{-# 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.Nat.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.Definitions.RawMagma
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
import qualified MAlonzo.Code.Algebra.Structures
import qualified MAlonzo.Code.Data.Irrelevant
import qualified MAlonzo.Code.Data.Nat.Base
import qualified MAlonzo.Code.Data.Sum.Base
import qualified MAlonzo.Code.Function.Base
import qualified MAlonzo.Code.Function.Bundles
import qualified MAlonzo.Code.Function.Metric.Nat.Bundles
import qualified MAlonzo.Code.Function.Metric.Structures
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.PropositionalEquality.Algebra
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.Binary.Structures.Biased
import qualified MAlonzo.Code.Relation.Nullary.Decidable
import qualified MAlonzo.Code.Relation.Nullary.Decidable.Core
import qualified MAlonzo.Code.Relation.Nullary.Negation.Core
import qualified MAlonzo.Code.Relation.Nullary.Reflects

-- Data.Nat.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.Nat.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.Nat.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.Nat.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.Nat.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.Nat.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.Nat.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.Nat.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.Nat.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.Nat.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.Nat.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.Nat.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.Nat.Properties._.Commutative
d_Commutative_32 :: (Integer -> Integer -> Integer) -> ()
d_Commutative_32 :: (Integer -> Integer -> Integer) -> ()
d_Commutative_32 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Congruent₁
d_Congruent'8321'_34 :: (Integer -> Integer) -> ()
d_Congruent'8321'_34 :: (Integer -> Integer) -> ()
d_Congruent'8321'_34 = (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Congruent₂
d_Congruent'8322'_36 :: (Integer -> Integer -> Integer) -> ()
d_Congruent'8322'_36 :: (Integer -> Integer -> Integer) -> ()
d_Congruent'8322'_36 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Conical
d_Conical_38 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Conical_38 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Conical_38 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Flexible
d_Flexible_40 :: (Integer -> Integer -> Integer) -> ()
d_Flexible_40 :: (Integer -> Integer -> Integer) -> ()
d_Flexible_40 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Idempotent
d_Idempotent_42 :: (Integer -> Integer -> Integer) -> ()
d_Idempotent_42 :: (Integer -> Integer -> Integer) -> ()
d_Idempotent_42 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.IdempotentFun
d_IdempotentFun_44 :: (Integer -> Integer) -> ()
d_IdempotentFun_44 :: (Integer -> Integer) -> ()
d_IdempotentFun_44 = (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Identical
d_Identical_46 :: (Integer -> Integer -> Integer) -> ()
d_Identical_46 :: (Integer -> Integer -> Integer) -> ()
d_Identical_46 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Identity
d_Identity_48 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Identity_48 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Identity_48 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Interchangable
d_Interchangable_50 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_Interchangable_50 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_Interchangable_50 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Inverse
d_Inverse_52 ::
  Integer ->
  (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_Inverse_52 :: Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_Inverse_52 = Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Invertible
d_Invertible_54 ::
  Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_Invertible_54 :: Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_Invertible_54 = Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
forall a. a
erased
-- Data.Nat.Properties._.Involutive
d_Involutive_56 :: (Integer -> Integer) -> ()
d_Involutive_56 :: (Integer -> Integer) -> ()
d_Involutive_56 = (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftAlternative
d_LeftAlternative_58 :: (Integer -> Integer -> Integer) -> ()
d_LeftAlternative_58 :: (Integer -> Integer -> Integer) -> ()
d_LeftAlternative_58 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftBol
d_LeftBol_60 :: (Integer -> Integer -> Integer) -> ()
d_LeftBol_60 :: (Integer -> Integer -> Integer) -> ()
d_LeftBol_60 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftCongruent
d_LeftCongruent_62 :: (Integer -> Integer -> Integer) -> ()
d_LeftCongruent_62 :: (Integer -> Integer -> Integer) -> ()
d_LeftCongruent_62 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftConical
d_LeftConical_64 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftConical_64 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftConical_64 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftDivides
d_LeftDivides_66 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_LeftDivides_66 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_LeftDivides_66 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftDividesʳ
d_LeftDivides'691'_68 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_LeftDivides'691'_68 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_LeftDivides'691'_68 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftDividesˡ
d_LeftDivides'737'_70 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_LeftDivides'737'_70 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_LeftDivides'737'_70 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftIdentity
d_LeftIdentity_72 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftIdentity_72 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftIdentity_72 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftInverse
d_LeftInverse_74 ::
  Integer ->
  (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_LeftInverse_74 :: Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_LeftInverse_74 = Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftInvertible
d_LeftInvertible_76 ::
  Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_LeftInvertible_76 :: Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_LeftInvertible_76 = Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftSemimedial
d_LeftSemimedial_78 :: (Integer -> Integer -> Integer) -> ()
d_LeftSemimedial_78 :: (Integer -> Integer -> Integer) -> ()
d_LeftSemimedial_78 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftZero
d_LeftZero_80 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftZero_80 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftZero_80 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Medial
d_Medial_82 :: (Integer -> Integer -> Integer) -> ()
d_Medial_82 :: (Integer -> Integer -> Integer) -> ()
d_Medial_82 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.MiddleBol
d_MiddleBol_84 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_MiddleBol_84 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> ()
d_MiddleBol_84 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> ()
forall a. a
erased
-- Data.Nat.Properties._.RightAlternative
d_RightAlternative_86 :: (Integer -> Integer -> Integer) -> ()
d_RightAlternative_86 :: (Integer -> Integer -> Integer) -> ()
d_RightAlternative_86 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightBol
d_RightBol_88 :: (Integer -> Integer -> Integer) -> ()
d_RightBol_88 :: (Integer -> Integer -> Integer) -> ()
d_RightBol_88 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightCongruent
d_RightCongruent_90 :: (Integer -> Integer -> Integer) -> ()
d_RightCongruent_90 :: (Integer -> Integer -> Integer) -> ()
d_RightCongruent_90 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightConical
d_RightConical_92 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_RightConical_92 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightConical_92 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightDivides
d_RightDivides_94 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_RightDivides_94 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_RightDivides_94 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightDividesʳ
d_RightDivides'691'_96 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_RightDivides'691'_96 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_RightDivides'691'_96 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightDividesˡ
d_RightDivides'737'_98 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> ()
d_RightDivides'737'_98 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
d_RightDivides'737'_98 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightIdentity
d_RightIdentity_100 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_RightIdentity_100 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightIdentity_100 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightInverse
d_RightInverse_102 ::
  Integer ->
  (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_RightInverse_102 :: Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
d_RightInverse_102 = Integer
-> (Integer -> Integer) -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightInvertible
d_RightInvertible_104 ::
  Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_RightInvertible_104 :: Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
d_RightInvertible_104 = Integer -> (Integer -> Integer -> Integer) -> Integer -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightSemimedial
d_RightSemimedial_106 :: (Integer -> Integer -> Integer) -> ()
d_RightSemimedial_106 :: (Integer -> Integer -> Integer) -> ()
d_RightSemimedial_106 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightZero
d_RightZero_108 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightZero_108 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightZero_108 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Selective
d_Selective_110 :: (Integer -> Integer -> Integer) -> ()
d_Selective_110 :: (Integer -> Integer -> Integer) -> ()
d_Selective_110 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.SelfInverse
d_SelfInverse_112 :: (Integer -> Integer) -> ()
d_SelfInverse_112 :: (Integer -> Integer) -> ()
d_SelfInverse_112 = (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Semimedial
d_Semimedial_114 :: (Integer -> Integer -> Integer) -> ()
d_Semimedial_114 :: (Integer -> Integer -> Integer) -> ()
d_Semimedial_114 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.StarDestructive
d_StarDestructive_116 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarDestructive_116 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarDestructive_116 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.StarExpansive
d_StarExpansive_118 ::
  Integer ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarExpansive_118 :: Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
d_StarExpansive_118 = Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
forall a. a
erased
-- Data.Nat.Properties._.StarLeftDestructive
d_StarLeftDestructive_120 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarLeftDestructive_120 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarLeftDestructive_120 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.StarLeftExpansive
d_StarLeftExpansive_122 ::
  Integer ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarLeftExpansive_122 :: Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
d_StarLeftExpansive_122 = Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
forall a. a
erased
-- Data.Nat.Properties._.StarRightDestructive
d_StarRightDestructive_124 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarRightDestructive_124 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarRightDestructive_124 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.StarRightExpansive
d_StarRightExpansive_126 ::
  Integer ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) -> (Integer -> Integer) -> ()
d_StarRightExpansive_126 :: Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
d_StarRightExpansive_126 = Integer
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> ()
forall a. a
erased
-- Data.Nat.Properties._.Zero
d_Zero_128 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Zero_128 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Zero_128 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup
d_IsAbelianGroup_132 :: p -> p -> p -> ()
d_IsAbelianGroup_132 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsAlternativeMagma
d_IsAlternativeMagma_136 :: p -> ()
d_IsAlternativeMagma_136 p
a0 = ()
-- Data.Nat.Properties._.IsBand
d_IsBand_140 :: p -> ()
d_IsBand_140 p
a0 = ()
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring
d_IsCancellativeCommutativeSemiring_144 :: p -> p -> p -> p -> ()
d_IsCancellativeCommutativeSemiring_144 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsCommutativeBand
d_IsCommutativeBand_148 :: p -> ()
d_IsCommutativeBand_148 p
a0 = ()
-- Data.Nat.Properties._.IsCommutativeMagma
d_IsCommutativeMagma_152 :: p -> ()
d_IsCommutativeMagma_152 p
a0 = ()
-- Data.Nat.Properties._.IsCommutativeMonoid
d_IsCommutativeMonoid_156 :: p -> p -> ()
d_IsCommutativeMonoid_156 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsCommutativeRing
d_IsCommutativeRing_160 :: p -> p -> p -> p -> p -> ()
d_IsCommutativeRing_160 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Nat.Properties._.IsCommutativeSemigroup
d_IsCommutativeSemigroup_164 :: p -> ()
d_IsCommutativeSemigroup_164 p
a0 = ()
-- Data.Nat.Properties._.IsCommutativeSemiring
d_IsCommutativeSemiring_168 :: p -> p -> p -> p -> ()
d_IsCommutativeSemiring_168 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne
d_IsCommutativeSemiringWithoutOne_172 :: p -> p -> p -> ()
d_IsCommutativeSemiringWithoutOne_172 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsFlexibleMagma
d_IsFlexibleMagma_176 :: p -> ()
d_IsFlexibleMagma_176 p
a0 = ()
-- Data.Nat.Properties._.IsGroup
d_IsGroup_180 :: p -> p -> p -> ()
d_IsGroup_180 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid
d_IsIdempotentCommutativeMonoid_184 :: p -> p -> ()
d_IsIdempotentCommutativeMonoid_184 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsIdempotentMagma
d_IsIdempotentMagma_188 :: p -> ()
d_IsIdempotentMagma_188 p
a0 = ()
-- Data.Nat.Properties._.IsIdempotentMonoid
d_IsIdempotentMonoid_192 :: p -> p -> ()
d_IsIdempotentMonoid_192 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsIdempotentSemiring
d_IsIdempotentSemiring_196 :: p -> p -> p -> p -> ()
d_IsIdempotentSemiring_196 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsInvertibleMagma
d_IsInvertibleMagma_200 :: p -> p -> p -> ()
d_IsInvertibleMagma_200 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsInvertibleUnitalMagma
d_IsInvertibleUnitalMagma_204 :: p -> p -> p -> ()
d_IsInvertibleUnitalMagma_204 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsKleeneAlgebra
d_IsKleeneAlgebra_208 :: p -> p -> p -> p -> p -> ()
d_IsKleeneAlgebra_208 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Nat.Properties._.IsLeftBolLoop
d_IsLeftBolLoop_212 :: p -> p -> p -> p -> ()
d_IsLeftBolLoop_212 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsLoop
d_IsLoop_216 :: p -> p -> p -> p -> ()
d_IsLoop_216 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsMagma
d_IsMagma_220 :: p -> ()
d_IsMagma_220 p
a0 = ()
-- Data.Nat.Properties._.IsMedialMagma
d_IsMedialMagma_224 :: p -> ()
d_IsMedialMagma_224 p
a0 = ()
-- Data.Nat.Properties._.IsMiddleBolLoop
d_IsMiddleBolLoop_228 :: p -> p -> p -> p -> ()
d_IsMiddleBolLoop_228 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsMonoid
d_IsMonoid_232 :: p -> p -> ()
d_IsMonoid_232 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsMoufangLoop
d_IsMoufangLoop_236 :: p -> p -> p -> p -> ()
d_IsMoufangLoop_236 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsNearSemiring
d_IsNearSemiring_240 :: p -> p -> p -> ()
d_IsNearSemiring_240 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsNearring
d_IsNearring_244 :: p -> p -> p -> p -> p -> ()
d_IsNearring_244 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Nat.Properties._.IsNonAssociativeRing
d_IsNonAssociativeRing_248 :: p -> p -> p -> p -> p -> ()
d_IsNonAssociativeRing_248 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Nat.Properties._.IsQuasigroup
d_IsQuasigroup_252 :: p -> p -> p -> ()
d_IsQuasigroup_252 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsQuasiring
d_IsQuasiring_256 :: p -> p -> p -> p -> ()
d_IsQuasiring_256 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsRightBolLoop
d_IsRightBolLoop_260 :: p -> p -> p -> p -> ()
d_IsRightBolLoop_260 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsRing
d_IsRing_264 :: p -> p -> p -> p -> p -> ()
d_IsRing_264 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Nat.Properties._.IsRingWithoutOne
d_IsRingWithoutOne_268 :: p -> p -> p -> p -> ()
d_IsRingWithoutOne_268 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsSelectiveMagma
d_IsSelectiveMagma_272 :: p -> ()
d_IsSelectiveMagma_272 p
a0 = ()
-- Data.Nat.Properties._.IsSemigroup
d_IsSemigroup_276 :: p -> ()
d_IsSemigroup_276 p
a0 = ()
-- Data.Nat.Properties._.IsSemimedialMagma
d_IsSemimedialMagma_280 :: p -> ()
d_IsSemimedialMagma_280 p
a0 = ()
