1582 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1582 :: T_IsMagma_178 -> Integer -> T__'8801'__12
d_refl_1582 = T_IsMagma_178 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMagma.reflexive
d_reflexive_1584 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1584 :: (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1584 = (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMagma.setoid
d_setoid_1586 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1586 :: (Integer -> Integer -> Integer) -> T_IsMagma_178 -> T_Setoid_46
d_setoid_1586 Integer -> Integer -> Integer
v0 T_IsMagma_178
v1
  = (T_IsMagma_178 -> T_Setoid_46) -> T_IsMagma_178 -> T_Setoid_46
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 T_IsMagma_178
v1
-- Data.Integer.Properties._.IsMagma.sym
d_sym_1588 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1588 :: T_IsMagma_178
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1588 = T_IsMagma_178
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMagma.trans
d_trans_1590 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1590 :: T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1590 = T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMagma.∙-cong
d_'8729''45'cong_1592 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1592 :: T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1592 = T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMagma.∙-congʳ
d_'8729''45'cong'691'_1594 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1594 :: (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1594 = (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMagma.∙-congˡ
d_'8729''45'cong'737'_1596 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1596 :: (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1596 = (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMedialMagma.isEquivalence
d_isEquivalence_1600 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1600 :: T_IsMedialMagma_370 -> T_IsEquivalence_28
d_isEquivalence_1600 T_IsMedialMagma_370
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsMedialMagma_370 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsMedialMagma_370 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_378 (T_IsMedialMagma_370 -> Any
forall a b. a -> b
coe T_IsMedialMagma_370
v0))
-- Data.Integer.Properties._.IsMedialMagma.isMagma
d_isMagma_1602 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1602 :: T_IsMedialMagma_370 -> T_IsMagma_178
d_isMagma_1602 T_IsMedialMagma_370
v0
  = (T_IsMedialMagma_370 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe T_IsMedialMagma_370 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_378 (T_IsMedialMagma_370 -> Any
forall a b. a -> b
coe T_IsMedialMagma_370
v0)
-- Data.Integer.Properties._.IsMedialMagma.isPartialEquivalence
d_isPartialEquivalence_1604 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1604 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1604 ~Integer -> Integer -> Integer
v0 T_IsMedialMagma_370
v1
  = T_IsMedialMagma_370 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1604 T_IsMedialMagma_370
v1
du_isPartialEquivalence_1604 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1604 :: T_IsMedialMagma_370 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1604 T_IsMedialMagma_370
v0
  = let v1 :: T_IsMagma_178
v1
          = T_IsMedialMagma_370 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_378 (T_IsMedialMagma_370 -> T_IsMedialMagma_370
forall a b. a -> b
coe T_IsMedialMagma_370
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
         T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
         ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v1)))
-- Data.Integer.Properties._.IsMedialMagma.medial
d_medial_1606 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_medial_1606 :: T_IsMedialMagma_370
-> Integer -> Integer -> Integer -> Integer -> T__'8801'__12
d_medial_1606 = T_IsMedialMagma_370
-> Integer -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMedialMagma.refl
d_refl_1608 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1608 :: T_IsMedialMagma_370 -> Integer -> T__'8801'__12
d_refl_1608 = T_IsMedialMagma_370 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMedialMagma.reflexive
d_reflexive_1610 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1610 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1610 = (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMedialMagma.setoid
d_setoid_1612 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1612 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370 -> T_Setoid_46
d_setoid_1612 ~Integer -> Integer -> Integer
v0 T_IsMedialMagma_370
v1 = T_IsMedialMagma_370 -> T_Setoid_46
du_setoid_1612 T_IsMedialMagma_370
v1
du_setoid_1612 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1612 :: T_IsMedialMagma_370 -> T_Setoid_46
du_setoid_1612 T_IsMedialMagma_370
v0
  = (T_IsMagma_178 -> T_Setoid_46) -> Any -> T_Setoid_46
forall a b. a -> b
coe
      T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
      ((T_IsMedialMagma_370 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsMedialMagma_370 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_378 (T_IsMedialMagma_370 -> Any
forall a b. a -> b
coe T_IsMedialMagma_370
v0))
-- Data.Integer.Properties._.IsMedialMagma.sym
d_sym_1614 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1614 :: T_IsMedialMagma_370
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1614 = T_IsMedialMagma_370