-- Data.Nat.Properties._.IsSemiring
d_IsSemiring_284 :: p -> p -> p -> p -> ()
d_IsSemiring_284 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero
d_IsSemiringWithoutAnnihilatingZero_288 :: p -> p -> p -> p -> ()
d_IsSemiringWithoutAnnihilatingZero_288 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsSemiringWithoutOne
d_IsSemiringWithoutOne_292 :: p -> p -> p -> ()
d_IsSemiringWithoutOne_292 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsSuccessorSet
d_IsSuccessorSet_296 :: p -> p -> ()
d_IsSuccessorSet_296 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsUnitalMagma
d_IsUnitalMagma_300 :: p -> p -> ()
d_IsUnitalMagma_300 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsAbelianGroup._//_
d__'47''47'__306 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer -> Integer -> Integer
d__'47''47'__306 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
d__'47''47'__306 Integer -> Integer -> Integer
v0 ~Integer
v1 Integer -> Integer
v2 ~T_IsAbelianGroup_1172
v3 = (Integer -> Integer -> Integer)
-> (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__306 Integer -> Integer -> Integer
v0 Integer -> Integer
v2
du__'47''47'__306 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__306 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__306 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.Nat.Properties._.IsAbelianGroup.assoc
d_assoc_308 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_308 :: T_IsAbelianGroup_1172
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_308 = T_IsAbelianGroup_1172
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.comm
d_comm_310 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_310 :: T_IsAbelianGroup_1172 -> Integer -> Integer -> T__'8801'__12
d_comm_310 = T_IsAbelianGroup_1172 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.identity
d_identity_312 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_312 :: T_IsAbelianGroup_1172 -> T_Σ_14
d_identity_312 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.Nat.Properties._.IsAbelianGroup.identityʳ
d_identity'691'_314 ::
  (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'_314 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
d_identity'691'_314 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.identityˡ
d_identity'737'_316 ::
  (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'_316 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
d_identity'737'_316 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.inverse
d_inverse_318 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_318 :: T_IsAbelianGroup_1172 -> T_Σ_14
d_inverse_318 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.Nat.Properties._.IsAbelianGroup.inverseʳ
d_inverse'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_inverse'691'_320 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
d_inverse'691'_320 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.inverseˡ
d_inverse'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_inverse'737'_322 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
d_inverse'737'_322 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.isCommutativeMagma
d_isCommutativeMagma_324 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_324 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_324 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_324 T_IsAbelianGroup_1172
v3
du_isCommutativeMagma_324 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_324 :: T_IsAbelianGroup_1172 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_324 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.Nat.Properties._.IsAbelianGroup.isCommutativeMonoid
d_isCommutativeMonoid_326 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_326 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_326 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.Nat.Properties._.IsAbelianGroup.isCommutativeSemigroup
d_isCommutativeSemigroup_328 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_328 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_328 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_328 T_IsAbelianGroup_1172
v3
du_isCommutativeSemigroup_328 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_328 :: T_IsAbelianGroup_1172 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_328 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.Nat.Properties._.IsAbelianGroup.isEquivalence
d_isEquivalence_330 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_330 :: T_IsAbelianGroup_1172 -> T_IsEquivalence_28
d_isEquivalence_330 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.Nat.Properties._.IsAbelianGroup.isGroup
d_isGroup_332 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074
d_isGroup_332 :: T_IsAbelianGroup_1172 -> T_IsGroup_1074
d_isGroup_332 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.Nat.Properties._.IsAbelianGroup.isInvertibleMagma
d_isInvertibleMagma_334 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_334 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsInvertibleMagma_958
d_isInvertibleMagma_334 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_334 T_IsAbelianGroup_1172
v3
du_isInvertibleMagma_334 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
du_isInvertibleMagma_334 :: T_IsAbelianGroup_1172 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_334 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.Nat.Properties._.IsAbelianGroup.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_336 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_336 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_336 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_336 T_IsAbelianGroup_1172
v3
du_isInvertibleUnitalMagma_336 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_336 :: T_IsAbelianGroup_1172 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_336 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.Nat.Properties._.IsAbelianGroup.isMagma
d_isMagma_338 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_338 :: T_IsAbelianGroup_1172 -> T_IsMagma_178
d_isMagma_338 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.Nat.Properties._.IsAbelianGroup.isMonoid
d_isMonoid_340 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_340 :: T_IsAbelianGroup_1172 -> T_IsMonoid_712
d_isMonoid_340 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.Nat.Properties._.IsAbelianGroup.isPartialEquivalence
d_isPartialEquivalence_342 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_342 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_342 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3
  = T_IsAbelianGroup_1172 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_342 T_IsAbelianGroup_1172
v3
du_isPartialEquivalence_342 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_342 :: T_IsAbelianGroup_1172 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_342 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.Nat.Properties._.IsAbelianGroup.isSemigroup
d_isSemigroup_344 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_344 :: T_IsAbelianGroup_1172 -> T_IsSemigroup_488
d_isSemigroup_344 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.Nat.Properties._.IsAbelianGroup.isUnitalMagma
d_isUnitalMagma_346 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_346 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_IsUnitalMagma_666
d_isUnitalMagma_346 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3 = T_IsAbelianGroup_1172 -> T_IsUnitalMagma_666
du_isUnitalMagma_346 T_IsAbelianGroup_1172
v3
du_isUnitalMagma_346 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_346 :: T_IsAbelianGroup_1172 -> T_IsUnitalMagma_666
du_isUnitalMagma_346 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.Nat.Properties._.IsAbelianGroup.refl
d_refl_348 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_348 :: T_IsAbelianGroup_1172 -> Integer -> T__'8801'__12
d_refl_348 = T_IsAbelianGroup_1172 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.reflexive
d_reflexive_350 ::
  (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_350 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_350 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.setoid
d_setoid_352 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_352 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> T_Setoid_46
d_setoid_352 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsAbelianGroup_1172
v3 = T_IsAbelianGroup_1172 -> T_Setoid_46
du_setoid_352 T_IsAbelianGroup_1172
v3
du_setoid_352 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_352 :: T_IsAbelianGroup_1172 -> T_Setoid_46
du_setoid_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_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.Nat.Properties._.IsAbelianGroup.sym
d_sym_354 ::
  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_354 :: T_IsAbelianGroup_1172
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_354 = T_IsAbelianGroup_1172
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.trans
d_trans_356 ::
  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_356 :: T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_356 = T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_358 ::
  (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'_358 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'691''45''8315''185'_358 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_360 ::
  (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'_360 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'737''45''8315''185'_360 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.⁻¹-cong
d_'8315''185''45'cong_362 ::
  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_362 :: T_IsAbelianGroup_1172
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_362 = T_IsAbelianGroup_1172
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.∙-cong
d_'8729''45'cong_364 ::
  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_364 :: T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_364 = T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.∙-congʳ
d_'8729''45'cong'691'_366 ::
  (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'_366 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_366 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.∙-congˡ
d_'8729''45'cong'737'_368 ::
  (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'_368 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_368 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsAbelianGroup_1172
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.alter
d_alter_372 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_alter_372 :: T_IsAlternativeMagma_290 -> T_Σ_14
d_alter_372 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.Nat.Properties._.IsAlternativeMagma.alternativeʳ
d_alternative'691'_374 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_alternative'691'_374 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> Integer -> Integer -> T__'8801'__12
d_alternative'691'_374 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.alternativeˡ
d_alternative'737'_376 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_alternative'737'_376 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> Integer -> Integer -> T__'8801'__12
d_alternative'737'_376 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.isEquivalence
d_isEquivalence_378 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_378 :: T_IsAlternativeMagma_290 -> T_IsEquivalence_28
d_isEquivalence_378 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.Nat.Properties._.IsAlternativeMagma.isMagma
d_isMagma_380 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_380 :: T_IsAlternativeMagma_290 -> T_IsMagma_178
d_isMagma_380 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.Nat.Properties._.IsAlternativeMagma.isPartialEquivalence
d_isPartialEquivalence_382 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_382 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_382 ~Integer -> Integer -> Integer
v0 T_IsAlternativeMagma_290
v1 = T_IsAlternativeMagma_290 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_382 T_IsAlternativeMagma_290
v1
du_isPartialEquivalence_382 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_382 :: T_IsAlternativeMagma_290 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_382 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.Nat.Properties._.IsAlternativeMagma.refl
d_refl_384 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_384 :: T_IsAlternativeMagma_290 -> Integer -> T__'8801'__12
d_refl_384 = T_IsAlternativeMagma_290 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.reflexive
d_reflexive_386 ::
  (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_386 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_386 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.setoid
d_setoid_388 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_388 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290 -> T_Setoid_46
d_setoid_388 ~Integer -> Integer -> Integer
v0 T_IsAlternativeMagma_290
v1 = T_IsAlternativeMagma_290 -> T_Setoid_46
du_setoid_388 T_IsAlternativeMagma_290
v1
du_setoid_388 ::
  MAlonzo.Code.Algebra.Structures.T_IsAlternativeMagma_290 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_388 :: T_IsAlternativeMagma_290 -> T_Setoid_46
du_setoid_388 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.Nat.Properties._.IsAlternativeMagma.sym
d_sym_390 ::
  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_390 :: T_IsAlternativeMagma_290
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_390 = T_IsAlternativeMagma_290
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.trans
d_trans_392 ::
  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_392 :: T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_392 = T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.∙-cong
d_'8729''45'cong_394 ::
  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_394 :: T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_394 = T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.∙-congʳ
d_'8729''45'cong'691'_396 ::
  (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'_396 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_396 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAlternativeMagma.∙-congˡ
d_'8729''45'cong'737'_398 ::
  (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'_398 :: (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_398 = (Integer -> Integer -> Integer)
-> T_IsAlternativeMagma_290
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.assoc
d_assoc_402 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_402 :: T_IsBand_526 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_402 = T_IsBand_526 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.idem
d_idem_404 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_404 :: T_IsBand_526 -> Integer -> T__'8801'__12
d_idem_404 = T_IsBand_526 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.isEquivalence
d_isEquivalence_406 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_406 :: T_IsBand_526 -> T_IsEquivalence_28
d_isEquivalence_406 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.Nat.Properties._.IsBand.isMagma
d_isMagma_408 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_408 :: T_IsBand_526 -> T_IsMagma_178
d_isMagma_408 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.Nat.Properties._.IsBand.isPartialEquivalence
d_isPartialEquivalence_410 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_410 :: (Integer -> Integer -> Integer)
-> T_IsBand_526 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_410 ~Integer -> Integer -> Integer
v0 T_IsBand_526
v1 = T_IsBand_526 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_410 T_IsBand_526
v1
du_isPartialEquivalence_410 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_410 :: T_IsBand_526 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_410 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.Nat.Properties._.IsBand.isSemigroup
d_isSemigroup_412 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_412 :: T_IsBand_526 -> T_IsSemigroup_488
d_isSemigroup_412 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.Nat.Properties._.IsBand.refl
d_refl_414 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_414 :: T_IsBand_526 -> Integer -> T__'8801'__12
d_refl_414 = T_IsBand_526 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.reflexive
d_reflexive_416 ::
  (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_416 :: (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_416 = (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.setoid
d_setoid_418 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_418 :: (Integer -> Integer -> Integer) -> T_IsBand_526 -> T_Setoid_46
d_setoid_418 ~Integer -> Integer -> Integer
v0 T_IsBand_526
v1 = T_IsBand_526 -> T_Setoid_46
du_setoid_418 T_IsBand_526
v1
du_setoid_418 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_526 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_418 :: T_IsBand_526 -> T_Setoid_46
du_setoid_418 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.Nat.Properties._.IsBand.sym
d_sym_420 ::
  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_420 :: T_IsBand_526
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_420 = T_IsBand_526
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.trans
d_trans_422 ::
  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_422 :: T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_422 = T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.∙-cong
d_'8729''45'cong_424 ::
  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_424 :: T_IsBand_526
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_424 = T_IsBand_526
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.∙-congʳ
d_'8729''45'cong'691'_426 ::
  (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'_426 :: (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_426 = (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.∙-congˡ
d_'8729''45'cong'737'_428 ::
  (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'_428 :: (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_428 = (Integer -> Integer -> Integer)
-> T_IsBand_526
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.*-assoc
d_'42''45'assoc_432 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_432 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_432 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.*-cancelʳ-nonZero
d_'42''45'cancel'691''45'nonZero_434 ::
  (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_434 :: (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_434 = (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.Nat.Properties._.IsCancellativeCommutativeSemiring.*-cancelˡ-nonZero
d_'42''45'cancel'737''45'nonZero_436 ::
  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_436 :: 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_436 = T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> (T__'8801'__12 -> T_Irrelevant_20)
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.*-comm
d_'42''45'comm_438 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_438 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_438 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.*-cong
d_'42''45'cong_440 ::
  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_440 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_440 = T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.∙-congʳ
d_'8729''45'cong'691'_442 ::
  (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'_442 :: (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'_442 = (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.Nat.Properties._.IsCancellativeCommutativeSemiring.∙-congˡ
d_'8729''45'cong'737'_444 ::
  (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'_444 :: (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'_444 = (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.Nat.Properties._.IsCancellativeCommutativeSemiring.*-identity
d_'42''45'identity_446 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_446 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Σ_14
d_'42''45'identity_446 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.Nat.Properties._.IsCancellativeCommutativeSemiring.identityʳ
d_identity'691'_448 ::
  (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'_448 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_identity'691'_448 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.identityˡ
d_identity'737'_450 ::
  (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'_450 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_identity'737'_450 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isCommutativeMagma
d_isCommutativeMagma_452 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_452 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_452 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_452 T_IsCancellativeCommutativeSemiring_1872
v4
du_isCommutativeMagma_452 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_452 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_452 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.Nat.Properties._.IsCancellativeCommutativeSemiring.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_454 ::
  (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_454 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_454 ~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_454 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isCommutativeMonoid_454 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_454 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_454 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.Nat.Properties._.IsCancellativeCommutativeSemiring.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_456 ::
  (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_456 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_456 ~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_456 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isCommutativeSemigroup_456 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_456 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_456 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.Nat.Properties._.IsCancellativeCommutativeSemiring.*-isMagma
d_'42''45'isMagma_458 ::
  (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_458 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsMagma_178
d_'42''45'isMagma_458 ~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_458 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isMagma_458 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_458 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsMagma_178
du_'42''45'isMagma_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_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.Nat.Properties._.IsCancellativeCommutativeSemiring.*-isMonoid
d_'42''45'isMonoid_460 ::
  (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_460 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsMonoid_712
d_'42''45'isMonoid_460 ~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_460 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isMonoid_460 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_460 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsMonoid_712
du_'42''45'isMonoid_460 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.Nat.Properties._.IsCancellativeCommutativeSemiring.*-isSemigroup
d_'42''45'isSemigroup_462 ::
  (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_462 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemigroup_488
d_'42''45'isSemigroup_462 ~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_462 T_IsCancellativeCommutativeSemiring_1872
v4
du_'42''45'isSemigroup_462 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_462 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsSemigroup_488
du_'42''45'isSemigroup_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_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.Nat.Properties._.IsCancellativeCommutativeSemiring.assoc
d_assoc_464 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_464 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_464 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.comm
d_comm_466 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_466 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12
d_comm_466 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.∙-cong
d_'8729''45'cong_468 ::
  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_468 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_468 = T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.∙-congʳ
d_'8729''45'cong'691'_470 ::
  (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'_470 :: (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'_470 = (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.Nat.Properties._.IsCancellativeCommutativeSemiring.∙-congˡ
d_'8729''45'cong'737'_472 ::
  (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'_472 :: (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'_472 = (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.Nat.Properties._.IsCancellativeCommutativeSemiring.identity
d_identity_474 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_474 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Σ_14
d_identity_474 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.Nat.Properties._.IsCancellativeCommutativeSemiring.identityʳ
d_identity'691'_476 ::
  (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'_476 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_identity'691'_476 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.identityˡ
d_identity'737'_478 ::
  (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'_478 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_identity'737'_478 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isCommutativeMagma
d_isCommutativeMagma_480 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_480 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_480 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_480 T_IsCancellativeCommutativeSemiring_1872
v4
du_isCommutativeMagma_480 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_480 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_480 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.Nat.Properties._.IsCancellativeCommutativeSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_482 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_482 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_482 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isCommutativeSemigroup
d_isCommutativeSemigroup_484 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_484 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_484 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_484 T_IsCancellativeCommutativeSemiring_1872
v4
du_isCommutativeSemigroup_484 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_484 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_484 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isMagma
d_isMagma_486 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_486 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsMagma_178
d_isMagma_486 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isMonoid
d_isMonoid_488 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_488 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsMonoid_712
d_isMonoid_488 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemigroup
d_isSemigroup_490 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_490 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsSemigroup_488
d_isSemigroup_490 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isUnitalMagma
d_isUnitalMagma_492 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_492 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsUnitalMagma_666
d_isUnitalMagma_492 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4 = T_IsCancellativeCommutativeSemiring_1872 -> T_IsUnitalMagma_666
du_isUnitalMagma_492 T_IsCancellativeCommutativeSemiring_1872
v4
du_isUnitalMagma_492 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_492 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsUnitalMagma_666
du_isUnitalMagma_492 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.Nat.Properties._.IsCancellativeCommutativeSemiring.distrib
d_distrib_494 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_494 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Σ_14
d_distrib_494 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.Nat.Properties._.IsCancellativeCommutativeSemiring.distribʳ
d_distrib'691'_496 ::
  (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'_496 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_496 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.distribˡ
d_distrib'737'_498 ::
  (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'_498 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_498 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isCommutativeSemiring
d_isCommutativeSemiring_500 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_500 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_500 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isCommutativeSemiringWithoutOne
d_isCommutativeSemiringWithoutOne_502 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_502 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_502 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_502 T_IsCancellativeCommutativeSemiring_1872
v4
du_isCommutativeSemiringWithoutOne_502 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_502 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_502 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isEquivalence
d_isEquivalence_504 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_504 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsEquivalence_28
d_isEquivalence_504 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isNearSemiring
d_isNearSemiring_506 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_506 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsNearSemiring_1260
d_isNearSemiring_506 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4 = T_IsCancellativeCommutativeSemiring_1872 -> T_IsNearSemiring_1260
du_isNearSemiring_506 T_IsCancellativeCommutativeSemiring_1872
v4
du_isNearSemiring_506 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_506 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsNearSemiring_1260
du_isNearSemiring_506 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isPartialEquivalence
d_isPartialEquivalence_508 ::
  (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_508 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_508 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsPartialEquivalence_16
du_isPartialEquivalence_508 T_IsCancellativeCommutativeSemiring_1872
v4
du_isPartialEquivalence_508 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_508 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsPartialEquivalence_16
du_isPartialEquivalence_508 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemiring
d_isSemiring_510 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_510 :: T_IsCancellativeCommutativeSemiring_1872 -> T_IsSemiring_1640
d_isSemiring_510 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_512 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_512 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_512 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.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_514 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_514 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_514 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4
  = T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_514 T_IsCancellativeCommutativeSemiring_1872
v4
du_isSemiringWithoutOne_514 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_514 :: T_IsCancellativeCommutativeSemiring_1872
-> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_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_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.Nat.Properties._.IsCancellativeCommutativeSemiring.refl
d_refl_516 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_516 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> T__'8801'__12
d_refl_516 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.reflexive
d_reflexive_518 ::
  (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_518 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_518 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.setoid
d_setoid_520 ::
  (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_520 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> T_Setoid_46
d_setoid_520 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1872
v4 = T_IsCancellativeCommutativeSemiring_1872 -> T_Setoid_46
du_setoid_520 T_IsCancellativeCommutativeSemiring_1872
v4
du_setoid_520 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_520 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Setoid_46
du_setoid_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_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.Nat.Properties._.IsCancellativeCommutativeSemiring.sym
d_sym_522 ::
  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_522 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_522 = T_IsCancellativeCommutativeSemiring_1872
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.trans
d_trans_524 ::
  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_524 :: T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_524 = T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.zero
d_zero_526 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1872 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_526 :: T_IsCancellativeCommutativeSemiring_1872 -> T_Σ_14
d_zero_526 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.Nat.Properties._.IsCancellativeCommutativeSemiring.zeroʳ
d_zero'691'_528 ::
  (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'_528 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_zero'691'_528 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.zeroˡ
d_zero'737'_530 ::
  (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'_530 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
d_zero'737'_530 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1872
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.assoc
d_assoc_534 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_534 :: T_IsCommutativeBand_612
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_534 = T_IsCommutativeBand_612
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.comm
d_comm_536 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_536 :: T_IsCommutativeBand_612 -> Integer -> Integer -> T__'8801'__12
d_comm_536 = T_IsCommutativeBand_612 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.idem