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMedialMagma.trans
d_trans_1616 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1616 :: T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1616 = T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMedialMagma.∙-cong
d_'8729''45'cong_1618 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1618 :: T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1618 = T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMedialMagma.∙-congʳ
d_'8729''45'cong'691'_1620 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1620 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1620 = (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMedialMagma.∙-congˡ
d_'8729''45'cong'737'_1622 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1622 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1622 = (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.//-cong
d_'47''47''45'cong_1626 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong_1626 :: T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong_1626 = T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.//-congʳ
d_'47''47''45'cong'691'_1628 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong'691'_1628 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong'691'_1628 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.//-congˡ
d_'47''47''45'cong'737'_1630 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong'737'_1630 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong'737'_1630 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.\\-cong
d_'92''92''45'cong_1632 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong_1632 :: T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong_1632 = T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.\\-congʳ
d_'92''92''45'cong'691'_1634 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong'691'_1634 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong'691'_1634 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.\\-congˡ
d_'92''92''45'cong'737'_1636 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong'737'_1636 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong'737'_1636 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.identity
d_identity_1638 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1638 :: T_IsMiddleBolLoop_3462 -> T_Σ_14
d_identity_1638 T_IsMiddleBolLoop_3462
v0
  = (T_IsLoop_3122 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsLoop_3122 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_3138
      ((T_IsMiddleBolLoop_3462 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0))
-- Data.Integer.Properties._.IsMiddleBolLoop.identityʳ
d_identity'691'_1640 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1640 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> T__'8801'__12
d_identity'691'_1640 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.identityˡ
d_identity'737'_1642 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1642 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> T__'8801'__12
d_identity'737'_1642 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.isEquivalence
d_isEquivalence_1644 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1644 :: T_IsMiddleBolLoop_3462 -> T_IsEquivalence_28
d_isEquivalence_1644 T_IsMiddleBolLoop_3462
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056
         ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
            T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
            ((T_IsMiddleBolLoop_3462 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0))))
-- Data.Integer.Properties._.IsMiddleBolLoop.isLoop
d_isLoop_1646 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122
d_isLoop_1646 :: T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
d_isLoop_1646 T_IsMiddleBolLoop_3462
v0
  = (T_IsMiddleBolLoop_3462 -> T_IsLoop_3122) -> Any -> T_IsLoop_3122
forall a b. a -> b
coe T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0)
-- Data.Integer.Properties._.IsMiddleBolLoop.isMagma
d_isMagma_1648 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1648 :: T_IsMiddleBolLoop_3462 -> T_IsMagma_178
d_isMagma_1648 T_IsMiddleBolLoop_3462
v0
  = (T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
         T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
         ((T_IsMiddleBolLoop_3462 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0)))
-- Data.Integer.Properties._.IsMiddleBolLoop.isPartialEquivalence
d_isPartialEquivalence_1650 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1650 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1650 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer -> Integer
v2 ~Integer
v3 T_IsMiddleBolLoop_3462
v4
  = T_IsMiddleBolLoop_3462 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1650 T_IsMiddleBolLoop_3462
v4
du_isPartialEquivalence_1650 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1650 :: T_IsMiddleBolLoop_3462 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1650 T_IsMiddleBolLoop_3462
v0
  = let v1 :: T_IsLoop_3122
v1
          = T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> T_IsMiddleBolLoop_3462
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsQuasigroup_3038
v2
             = T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> T_IsLoop_3122
forall a b. a -> b
coe T_IsLoop_3122
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3