d_idem_538 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_538 :: T_IsCommutativeBand_612 -> Integer -> T__'8801'__12
d_idem_538 = T_IsCommutativeBand_612 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.isBand
d_isBand_540 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_540 :: T_IsCommutativeBand_612 -> T_IsBand_526
d_isBand_540 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.Nat.Properties._.IsCommutativeBand.isCommutativeMagma
d_isCommutativeMagma_542 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_542 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612 -> T_IsCommutativeMagma_214
d_isCommutativeMagma_542 ~Integer -> Integer -> Integer
v0 T_IsCommutativeBand_612
v1 = T_IsCommutativeBand_612 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_542 T_IsCommutativeBand_612
v1
du_isCommutativeMagma_542 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_542 :: T_IsCommutativeBand_612 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_542 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.Nat.Properties._.IsCommutativeBand.isCommutativeSemigroup
d_isCommutativeSemigroup_544 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_544 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612 -> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_544 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.Nat.Properties._.IsCommutativeBand.isEquivalence
d_isEquivalence_546 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_546 :: T_IsCommutativeBand_612 -> T_IsEquivalence_28
d_isEquivalence_546 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.Nat.Properties._.IsCommutativeBand.isMagma
d_isMagma_548 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_548 :: T_IsCommutativeBand_612 -> T_IsMagma_178
d_isMagma_548 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.Nat.Properties._.IsCommutativeBand.isPartialEquivalence
d_isPartialEquivalence_550 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_550 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_550 ~Integer -> Integer -> Integer
v0 T_IsCommutativeBand_612
v1 = T_IsCommutativeBand_612 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_550 T_IsCommutativeBand_612
v1
du_isPartialEquivalence_550 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_550 :: T_IsCommutativeBand_612 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_550 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.Nat.Properties._.IsCommutativeBand.isSemigroup
d_isSemigroup_552 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_552 :: T_IsCommutativeBand_612 -> T_IsSemigroup_488
d_isSemigroup_552 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.Nat.Properties._.IsCommutativeBand.refl
d_refl_554 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_554 :: T_IsCommutativeBand_612 -> Integer -> T__'8801'__12
d_refl_554 = T_IsCommutativeBand_612 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.reflexive
d_reflexive_556 ::
  (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_556 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_556 = (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.setoid
d_setoid_558 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_558 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612 -> T_Setoid_46
d_setoid_558 ~Integer -> Integer -> Integer
v0 T_IsCommutativeBand_612
v1 = T_IsCommutativeBand_612 -> T_Setoid_46
du_setoid_558 T_IsCommutativeBand_612
v1
du_setoid_558 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_558 :: T_IsCommutativeBand_612 -> T_Setoid_46
du_setoid_558 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.Nat.Properties._.IsCommutativeBand.sym
d_sym_560 ::
  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_560 :: T_IsCommutativeBand_612
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_560 = T_IsCommutativeBand_612
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.trans
d_trans_562 ::
  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_562 :: T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_562 = T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.∙-cong
d_'8729''45'cong_564 ::
  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_564 :: T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_564 = T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.∙-congʳ
d_'8729''45'cong'691'_566 ::
  (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'_566 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_566 = (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeBand.∙-congˡ
d_'8729''45'cong'737'_568 ::
  (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'_568 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_568 = (Integer -> Integer -> Integer)
-> T_IsCommutativeBand_612
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.comm
d_comm_572 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_572 :: T_IsCommutativeMagma_214 -> Integer -> Integer -> T__'8801'__12
d_comm_572 = T_IsCommutativeMagma_214 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.isEquivalence
d_isEquivalence_574 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_574 :: T_IsCommutativeMagma_214 -> T_IsEquivalence_28
d_isEquivalence_574 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.Nat.Properties._.IsCommutativeMagma.isMagma
d_isMagma_576 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_576 :: T_IsCommutativeMagma_214 -> T_IsMagma_178
d_isMagma_576 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.Nat.Properties._.IsCommutativeMagma.isPartialEquivalence
d_isPartialEquivalence_578 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_578 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_578 ~Integer -> Integer -> Integer
v0 T_IsCommutativeMagma_214
v1 = T_IsCommutativeMagma_214 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_578 T_IsCommutativeMagma_214
v1
du_isPartialEquivalence_578 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_578 :: T_IsCommutativeMagma_214 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_578 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.Nat.Properties._.IsCommutativeMagma.refl
d_refl_580 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_580 :: T_IsCommutativeMagma_214 -> Integer -> T__'8801'__12
d_refl_580 = T_IsCommutativeMagma_214 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.reflexive
d_reflexive_582 ::
  (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_582 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_582 = (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.setoid
d_setoid_584 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_584 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214 -> T_Setoid_46
d_setoid_584 ~Integer -> Integer -> Integer
v0 T_IsCommutativeMagma_214
v1 = T_IsCommutativeMagma_214 -> T_Setoid_46
du_setoid_584 T_IsCommutativeMagma_214
v1
du_setoid_584 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_584 :: T_IsCommutativeMagma_214 -> T_Setoid_46
du_setoid_584 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.Nat.Properties._.IsCommutativeMagma.sym
d_sym_586 ::
  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_586 :: T_IsCommutativeMagma_214
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_586 = T_IsCommutativeMagma_214
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.trans
d_trans_588 ::
  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_588 :: T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_588 = T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.∙-cong
d_'8729''45'cong_590 ::
  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_590 :: T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_590 = T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.∙-congʳ
d_'8729''45'cong'691'_592 ::
  (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'_592 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_592 = (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.∙-congˡ
d_'8729''45'cong'737'_594 ::
  (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'_594 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_594 = (Integer -> Integer -> Integer)
-> T_IsCommutativeMagma_214
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.assoc
d_assoc_598 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_598 :: T_IsCommutativeMonoid_764
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_598 = T_IsCommutativeMonoid_764
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.comm
d_comm_600 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_600 :: T_IsCommutativeMonoid_764 -> Integer -> Integer -> T__'8801'__12
d_comm_600 = T_IsCommutativeMonoid_764 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.identity
d_identity_602 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_602 :: T_IsCommutativeMonoid_764 -> T_Σ_14
d_identity_602 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.Nat.Properties._.IsCommutativeMonoid.identityʳ
d_identity'691'_604 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_604 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
d_identity'691'_604 = (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.identityˡ
d_identity'737'_606 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_606 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
d_identity'737'_606 = (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.isCommutativeMagma
d_isCommutativeMagma_608 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_608 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> T_IsCommutativeMagma_214
d_isCommutativeMagma_608 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsCommutativeMonoid_764
v2 = T_IsCommutativeMonoid_764 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_608 T_IsCommutativeMonoid_764
v2
du_isCommutativeMagma_608 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_608 :: T_IsCommutativeMonoid_764 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_608 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.Nat.Properties._.IsCommutativeMonoid.isCommutativeSemigroup
d_isCommutativeSemigroup_610 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_610 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_610 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.Nat.Properties._.IsCommutativeMonoid.isEquivalence
d_isEquivalence_612 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_612 :: T_IsCommutativeMonoid_764 -> T_IsEquivalence_28
d_isEquivalence_612 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.Nat.Properties._.IsCommutativeMonoid.isMagma
d_isMagma_614 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_614 :: T_IsCommutativeMonoid_764 -> T_IsMagma_178
d_isMagma_614 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.Nat.Properties._.IsCommutativeMonoid.isMonoid
d_isMonoid_616 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_616 :: T_IsCommutativeMonoid_764 -> T_IsMonoid_712
d_isMonoid_616 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.Nat.Properties._.IsCommutativeMonoid.isPartialEquivalence
d_isPartialEquivalence_618 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_618 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_618 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsCommutativeMonoid_764
v2
  = T_IsCommutativeMonoid_764 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_618 T_IsCommutativeMonoid_764
v2
du_isPartialEquivalence_618 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_618 :: T_IsCommutativeMonoid_764 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_618 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.Nat.Properties._.IsCommutativeMonoid.isSemigroup
d_isSemigroup_620 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_620 :: T_IsCommutativeMonoid_764 -> T_IsSemigroup_488
d_isSemigroup_620 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.Nat.Properties._.IsCommutativeMonoid.isUnitalMagma
d_isUnitalMagma_622 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_622 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> T_IsUnitalMagma_666
d_isUnitalMagma_622 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsCommutativeMonoid_764
v2 = T_IsCommutativeMonoid_764 -> T_IsUnitalMagma_666
du_isUnitalMagma_622 T_IsCommutativeMonoid_764
v2
du_isUnitalMagma_622 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_622 :: T_IsCommutativeMonoid_764 -> T_IsUnitalMagma_666
du_isUnitalMagma_622 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.Nat.Properties._.IsCommutativeMonoid.refl
d_refl_624 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_624 :: T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
d_refl_624 = T_IsCommutativeMonoid_764 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.reflexive
d_reflexive_626 ::
  (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_626 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_626 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.setoid
d_setoid_628 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_628 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsCommutativeMonoid_764 -> T_Setoid_46
d_setoid_628 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsCommutativeMonoid_764
v2 = T_IsCommutativeMonoid_764 -> T_Setoid_46
du_setoid_628 T_IsCommutativeMonoid_764
v2
du_setoid_628 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_628 :: T_IsCommutativeMonoid_764 -> T_Setoid_46
du_setoid_628 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.Nat.Properties._.IsCommutativeMonoid.sym
d_sym_630 ::
  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_630 :: T_IsCommutativeMonoid_764
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_630 = T_IsCommutativeMonoid_764
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.trans
d_trans_632 ::
  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_632 :: T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_632 = T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.∙-cong
d_'8729''45'cong_634 ::
  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_634 :: T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_634 = T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.∙-congʳ
d_'8729''45'cong'691'_636 ::
  (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'_636 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_636 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.∙-congˡ
d_'8729''45'cong'737'_638 ::
  (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'_638 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_638 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_764
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing._//_
d__'47''47'__642 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer -> Integer -> Integer
d__'47''47'__642 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
d__'47''47'__642 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'__642 Integer -> Integer -> Integer
v0 Integer -> Integer
v2
du__'47''47'__642 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__642 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer) -> Integer -> Integer -> Integer
du__'47''47'__642 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.Nat.Properties._.IsCommutativeRing.*-assoc
d_'42''45'assoc_644 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_644 :: T_IsCommutativeRing_2888
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_644 = T_IsCommutativeRing_2888
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.*-comm
d_'42''45'comm_646 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_646 :: T_IsCommutativeRing_2888 -> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_646 = T_IsCommutativeRing_2888 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.*-cong
d_'42''45'cong_648 ::
  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_648 :: T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_648 = T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.∙-congʳ
d_'8729''45'cong'691'_650 ::
  (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'_650 :: (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'_650 = (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.Nat.Properties._.IsCommutativeRing.∙-congˡ
d_'8729''45'cong'737'_652 ::
  (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'_652 :: (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'_652 = (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.Nat.Properties._.IsCommutativeRing.*-identity
d_'42''45'identity_654 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_654 :: T_IsCommutativeRing_2888 -> T_Σ_14
d_'42''45'identity_654 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.Nat.Properties._.IsCommutativeRing.identityʳ
d_identity'691'_656 ::
  (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'_656 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_identity'691'_656 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.identityˡ
d_identity'737'_658 ::
  (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'_658 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_identity'737'_658 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.isCommutativeMagma
d_isCommutativeMagma_660 ::
  (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_660 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_660 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_660 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isCommutativeMagma_660 ::
  (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_660 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMagma_214
du_isCommutativeMagma_660 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.Nat.Properties._.IsCommutativeRing.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_662 ::
  (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_662 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_662 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_662 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isCommutativeMonoid_662 ::
  (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_662 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMonoid_764
du_'42''45'isCommutativeMonoid_662 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.Nat.Properties._.IsCommutativeRing.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_664 ::
  (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_664 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_664 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_664 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isCommutativeSemigroup_664 ::
  (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_664 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_664 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.Nat.Properties._.IsCommutativeRing.*-isMagma
d_'42''45'isMagma_666 ::
  (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_666 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMagma_178
d_'42''45'isMagma_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_IsMagma_178
du_'42''45'isMagma_666 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isMagma_666 ::
  (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_666 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMagma_178
du_'42''45'isMagma_666 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.Nat.Properties._.IsCommutativeRing.*-isMonoid
d_'42''45'isMonoid_668 ::
  (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_668 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMonoid_712
d_'42''45'isMonoid_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_IsMonoid_712
du_'42''45'isMonoid_668 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isMonoid_668 ::
  (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_668 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMonoid_712
du_'42''45'isMonoid_668 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.Nat.Properties._.IsCommutativeRing.*-isSemigroup
d_'42''45'isSemigroup_670 ::
  (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_670 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemigroup_488
d_'42''45'isSemigroup_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_IsSemigroup_488
du_'42''45'isSemigroup_670 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_'42''45'isSemigroup_670 ::
  (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_670 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemigroup_488
du_'42''45'isSemigroup_670 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.Nat.Properties._.IsCommutativeRing.assoc
d_assoc_672 ::
  (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_672 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_assoc_672 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.comm
d_comm_674 ::
  (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_674 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
d_comm_674 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.∙-cong
d_'8729''45'cong_676 ::
  (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_676 :: (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_676 = (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.Nat.Properties._.IsCommutativeRing.∙-congʳ
d_'8729''45'cong'691'_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 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong'691'_678 :: (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'_678 = (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.Nat.Properties._.IsCommutativeRing.∙-congˡ
d_'8729''45'cong'737'_680 ::
  (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'_680 :: (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'_680 = (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.Nat.Properties._.IsCommutativeRing.identity
d_identity_682 ::
  (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_682 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
d_identity_682 ~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_682 T_IsCommutativeRing_2888
v5
du_identity_682 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_identity_682 :: T_IsCommutativeRing_2888 -> T_Σ_14
du_identity_682 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.Nat.Properties._.IsCommutativeRing.identityʳ
d_identity'691'_684 ::
  (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'_684 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_identity'691'_684 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.identityˡ
d_identity'737'_686 ::
  (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'_686 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_identity'737'_686 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.+-isAbelianGroup
d_'43''45'isAbelianGroup_688 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_688 :: T_IsCommutativeRing_2888 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_688 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.Nat.Properties._.IsCommutativeRing.isCommutativeMagma
d_isCommutativeMagma_690 ::
  (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_690 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_690 ~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_690 T_IsCommutativeRing_2888
v5
du_isCommutativeMagma_690 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_690 :: T_IsCommutativeRing_2888 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_690 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.Nat.Properties._.IsCommutativeRing.isCommutativeMonoid
d_isCommutativeMonoid_692 ::
  (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_692 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_692 ~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_692 T_IsCommutativeRing_2888
v5
du_isCommutativeMonoid_692 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_isCommutativeMonoid_692 :: T_IsCommutativeRing_2888 -> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_692 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.Nat.Properties._.IsCommutativeRing.isCommutativeSemigroup
d_isCommutativeSemigroup_694 ::
  (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_694 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_694 ~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_694 T_IsCommutativeRing_2888
v5
du_isCommutativeSemigroup_694 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_694 :: T_IsCommutativeRing_2888 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_694 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.Nat.Properties._.IsCommutativeRing.isGroup
d_isGroup_696 ::
  (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_696 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsGroup_1074
d_isGroup_696 ~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_696 T_IsCommutativeRing_2888
v5
du_isGroup_696 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074
du_isGroup_696 :: T_IsCommutativeRing_2888 -> T_IsGroup_1074
du_isGroup_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_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.Nat.Properties._.IsCommutativeRing.isInvertibleMagma
d_isInvertibleMagma_698 ::
  (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_698 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsInvertibleMagma_958
d_isInvertibleMagma_698 ~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_698 T_IsCommutativeRing_2888
v5
du_isInvertibleMagma_698 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
du_isInvertibleMagma_698 :: T_IsCommutativeRing_2888 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_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_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.Nat.Properties._.IsCommutativeRing.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_700 ::
  (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_700 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_700 ~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_700 T_IsCommutativeRing_2888
v5
du_isInvertibleUnitalMagma_700 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_700 :: T_IsCommutativeRing_2888 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_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_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.Nat.Properties._.IsCommutativeRing.isMagma
d_isMagma_702 ::
  (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_702 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMagma_178
d_isMagma_702 ~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_702 T_IsCommutativeRing_2888
v5
du_isMagma_702 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_isMagma_702 :: T_IsCommutativeRing_2888 -> T_IsMagma_178
du_isMagma_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_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.Nat.Properties._.IsCommutativeRing.isMonoid
d_isMonoid_704 ::
  (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_704 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsMonoid_712
d_isMonoid_704 ~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_704 T_IsCommutativeRing_2888
v5
du_isMonoid_704 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_isMonoid_704 :: T_IsCommutativeRing_2888 -> T_IsMonoid_712
du_isMonoid_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_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.Nat.Properties._.IsCommutativeRing.isSemigroup
d_isSemigroup_706 ::
  (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_706 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemigroup_488
d_isSemigroup_706 ~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_706 T_IsCommutativeRing_2888
v5
du_isSemigroup_706 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_isSemigroup_706 :: T_IsCommutativeRing_2888 -> T_IsSemigroup_488
du_isSemigroup_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_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.Nat.Properties._.IsCommutativeRing.isUnitalMagma
d_isUnitalMagma_708 ::
  (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_708 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsUnitalMagma_666
d_isUnitalMagma_708 ~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_708 T_IsCommutativeRing_2888
v5
du_isUnitalMagma_708 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_708 :: T_IsCommutativeRing_2888 -> T_IsUnitalMagma_666
du_isUnitalMagma_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_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.Nat.Properties._.IsCommutativeRing.⁻¹-cong
d_'8315''185''45'cong_710 ::
  (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_710 :: (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_710 = (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.Nat.Properties._.IsCommutativeRing.inverse
d_inverse_712 ::
  (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_712 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
d_inverse_712 ~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_712 T_IsCommutativeRing_2888
v5
du_inverse_712 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_inverse_712 :: T_IsCommutativeRing_2888 -> T_Σ_14
du_inverse_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_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.Nat.Properties._.IsCommutativeRing.inverseʳ
d_inverse'691'_714 ::
  (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'_714 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_inverse'691'_714 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.inverseˡ
d_inverse'737'_716 ::
  (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'_716 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_inverse'737'_716 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.distrib
d_distrib_718 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_718 :: T_IsCommutativeRing_2888 -> T_Σ_14
d_distrib_718 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.Nat.Properties._.IsCommutativeRing.distribʳ
d_distrib'691'_720 ::
  (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'_720 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_720 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.distribˡ
d_distrib'737'_722 ::
  (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'_722 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_722 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.isCommutativeSemiring
d_isCommutativeSemiring_724 ::
  (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_724 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_724 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.Nat.Properties._.IsCommutativeRing.isCommutativeSemiringWithoutOne
d_isCommutativeSemiringWithoutOne_726 ::
  (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_726 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_726 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_726 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isCommutativeSemiringWithoutOne_726 ::
  (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_726 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsCommutativeSemiringWithoutOne_1438
du_isCommutativeSemiringWithoutOne_726 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.Nat.Properties._.IsCommutativeRing.isEquivalence
d_isEquivalence_728 ::
  (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_728 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsEquivalence_28
d_isEquivalence_728 ~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_728 T_IsCommutativeRing_2888
v5
du_isEquivalence_728 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
du_isEquivalence_728 :: T_IsCommutativeRing_2888 -> T_IsEquivalence_28
du_isEquivalence_728 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.Nat.Properties._.IsCommutativeRing.isNearSemiring
d_isNearSemiring_730 ::
  (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_730 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsNearSemiring_1260
d_isNearSemiring_730 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_730 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isNearSemiring_730 ::
  (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_730 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsNearSemiring_1260
du_isNearSemiring_730 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.Nat.Properties._.IsCommutativeRing.isPartialEquivalence
d_isPartialEquivalence_732 ::
  (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_732 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_732 ~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_732 T_IsCommutativeRing_2888
v5
du_isPartialEquivalence_732 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_732 :: T_IsCommutativeRing_2888 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_732 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.Nat.Properties._.IsCommutativeRing.isRing
d_isRing_734 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740
d_isRing_734 :: T_IsCommutativeRing_2888 -> T_IsRing_2740
d_isRing_734 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.Nat.Properties._.IsCommutativeRing.isRingWithoutOne
d_isRingWithoutOne_736 ::
  (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_736 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsRingWithoutOne_2368
d_isRingWithoutOne_736 ~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_736 T_IsCommutativeRing_2888
v5
du_isRingWithoutOne_736 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsRingWithoutOne_2368
du_isRingWithoutOne_736 :: T_IsCommutativeRing_2888 -> T_IsRingWithoutOne_2368
du_isRingWithoutOne_736 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.Nat.Properties._.IsCommutativeRing.isSemiring
d_isSemiring_738 ::
  (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_738 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiring_1640
d_isSemiring_738 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_738 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isSemiring_738 ::
  (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_738 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiring_1640
du_isSemiring_738 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.Nat.Properties._.IsCommutativeRing.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_740 ::
  (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_740 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_740 ~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_740 T_IsCommutativeRing_2888
v5
du_isSemiringWithoutAnnihilatingZero_740 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
du_isSemiringWithoutAnnihilatingZero_740 :: T_IsCommutativeRing_2888
-> T_IsSemiringWithoutAnnihilatingZero_1536
du_isSemiringWithoutAnnihilatingZero_740 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.Nat.Properties._.IsCommutativeRing.isSemiringWithoutOne
d_isSemiringWithoutOne_742 ::
  (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_742 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_742 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_742 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_isSemiringWithoutOne_742 ::