                = T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056 (T_IsQuasigroup_3038 -> T_IsQuasigroup_3038
forall a b. a -> b
coe T_IsQuasigroup_3038
v2) in
          Any -> Any
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Data.Integer.Properties._.IsMiddleBolLoop.isQuasigroup
d_isQuasigroup_1652 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Algebra.Structures.T_IsQuasigroup_3038
d_isQuasigroup_1652 :: T_IsMiddleBolLoop_3462 -> T_IsQuasigroup_3038
d_isQuasigroup_1652 T_IsMiddleBolLoop_3462
v0
  = (T_IsLoop_3122 -> T_IsQuasigroup_3038)
-> Any -> T_IsQuasigroup_3038
forall a b. a -> b
coe
      T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
      ((T_IsMiddleBolLoop_3462 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0))
-- Data.Integer.Properties._.IsMiddleBolLoop.leftDivides
d_leftDivides_1654 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_leftDivides_1654 :: T_IsMiddleBolLoop_3462 -> T_Σ_14
d_leftDivides_1654 T_IsMiddleBolLoop_3462
v0
  = (T_IsQuasigroup_3038 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_leftDivides_3062
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
         T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
         ((T_IsMiddleBolLoop_3462 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0)))
-- Data.Integer.Properties._.IsMiddleBolLoop.leftDividesʳ
d_leftDivides'691'_1656 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftDivides'691'_1656 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'691'_1656 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.leftDividesˡ
d_leftDivides'737'_1658 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftDivides'737'_1658 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'737'_1658 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.middleBol
d_middleBol_1660 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_middleBol_1660 :: T_IsMiddleBolLoop_3462
-> Integer -> Integer -> Integer -> T__'8801'__12
d_middleBol_1660 = T_IsMiddleBolLoop_3462
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.refl
d_refl_1662 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1662 :: T_IsMiddleBolLoop_3462 -> Integer -> T__'8801'__12
d_refl_1662 = T_IsMiddleBolLoop_3462 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.reflexive
d_reflexive_1664 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1664 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1664 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.rightDivides
d_rightDivides_1666 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_rightDivides_1666 :: T_IsMiddleBolLoop_3462 -> T_Σ_14
d_rightDivides_1666 T_IsMiddleBolLoop_3462
v0
  = (T_IsQuasigroup_3038 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_rightDivides_3064
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
         T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
         ((T_IsMiddleBolLoop_3462 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> Any
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0)))
-- Data.Integer.Properties._.IsMiddleBolLoop.rightDividesʳ
d_rightDivides'691'_1668 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_rightDivides'691'_1668 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'691'_1668 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.rightDividesˡ
d_rightDivides'737'_1670 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_rightDivides'737'_1670 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'737'_1670 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.setoid
d_setoid_1672 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1672 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> T_Setoid_46
d_setoid_1672 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer -> Integer
v2 ~Integer
v3 T_IsMiddleBolLoop_3462
v4 = T_IsMiddleBolLoop_3462 -> T_Setoid_46
du_setoid_1672 T_IsMiddleBolLoop_3462
v4
du_setoid_1672 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1672 :: T_IsMiddleBolLoop_3462 -> T_Setoid_46
du_setoid_1672 T_IsMiddleBolLoop_3462
v0
  = let v1 :: T_IsLoop_3122
v1
          = T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3476 (T_IsMiddleBolLoop_3462 -> T_IsMiddleBolLoop_3462
forall a b. a -> b
coe T_IsMiddleBolLoop_3462
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsQuasigroup_3038
v2
             = T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> T_IsLoop_3122
forall a b. a -> b
coe T_IsLoop_3122
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056 (T_IsQuasigroup_3038 -> Any
forall a b. a -> b
coe T_IsQuasigroup_3038
v2))))
-- Data.Integer.Properties._.IsMiddleBolLoop.sym
d_sym_1674 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1674 :: T_IsMiddleBolLoop_3462
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1674 = T_IsMiddleBolLoop_3462
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.trans
d_trans_1676 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1676 :: T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1676 = T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.∙-cong
d_'8729''45'cong_1678 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1678 :: T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1678 = T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.∙-congʳ
d_'8729''45'cong'691'_1680 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1680 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1680 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMiddleBolLoop.∙-congˡ
d_'8729''45'cong'737'_1682 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1682 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1682 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.assoc
d_assoc_1686 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1686 :: T_IsMonoid_712 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1686 = T_IsMonoid_712 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.identity
d_identity_1688 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1688 :: T_IsMonoid_712 -> T_Σ_14
d_identity_1688 T_IsMonoid_712
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724 (T_IsMonoid_712 -> Any
forall a b. a -> b
coe T_IsMonoid_712
v0)
-- Data.Integer.Properties._.IsMonoid.identityʳ
d_identity'691'_1690 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1690 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> Integer -> T__'8801'__12
d_identity'691'_1690 = (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.identityˡ
d_identity'737'_1692 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1692 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> Integer -> T__'8801'__12
d_identity'737'_1692 = (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.isEquivalence
d_isEquivalence_1694 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1694 :: T_IsMonoid_712 -> T_IsEquivalence_28
d_isEquivalence_1694 T_IsMonoid_712
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> Any
forall a b. a -> b
coe T_IsMonoid_712
v0)))
-- Data.Integer.Properties._.IsMonoid.isMagma
d_isMagma_1696 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1696 :: T_IsMonoid_712 -> T_IsMagma_178
d_isMagma_1696 T_IsMonoid_712
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> Any
forall a b. a -> b
coe T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> Any
forall a b. a -> b
coe T_IsMonoid_712
v0))
-- Data.Integer.Properties._.IsMonoid.isPartialEquivalence
d_isPartialEquivalence_1698 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1698 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1698 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsMonoid_712
v2
  = T_IsMonoid_712 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1698 T_IsMonoid_712
v2
du_isPartialEquivalence_1698 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1698 :: T_IsMonoid_712 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1698 T_IsMonoid_712
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
            T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
            ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
               T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v2))))
-- Data.Integer.Properties._.IsMonoid.isSemigroup
d_isSemigroup_1700 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1700 :: T_IsMonoid_712 -> T_IsSemigroup_488
d_isSemigroup_1700 T_IsMonoid_712
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488) -> Any -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> Any
forall a b. a -> b
coe T_IsMonoid_712
v0)
-- Data.Integer.Properties._.IsMonoid.isUnitalMagma
d_isUnitalMagma_1702 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1702 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> T_IsUnitalMagma_666
d_isUnitalMagma_1702 Integer -> Integer -> Integer
v0 Integer
v1 T_IsMonoid_712
v2
  = (T_IsMonoid_712 -> T_IsUnitalMagma_666)
-> T_IsMonoid_712 -> T_IsUnitalMagma_666
forall a b. a -> b
coe T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756 T_IsMonoid_712
v2
-- Data.Integer.Properties._.IsMonoid.refl
d_refl_1704 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1704 :: T_IsMonoid_712 -> Integer -> T__'8801'__12
d_refl_1704 = T_IsMonoid_712 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.reflexive
d_reflexive_1706 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1706 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1706 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.setoid
d_setoid_1708 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1708 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> T_Setoid_46
d_setoid_1708 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsMonoid_712
v2 = T_IsMonoid_712 -> T_Setoid_46
du_setoid_1708 T_IsMonoid_712
v2
du_setoid_1708 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1708 :: T_IsMonoid_712 -> T_Setoid_46
du_setoid_1708 T_IsMonoid_712
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v0) in
    Any -> T_Setoid_46
forall a b. a -> b
coe
      ((T_IsMagma_178 -> T_Setoid_46) -> Any -> Any
forall a b. a -> b
coe
         T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> Any
forall a b. a -> b
coe T_IsSemigroup_488
v1)))
-- Data.Integer.Properties._.IsMonoid.sym
d_sym_1710 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_sym_1710 :: T_IsMonoid_712
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1710 = T_IsMonoid_712
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.trans
d_trans_1712 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_trans_1712 :: T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1712 = T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.∙-cong
d_'8729''45'cong_1714 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1714 :: T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1714 = T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.∙-congʳ