  (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_742 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_742 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.Nat.Properties._.IsCommutativeRing.refl
d_refl_744 ::
  (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_744 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_refl_744 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.reflexive
d_reflexive_746 ::
  (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_746 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_746 = (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.Nat.Properties._.IsCommutativeRing.setoid
d_setoid_748 ::
  (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_748 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_Setoid_46
d_setoid_748 ~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_748 T_IsCommutativeRing_2888
v5
du_setoid_748 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_2888 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_748 :: T_IsCommutativeRing_2888 -> T_Setoid_46
du_setoid_748 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.Nat.Properties._.IsCommutativeRing.sym
d_sym_750 ::
  (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_750 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_sym_750 = (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.Nat.Properties._.IsCommutativeRing.trans
d_trans_752 ::
  (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_752 :: (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_752 = (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.Nat.Properties._.IsCommutativeRing.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_754 ::
  (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'_754 :: (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'_754 = (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.Nat.Properties._.IsCommutativeRing.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_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_unique'737''45''8315''185'_756 :: (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'_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.Nat.Properties._.IsCommutativeRing.zero
d_zero_758 ::
  (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_758 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
d_zero_758 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_758 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 T_IsCommutativeRing_2888
v5
du_zero_758 ::
  (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_758 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> T_IsCommutativeRing_2888
-> T_Σ_14
du_zero_758 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.Nat.Properties._.IsCommutativeRing.zeroʳ
d_zero'691'_760 ::
  (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'_760 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_zero'691'_760 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.zeroˡ
d_zero'737'_762 ::
  (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'_762 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
d_zero'737'_762 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_2888
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.assoc
d_assoc_766 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_766 :: T_IsCommutativeSemigroup_568
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_766 = T_IsCommutativeSemigroup_568
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.comm
d_comm_768 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_768 :: T_IsCommutativeSemigroup_568 -> Integer -> Integer -> T__'8801'__12
d_comm_768 = T_IsCommutativeSemigroup_568 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.isCommutativeMagma
d_isCommutativeMagma_770 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_770 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
d_isCommutativeMagma_770 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.Nat.Properties._.IsCommutativeSemigroup.isEquivalence
d_isEquivalence_772 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_772 :: T_IsCommutativeSemigroup_568 -> T_IsEquivalence_28
d_isEquivalence_772 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.Nat.Properties._.IsCommutativeSemigroup.isMagma
d_isMagma_774 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_774 :: T_IsCommutativeSemigroup_568 -> T_IsMagma_178
d_isMagma_774 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.Nat.Properties._.IsCommutativeSemigroup.isPartialEquivalence
d_isPartialEquivalence_776 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_776 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_776 ~Integer -> Integer -> Integer
v0 T_IsCommutativeSemigroup_568
v1 = T_IsCommutativeSemigroup_568 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_776 T_IsCommutativeSemigroup_568
v1
du_isPartialEquivalence_776 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_776 :: T_IsCommutativeSemigroup_568 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_776 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.Nat.Properties._.IsCommutativeSemigroup.isSemigroup
d_isSemigroup_778 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_778 :: T_IsCommutativeSemigroup_568 -> T_IsSemigroup_488
d_isSemigroup_778 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.Nat.Properties._.IsCommutativeSemigroup.refl
d_refl_780 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_780 :: T_IsCommutativeSemigroup_568 -> Integer -> T__'8801'__12
d_refl_780 = T_IsCommutativeSemigroup_568 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.reflexive
d_reflexive_782 ::
  (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_782 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_782 = (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.setoid
d_setoid_784 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_784 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568 -> T_Setoid_46
d_setoid_784 ~Integer -> Integer -> Integer
v0 T_IsCommutativeSemigroup_568
v1 = T_IsCommutativeSemigroup_568 -> T_Setoid_46
du_setoid_784 T_IsCommutativeSemigroup_568
v1
du_setoid_784 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_784 :: T_IsCommutativeSemigroup_568 -> T_Setoid_46
du_setoid_784 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.Nat.Properties._.IsCommutativeSemigroup.sym
d_sym_786 ::
  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_786 :: T_IsCommutativeSemigroup_568
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_786 = T_IsCommutativeSemigroup_568
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.trans
d_trans_788 ::
  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_788 :: T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_788 = T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.∙-cong
d_'8729''45'cong_790 ::
  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_790 :: T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_790 = T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.∙-congʳ
d_'8729''45'cong'691'_792 ::
  (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'_792 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_792 = (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.∙-congˡ
d_'8729''45'cong'737'_794 ::
  (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'_794 :: (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_794 = (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_568
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.*-assoc
d_'42''45'assoc_798 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_798 :: T_IsCommutativeSemiring_1750
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_798 = T_IsCommutativeSemiring_1750
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.*-comm
d_'42''45'comm_800 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_800 :: T_IsCommutativeSemiring_1750 -> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_800 = T_IsCommutativeSemiring_1750 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.*-cong
d_'42''45'cong_802 ::
  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_802 :: T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_802 = T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.∙-congʳ
d_'8729''45'cong'691'_804 ::
  (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'_804 :: (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'_804 = (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.Nat.Properties._.IsCommutativeSemiring.∙-congˡ
d_'8729''45'cong'737'_806 ::
  (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'_806 :: (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'_806 = (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.Nat.Properties._.IsCommutativeSemiring.*-identity
d_'42''45'identity_808 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_808 :: T_IsCommutativeSemiring_1750 -> T_Σ_14
d_'42''45'identity_808 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.Nat.Properties._.IsCommutativeSemiring.identityʳ
d_identity'691'_810 ::
  (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'_810 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_identity'691'_810 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.identityˡ
d_identity'737'_812 ::
  (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'_812 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_identity'737'_812 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.isCommutativeMagma
d_isCommutativeMagma_814 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_814 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_814 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_814 T_IsCommutativeSemiring_1750
v4
du_isCommutativeMagma_814 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_814 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_814 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.Nat.Properties._.IsCommutativeSemiring.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_816 ::
  (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_816 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_816 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.Nat.Properties._.IsCommutativeSemiring.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_818 ::
  (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_818 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_818 ~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_818 T_IsCommutativeSemiring_1750
v4
du_'42''45'isCommutativeSemigroup_818 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_818 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemigroup_568
du_'42''45'isCommutativeSemigroup_818 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.Nat.Properties._.IsCommutativeSemiring.*-isMagma
d_'42''45'isMagma_820 ::
  (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_820 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsMagma_178
d_'42''45'isMagma_820 ~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_820 T_IsCommutativeSemiring_1750
v4
du_'42''45'isMagma_820 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_820 :: T_IsCommutativeSemiring_1750 -> T_IsMagma_178
du_'42''45'isMagma_820 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.Nat.Properties._.IsCommutativeSemiring.*-isMonoid
d_'42''45'isMonoid_822 ::
  (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_822 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsMonoid_712
d_'42''45'isMonoid_822 ~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_822 T_IsCommutativeSemiring_1750
v4
du_'42''45'isMonoid_822 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_822 :: T_IsCommutativeSemiring_1750 -> T_IsMonoid_712
du_'42''45'isMonoid_822 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.Nat.Properties._.IsCommutativeSemiring.*-isSemigroup
d_'42''45'isSemigroup_824 ::
  (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_824 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsSemigroup_488
d_'42''45'isSemigroup_824 ~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_824 T_IsCommutativeSemiring_1750
v4
du_'42''45'isSemigroup_824 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_824 :: T_IsCommutativeSemiring_1750 -> T_IsSemigroup_488
du_'42''45'isSemigroup_824 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.Nat.Properties._.IsCommutativeSemiring.assoc
d_assoc_826 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_826 :: T_IsCommutativeSemiring_1750
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_826 = T_IsCommutativeSemiring_1750
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.comm
d_comm_828 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_828 :: T_IsCommutativeSemiring_1750 -> Integer -> Integer -> T__'8801'__12
d_comm_828 = T_IsCommutativeSemiring_1750 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.∙-cong
d_'8729''45'cong_830 ::
  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_830 :: T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_830 = T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.∙-congʳ
d_'8729''45'cong'691'_832 ::
  (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'_832 :: (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'_832 = (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.Nat.Properties._.IsCommutativeSemiring.∙-congˡ
d_'8729''45'cong'737'_834 ::
  (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'_834 :: (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'_834 = (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.Nat.Properties._.IsCommutativeSemiring.identity
d_identity_836 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_836 :: T_IsCommutativeSemiring_1750 -> T_Σ_14
d_identity_836 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.Nat.Properties._.IsCommutativeSemiring.identityʳ
d_identity'691'_838 ::
  (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'_838 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_identity'691'_838 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.identityˡ
d_identity'737'_840 ::
  (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'_840 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_identity'737'_840 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.isCommutativeMagma
d_isCommutativeMagma_842 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_842 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_842 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_842 T_IsCommutativeSemiring_1750
v4
du_isCommutativeMagma_842 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_842 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_842 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.Nat.Properties._.IsCommutativeSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_844 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_844 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_844 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.Nat.Properties._.IsCommutativeSemiring.isCommutativeSemigroup
d_isCommutativeSemigroup_846 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_846 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_846 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_846 T_IsCommutativeSemiring_1750
v4
du_isCommutativeSemigroup_846 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_846 :: T_IsCommutativeSemiring_1750 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_846 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.Nat.Properties._.IsCommutativeSemiring.isMagma
d_isMagma_848 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_848 :: T_IsCommutativeSemiring_1750 -> T_IsMagma_178
d_isMagma_848 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.Nat.Properties._.IsCommutativeSemiring.isMonoid
d_isMonoid_850 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_850 :: T_IsCommutativeSemiring_1750 -> T_IsMonoid_712
d_isMonoid_850 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.Nat.Properties._.IsCommutativeSemiring.isSemigroup
d_isSemigroup_852 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_852 :: T_IsCommutativeSemiring_1750 -> T_IsSemigroup_488
d_isSemigroup_852 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.Nat.Properties._.IsCommutativeSemiring.isUnitalMagma
d_isUnitalMagma_854 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_854 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsUnitalMagma_666
d_isUnitalMagma_854 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4 = T_IsCommutativeSemiring_1750 -> T_IsUnitalMagma_666
du_isUnitalMagma_854 T_IsCommutativeSemiring_1750
v4
du_isUnitalMagma_854 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_854 :: T_IsCommutativeSemiring_1750 -> T_IsUnitalMagma_666
du_isUnitalMagma_854 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.Nat.Properties._.IsCommutativeSemiring.distrib
d_distrib_856 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_856 :: T_IsCommutativeSemiring_1750 -> T_Σ_14
d_distrib_856 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.Nat.Properties._.IsCommutativeSemiring.distribʳ
d_distrib'691'_858 ::
  (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'_858 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_858 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.distribˡ
d_distrib'737'_860 ::
  (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'_860 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_860 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.isCommutativeSemiringWithoutOne
d_isCommutativeSemiringWithoutOne_862 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_862 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsCommutativeSemiringWithoutOne_1438
d_isCommutativeSemiringWithoutOne_862 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.Nat.Properties._.IsCommutativeSemiring.isEquivalence
d_isEquivalence_864 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_864 :: T_IsCommutativeSemiring_1750 -> T_IsEquivalence_28
d_isEquivalence_864 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.Nat.Properties._.IsCommutativeSemiring.isNearSemiring
d_isNearSemiring_866 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_866 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsNearSemiring_1260
d_isNearSemiring_866 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4 = T_IsCommutativeSemiring_1750 -> T_IsNearSemiring_1260
du_isNearSemiring_866 T_IsCommutativeSemiring_1750
v4
du_isNearSemiring_866 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_866 :: T_IsCommutativeSemiring_1750 -> T_IsNearSemiring_1260
du_isNearSemiring_866 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.Nat.Properties._.IsCommutativeSemiring.isPartialEquivalence
d_isPartialEquivalence_868 ::
  (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_868 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_868 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_868 T_IsCommutativeSemiring_1750
v4
du_isPartialEquivalence_868 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_868 :: T_IsCommutativeSemiring_1750 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_868 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.Nat.Properties._.IsCommutativeSemiring.isSemiring
d_isSemiring_870 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_870 :: T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
d_isSemiring_870 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.Nat.Properties._.IsCommutativeSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_872 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_872 :: T_IsCommutativeSemiring_1750
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_872 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.Nat.Properties._.IsCommutativeSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_874 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_874 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_874 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4
  = T_IsCommutativeSemiring_1750 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_874 T_IsCommutativeSemiring_1750
v4
du_isSemiringWithoutOne_874 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_874 :: T_IsCommutativeSemiring_1750 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_874 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.Nat.Properties._.IsCommutativeSemiring.refl
d_refl_876 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_876 :: T_IsCommutativeSemiring_1750 -> Integer -> T__'8801'__12
d_refl_876 = T_IsCommutativeSemiring_1750 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.reflexive
d_reflexive_878 ::
  (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_878 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_878 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.setoid
d_setoid_880 ::
  (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_880 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> T_Setoid_46
d_setoid_880 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1750
v4 = T_IsCommutativeSemiring_1750 -> T_Setoid_46
du_setoid_880 T_IsCommutativeSemiring_1750
v4
du_setoid_880 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_880 :: T_IsCommutativeSemiring_1750 -> T_Setoid_46
du_setoid_880 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.Nat.Properties._.IsCommutativeSemiring.sym
d_sym_882 ::
  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_882 :: T_IsCommutativeSemiring_1750
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_882 = T_IsCommutativeSemiring_1750
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.trans
d_trans_884 ::
  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_884 :: T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_884 = T_IsCommutativeSemiring_1750
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.zero
d_zero_886 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_886 :: T_IsCommutativeSemiring_1750 -> T_Σ_14
d_zero_886 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.Nat.Properties._.IsCommutativeSemiring.zeroʳ
d_zero'691'_888 ::
  (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'_888 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_zero'691'_888 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.zeroˡ
d_zero'737'_890 ::
  (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'_890 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
d_zero'737'_890 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1750
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-assoc
d_'42''45'assoc_894 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_894 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_894 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-comm
d_'42''45'comm_896 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_896 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_896 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-cong
d_'42''45'cong_898 ::
  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_898 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_898 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.∙-congʳ
d_'8729''45'cong'691'_900 ::
  (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'_900 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_900 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.∙-congˡ
d_'8729''45'cong'737'_902 ::
  (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'_902 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_902 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isCommutativeMagma
d_isCommutativeMagma_904 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_904 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_904 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_904 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isCommutativeMagma_904 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_904 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_904 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_906 ::
  (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_906 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
d_'42''45'isCommutativeSemigroup_906 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-isMagma
d_'42''45'isMagma_908 ::
  (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_908 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsMagma_178
d_'42''45'isMagma_908 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3 = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsMagma_178
du_'42''45'isMagma_908 T_IsCommutativeSemiringWithoutOne_1438
v3
du_'42''45'isMagma_908 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_908 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsMagma_178
du_'42''45'isMagma_908 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-isSemigroup
d_'42''45'isSemigroup_910 ::
  (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_910 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemigroup_488
d_'42''45'isSemigroup_910 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsSemigroup_488
du_'42''45'isSemigroup_910 T_IsCommutativeSemiringWithoutOne_1438
v3
du_'42''45'isSemigroup_910 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_910 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsSemigroup_488
du_'42''45'isSemigroup_910 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.assoc
d_assoc_912 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_912 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_912 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.comm
d_comm_914 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_914 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12
d_comm_914 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.∙-cong
d_'8729''45'cong_916 ::
  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_916 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_916 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.∙-congʳ
d_'8729''45'cong'691'_918 ::
  (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'_918 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_918 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.∙-congˡ
d_'8729''45'cong'737'_920 ::
  (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'_920 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_920 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.identity
d_identity_922 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_922 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_Σ_14
d_identity_922 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.identityʳ
d_identity'691'_924 ::
  (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'_924 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
d_identity'691'_924 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.identityˡ
d_identity'737'_926 ::
  (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'_926 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
d_identity'737'_926 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isCommutativeMagma
d_isCommutativeMagma_928 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_928 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_928 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_928 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isCommutativeMagma_928 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_928 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_928 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_930 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_930 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_930 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.isCommutativeSemigroup
d_isCommutativeSemigroup_932 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_932 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_932 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_932 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isCommutativeSemigroup_932 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_932 :: T_IsCommutativeSemiringWithoutOne_1438
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_932 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.isMonoid
d_isMonoid_934 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_934 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsMonoid_712
d_isMonoid_934 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.distrib
d_distrib_936 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_936 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_Σ_14
d_distrib_936 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.distribʳ
d_distrib'691'_938 ::
  (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'_938 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_938 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.distribˡ
d_distrib'737'_940 ::
  (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'_940 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_940 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isEquivalence
d_isEquivalence_942 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_942 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsEquivalence_28
d_isEquivalence_942 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.isNearSemiring
d_isNearSemiring_944 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_944 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsNearSemiring_1260
d_isNearSemiring_944 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3 = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsNearSemiring_1260
du_isNearSemiring_944 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isNearSemiring_944 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_944 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsNearSemiring_1260
du_isNearSemiring_944 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.isPartialEquivalence
d_isPartialEquivalence_946 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_946 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_946 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3
  = T_IsCommutativeSemiringWithoutOne_1438 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_946 T_IsCommutativeSemiringWithoutOne_1438
v3
du_isPartialEquivalence_946 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_946 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_946 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.isSemiringWithoutOne
d_isSemiringWithoutOne_948 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_948 :: T_IsCommutativeSemiringWithoutOne_1438
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_948 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.refl
d_refl_950 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_950 :: T_IsCommutativeSemiringWithoutOne_1438 -> Integer -> T__'8801'__12
d_refl_950 = T_IsCommutativeSemiringWithoutOne_1438 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.reflexive
d_reflexive_952 ::
  (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_952 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_952 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.setoid
d_setoid_954 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_954 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> T_Setoid_46
d_setoid_954 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1438
v3 = T_IsCommutativeSemiringWithoutOne_1438 -> T_Setoid_46
du_setoid_954 T_IsCommutativeSemiringWithoutOne_1438
v3
du_setoid_954 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_954 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_Setoid_46
du_setoid_954 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.sym
d_sym_956 ::
  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_956 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_956 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.trans
d_trans_958 ::
  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_958 :: T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_958 = T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.zero
d_zero_960 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1438 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_960 :: T_IsCommutativeSemiringWithoutOne_1438 -> T_Σ_14
d_zero_960 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.Nat.Properties._.IsCommutativeSemiringWithoutOne.zeroʳ
d_zero'691'_962 ::
  (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'_962 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
d_zero'691'_962 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.zeroˡ
d_zero'737'_964 ::
  (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'_964 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
d_zero'737'_964 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1438
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsFlexibleMagma.flex
d_flex_968 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_flex_968 :: T_IsFlexibleMagma_332 -> Integer -> Integer -> T__'8801'__12
d_flex_968 = T_IsFlexibleMagma_332 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsFlexibleMagma.isEquivalence
d_isEquivalence_970 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_970 :: T_IsFlexibleMagma_332 -> T_IsEquivalence_28
d_isEquivalence_970 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.Nat.Properties._.IsFlexibleMagma.isMagma
d_isMagma_972 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_972 :: T_IsFlexibleMagma_332 -> T_IsMagma_178
d_isMagma_972 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.Nat.Properties._.IsFlexibleMagma.isPartialEquivalence
d_isPartialEquivalence_974 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_974 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_974 ~Integer -> Integer -> Integer
v0 T_IsFlexibleMagma_332
v1 = T_IsFlexibleMagma_332 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_974 T_IsFlexibleMagma_332
v1
du_isPartialEquivalence_974 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_974 :: T_IsFlexibleMagma_332 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_974 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.Nat.Properties._.IsFlexibleMagma.refl
d_refl_976 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_976 :: T_IsFlexibleMagma_332 -> Integer -> T__'8801'__12
d_refl_976 = T_IsFlexibleMagma_332 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsFlexibleMagma.reflexive
d_reflexive_978 ::
  (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_978 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_978 = (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsFlexibleMagma.setoid
d_setoid_980 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_980 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332 -> T_Setoid_46
d_setoid_980 ~Integer -> Integer -> Integer
v0 T_IsFlexibleMagma_332
v1 = T_IsFlexibleMagma_332 -> T_Setoid_46
du_setoid_980 T_IsFlexibleMagma_332
v1
du_setoid_980 ::
  MAlonzo.Code.Algebra.Structures.T_IsFlexibleMagma_332 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_980 :: T_IsFlexibleMagma_332 -> T_Setoid_46
du_setoid_980 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.Nat.Properties._.IsFlexibleMagma.sym