d_'8729''45'cong'691'_1716 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_1716 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1716 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMonoid.∙-congˡ
d_'8729''45'cong'737'_1718 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'737'_1718 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1718 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.//-cong
d_'47''47''45'cong_1722 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong_1722 :: T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong_1722 = T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.//-congʳ
d_'47''47''45'cong'691'_1724 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong'691'_1724 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong'691'_1724 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.//-congˡ
d_'47''47''45'cong'737'_1726 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'47''47''45'cong'737'_1726 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong'737'_1726 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.\\-cong
d_'92''92''45'cong_1728 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong_1728 :: T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong_1728 = T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.\\-congʳ
d_'92''92''45'cong'691'_1730 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong'691'_1730 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong'691'_1730 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.\\-congˡ
d_'92''92''45'cong'737'_1732 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'92''92''45'cong'737'_1732 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong'737'_1732 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.identical
d_identical_1734 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identical_1734 :: T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
d_identical_1734 = T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.identity
d_identity_1736 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1736 :: T_IsMoufangLoop_3370 -> T_Σ_14
d_identity_1736 T_IsMoufangLoop_3370
v0
  = (T_IsLoop_3122 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsLoop_3122 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_3138
      ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe
         T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216
         ((T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202) -> Any -> Any
forall a b. a -> b
coe
            T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
MAlonzo.Code.Algebra.Structures.d_isLeftBolLoop_3386 (T_IsMoufangLoop_3370 -> Any
forall a b. a -> b
coe T_IsMoufangLoop_3370
v0)))
-- Data.Integer.Properties._.IsMoufangLoop.identityʳ
d_identity'691'_1738 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1738 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> T__'8801'__12
d_identity'691'_1738 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.identityˡ
d_identity'737'_1740 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1740 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> T__'8801'__12
d_identity'737'_1740 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.isEquivalence
d_isEquivalence_1742 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1742 :: T_IsMoufangLoop_3370 -> T_IsEquivalence_28
d_isEquivalence_1742 T_IsMoufangLoop_3370
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> Any
forall a b. a -> b
coe
         T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056
         ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
            T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
            ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe
               T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216
               ((T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202) -> Any -> Any
forall a b. a -> b
coe
                  T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
MAlonzo.Code.Algebra.Structures.d_isLeftBolLoop_3386 (T_IsMoufangLoop_3370 -> Any
forall a b. a -> b
coe T_IsMoufangLoop_3370
v0)))))
-- Data.Integer.Properties._.IsMoufangLoop.isLeftBolLoop
d_isLeftBolLoop_1744 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202
d_isLeftBolLoop_1744 :: T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
d_isLeftBolLoop_1744 T_IsMoufangLoop_3370
v0
  = (T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202)
-> Any -> T_IsLeftBolLoop_3202
forall a b. a -> b
coe T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
MAlonzo.Code.Algebra.Structures.d_isLeftBolLoop_3386 (T_IsMoufangLoop_3370 -> Any
forall a b. a -> b
coe T_IsMoufangLoop_3370
v0)
-- Data.Integer.Properties._.IsMoufangLoop.isLoop
d_isLoop_1746 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122
d_isLoop_1746 :: T_IsMoufangLoop_3370 -> T_IsLoop_3122
d_isLoop_1746 T_IsMoufangLoop_3370
v0
  = (T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> T_IsLoop_3122
forall a b. a -> b
coe
      T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216
      ((T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202) -> Any -> Any
forall a b. a -> b
coe T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
MAlonzo.Code.Algebra.Structures.d_isLeftBolLoop_3386 (T_IsMoufangLoop_3370 -> Any
forall a b. a -> b
coe T_IsMoufangLoop_3370
v0))
-- Data.Integer.Properties._.IsMoufangLoop.isMagma
d_isMagma_1748 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1748 :: T_IsMoufangLoop_3370 -> T_IsMagma_178
d_isMagma_1748 T_IsMoufangLoop_3370
v0
  = (T_IsQuasigroup_3038 -> T_IsMagma_178) -> Any -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
         T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
         ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe
            T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216
            ((T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202) -> Any -> Any
forall a b. a -> b
coe
               T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
MAlonzo.Code.Algebra.Structures.d_isLeftBolLoop_3386 (T_IsMoufangLoop_3370 -> Any
forall a b. a -> b
coe T_IsMoufangLoop_3370
v0))))
-- Data.Integer.Properties._.IsMoufangLoop.isPartialEquivalence
d_isPartialEquivalence_1750 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1750 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1750 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer -> Integer
v2 ~Integer
v3 T_IsMoufangLoop_3370
v4
  = T_IsMoufangLoop_3370 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1750 T_IsMoufangLoop_3370
v4
du_isPartialEquivalence_1750 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1750 :: T_IsMoufangLoop_3370 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1750 T_IsMoufangLoop_3370
v0
  = let v1 :: T_IsLeftBolLoop_3202
v1
          = T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
MAlonzo.Code.Algebra.Structures.d_isLeftBolLoop_3386 (T_IsMoufangLoop_3370 -> T_IsMoufangLoop_3370
forall a b. a -> b
coe T_IsMoufangLoop_3370
v0) in
    Any -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsLoop_3122
v2 = T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216 (T_IsLeftBolLoop_3202 -> T_IsLeftBolLoop_3202
forall a b. a -> b
coe T_IsLeftBolLoop_3202
v1) in
       Any -> Any
forall a b. a -> b
coe
         (let v3 :: T_IsQuasigroup_3038
v3
                = T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136 (T_IsLoop_3122 -> T_IsLoop_3122
forall a b. a -> b
coe T_IsLoop_3122
v2) in
          Any -> Any
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_178
v4
                   = T_IsQuasigroup_3038 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_3056 (T_IsQuasigroup_3038 -> T_IsQuasigroup_3038
forall a b. a -> b
coe T_IsQuasigroup_3038
v3) in
             Any -> Any
forall a b. a -> b
coe
               ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16) -> Any -> Any
forall a b. a -> b
coe
                  T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                  ((T_IsMagma_178 -> T_IsEquivalence_28) -> Any -> Any
forall a b. a -> b
coe
                     T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> Any
forall a b. a -> b
coe T_IsMagma_178
v4))))))
-- Data.Integer.Properties._.IsMoufangLoop.isQuasigroup
d_isQuasigroup_1752 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Algebra.Structures.T_IsQuasigroup_3038
d_isQuasigroup_1752 :: T_IsMoufangLoop_3370 -> T_IsQuasigroup_3038
d_isQuasigroup_1752 T_IsMoufangLoop_3370
v0
  = (T_IsLoop_3122 -> T_IsQuasigroup_3038)
-> Any -> T_IsQuasigroup_3038
forall a b. a -> b
coe
      T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
      ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe
         T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216
         ((T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202) -> Any -> Any
forall a b. a -> b
coe
            T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
MAlonzo.Code.Algebra.Structures.d_isLeftBolLoop_3386 (T_IsMoufangLoop_3370 -> Any
forall a b. a -> b
coe T_IsMoufangLoop_3370
v0)))
-- Data.Integer.Properties._.IsMoufangLoop.leftBol
d_leftBol_1754 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftBol_1754 :: T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
d_leftBol_1754 = T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.leftDivides
d_leftDivides_1756 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_leftDivides_1756 :: T_IsMoufangLoop_3370 -> T_Σ_14
d_leftDivides_1756 T_IsMoufangLoop_3370
v0
  = (T_IsQuasigroup_3038 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsQuasigroup_3038 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_leftDivides_3062
      ((T_IsLoop_3122 -> T_IsQuasigroup_3038) -> Any -> Any
forall a b. a -> b
coe
         T_IsLoop_3122 -> T_IsQuasigroup_3038
MAlonzo.Code.Algebra.Structures.d_isQuasigroup_3136
         ((T_IsLeftBolLoop_3202 -> T_IsLoop_3122) -> Any -> Any
forall a b. a -> b
coe
            T_IsLeftBolLoop_3202 -> T_IsLoop_3122
MAlonzo.Code.Algebra.Structures.d_isLoop_3216
            ((T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202) -> Any -> Any
forall a b. a -> b
coe
               T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
MAlonzo.Code.Algebra.Structures.d_isLeftBolLoop_3386 (T_IsMoufangLoop_3370 -> Any
forall a b. a -> b
coe T_IsMoufangLoop_3370
v0))))
-- Data.Integer.Properties._.IsMoufangLoop.leftDividesʳ
d_leftDivides'691'_1758 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftDivides'691'_1758 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'691'_1758 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.leftDividesˡ
d_leftDivides'737'_1760 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftDivides'737'_1760 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'737'_1760 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Integer.Properties._.IsMoufangLoop.refl
d_refl_1762 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'_