d_sym_982 ::
  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_982 :: T_IsFlexibleMagma_332
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_982 = T_IsFlexibleMagma_332
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsFlexibleMagma.trans
d_trans_984 ::
  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_984 :: T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_984 = T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsFlexibleMagma.∙-cong
d_'8729''45'cong_986 ::
  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_986 :: T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_986 = T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsFlexibleMagma.∙-congʳ
d_'8729''45'cong'691'_988 ::
  (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'_988 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_988 = (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsFlexibleMagma.∙-congˡ
d_'8729''45'cong'737'_990 ::
  (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'_990 :: (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_990 = (Integer -> Integer -> Integer)
-> T_IsFlexibleMagma_332
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup._-_
d__'45'__994 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> Integer -> Integer
d__'45'__994 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
d__'45'__994 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.Nat.Properties._.IsGroup._//_
d__'47''47'__996 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> Integer -> Integer
d__'47''47'__996 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
d__'47''47'__996 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.Nat.Properties._.IsGroup._\\_
d__'92''92'__998 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> Integer -> Integer
d__'92''92'__998 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
d__'92''92'__998 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.Nat.Properties._.IsGroup.assoc
d_assoc_1000 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1000 :: T_IsGroup_1074 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1000 = T_IsGroup_1074 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.identity
d_identity_1002 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1002 :: T_IsGroup_1074 -> T_Σ_14
d_identity_1002 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.Nat.Properties._.IsGroup.identityʳ
d_identity'691'_1004 ::
  (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'_1004 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
d_identity'691'_1004 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.identityˡ
d_identity'737'_1006 ::
  (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'_1006 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
d_identity'737'_1006 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.inverse
d_inverse_1008 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_1008 :: T_IsGroup_1074 -> T_Σ_14
d_inverse_1008 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.Nat.Properties._.IsGroup.inverseʳ
d_inverse'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_inverse'691'_1010 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
d_inverse'691'_1010 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.inverseˡ
d_inverse'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_inverse'737'_1012 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
d_inverse'737'_1012 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.isEquivalence
d_isEquivalence_1014 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1014 :: T_IsGroup_1074 -> T_IsEquivalence_28
d_isEquivalence_1014 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.Nat.Properties._.IsGroup.isInvertibleMagma
d_isInvertibleMagma_1016 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_1016 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> T_IsInvertibleMagma_958
d_isInvertibleMagma_1016 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.Nat.Properties._.IsGroup.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_1018 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_1018 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_1018 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.Nat.Properties._.IsGroup.isMagma
d_isMagma_1020 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1020 :: T_IsGroup_1074 -> T_IsMagma_178
d_isMagma_1020 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.Nat.Properties._.IsGroup.isMonoid
d_isMonoid_1022 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1022 :: T_IsGroup_1074 -> T_IsMonoid_712
d_isMonoid_1022 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.Nat.Properties._.IsGroup.isPartialEquivalence
d_isPartialEquivalence_1024 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1024 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1024 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsGroup_1074
v3
  = T_IsGroup_1074 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1024 T_IsGroup_1074
v3
du_isPartialEquivalence_1024 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1024 :: T_IsGroup_1074 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1024 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.Nat.Properties._.IsGroup.isSemigroup
d_isSemigroup_1026 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1026 :: T_IsGroup_1074 -> T_IsSemigroup_488
d_isSemigroup_1026 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.Nat.Properties._.IsGroup.isUnitalMagma
d_isUnitalMagma_1028 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1028 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> T_IsUnitalMagma_666
d_isUnitalMagma_1028 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsGroup_1074
v3 = T_IsGroup_1074 -> T_IsUnitalMagma_666
du_isUnitalMagma_1028 T_IsGroup_1074
v3
du_isUnitalMagma_1028 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1028 :: T_IsGroup_1074 -> T_IsUnitalMagma_666
du_isUnitalMagma_1028 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.Nat.Properties._.IsGroup.refl
d_refl_1030 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1030 :: T_IsGroup_1074 -> Integer -> T__'8801'__12
d_refl_1030 = T_IsGroup_1074 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.reflexive
d_reflexive_1032 ::
  (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_1032 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1032 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.setoid
d_setoid_1034 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1034 :: (Integer -> Integer -> Integer)
-> Integer -> (Integer -> Integer) -> T_IsGroup_1074 -> T_Setoid_46
d_setoid_1034 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsGroup_1074
v3 = T_IsGroup_1074 -> T_Setoid_46
du_setoid_1034 T_IsGroup_1074
v3
du_setoid_1034 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1034 :: T_IsGroup_1074 -> T_Setoid_46
du_setoid_1034 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.Nat.Properties._.IsGroup.sym
d_sym_1036 ::
  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_1036 :: T_IsGroup_1074
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1036 = T_IsGroup_1074
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.trans
d_trans_1038 ::
  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_1038 :: T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1038 = T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_1040 ::
  (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'_1040 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'691''45''8315''185'_1040 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_1042 ::
  (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'_1042 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_unique'737''45''8315''185'_1042 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.⁻¹-cong
d_'8315''185''45'cong_1044 ::
  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_1044 :: T_IsGroup_1074
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_1044 = T_IsGroup_1074
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.∙-cong
d_'8729''45'cong_1046 ::
  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_1046 :: T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1046 = T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.∙-congʳ
d_'8729''45'cong'691'_1048 ::
  (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'_1048 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1048 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.∙-congˡ
d_'8729''45'cong'737'_1050 ::
  (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'_1050 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1050 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsGroup_1074
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.assoc
d_assoc_1054 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1054 :: T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1054 = T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.comm
d_comm_1056 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1056 :: T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> T__'8801'__12
d_comm_1056 = T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.idem
d_idem_1058 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_1058 :: T_IsIdempotentCommutativeMonoid_884 -> Integer -> T__'8801'__12
d_idem_1058 = T_IsIdempotentCommutativeMonoid_884 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.identity
d_identity_1060 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1060 :: T_IsIdempotentCommutativeMonoid_884 -> T_Σ_14
d_identity_1060 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.Nat.Properties._.IsIdempotentCommutativeMonoid.identityʳ
d_identity'691'_1062 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1062 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> T__'8801'__12
d_identity'691'_1062 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.identityˡ
d_identity'737'_1064 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1064 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> T__'8801'__12
d_identity'737'_1064 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.isBand
d_isBand_1066 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_1066 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentCommutativeMonoid_884 -> T_IsBand_526
d_isBand_1066 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2 = T_IsIdempotentCommutativeMonoid_884 -> T_IsBand_526
du_isBand_1066 T_IsIdempotentCommutativeMonoid_884
v2
du_isBand_1066 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
du_isBand_1066 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsBand_526
du_isBand_1066 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isCommutativeBand
d_isCommutativeBand_1068 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
d_isCommutativeBand_1068 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsCommutativeBand_612
d_isCommutativeBand_1068 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isCommutativeMagma
d_isCommutativeMagma_1070 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_1070 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_1070 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2
  = T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1070 T_IsIdempotentCommutativeMonoid_884
v2
du_isCommutativeMagma_1070 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_1070 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1070 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isCommutativeMonoid
d_isCommutativeMonoid_1072 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_1072 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_1072 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isCommutativeSemigroup
d_isCommutativeSemigroup_1074 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1074 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1074 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2
  = T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1074 T_IsIdempotentCommutativeMonoid_884
v2
du_isCommutativeSemigroup_1074 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1074 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1074 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isEquivalence
d_isEquivalence_1076 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1076 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsEquivalence_28
d_isEquivalence_1076 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isIdempotentMonoid
d_isIdempotentMonoid_1078 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1078 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1078 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isMagma
d_isMagma_1080 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1080 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsMagma_178
d_isMagma_1080 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isMonoid
d_isMonoid_1082 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1082 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsMonoid_712
d_isMonoid_1082 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isPartialEquivalence
d_isPartialEquivalence_1084 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1084 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1084 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2
  = T_IsIdempotentCommutativeMonoid_884 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1084 T_IsIdempotentCommutativeMonoid_884
v2
du_isPartialEquivalence_1084 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1084 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1084 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isSemigroup
d_isSemigroup_1086 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1086 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsSemigroup_488
d_isSemigroup_1086 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.Nat.Properties._.IsIdempotentCommutativeMonoid.isUnitalMagma
d_isUnitalMagma_1088 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1088 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> T_IsUnitalMagma_666
d_isUnitalMagma_1088 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2 = T_IsIdempotentCommutativeMonoid_884 -> T_IsUnitalMagma_666
du_isUnitalMagma_1088 T_IsIdempotentCommutativeMonoid_884
v2
du_isUnitalMagma_1088 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1088 :: T_IsIdempotentCommutativeMonoid_884 -> T_IsUnitalMagma_666
du_isUnitalMagma_1088 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.Nat.Properties._.IsIdempotentCommutativeMonoid.refl
d_refl_1090 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1090 :: T_IsIdempotentCommutativeMonoid_884 -> Integer -> T__'8801'__12
d_refl_1090 = T_IsIdempotentCommutativeMonoid_884 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.reflexive
d_reflexive_1092 ::
  (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_1092 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1092 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.setoid
d_setoid_1094 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1094 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentCommutativeMonoid_884 -> T_Setoid_46
d_setoid_1094 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentCommutativeMonoid_884
v2 = T_IsIdempotentCommutativeMonoid_884 -> T_Setoid_46
du_setoid_1094 T_IsIdempotentCommutativeMonoid_884
v2
du_setoid_1094 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1094 :: T_IsIdempotentCommutativeMonoid_884 -> T_Setoid_46
du_setoid_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_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.Nat.Properties._.IsIdempotentCommutativeMonoid.sym
d_sym_1096 ::
  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_1096 :: T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1096 = T_IsIdempotentCommutativeMonoid_884
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.trans
d_trans_1098 ::
  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_1098 :: T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1098 = T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.∙-cong
d_'8729''45'cong_1100 ::
  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_1100 :: T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1100 = T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.∙-congʳ
d_'8729''45'cong'691'_1102 ::
  (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'_1102 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1102 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.∙-congˡ
d_'8729''45'cong'737'_1104 ::
  (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'_1104 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1104 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentCommutativeMonoid_884
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMagma.idem
d_idem_1108 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_1108 :: T_IsIdempotentMagma_252 -> Integer -> T__'8801'__12
d_idem_1108 = T_IsIdempotentMagma_252 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMagma.isEquivalence
d_isEquivalence_1110 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1110 :: T_IsIdempotentMagma_252 -> T_IsEquivalence_28
d_isEquivalence_1110 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.Nat.Properties._.IsIdempotentMagma.isMagma
d_isMagma_1112 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1112 :: T_IsIdempotentMagma_252 -> T_IsMagma_178
d_isMagma_1112 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.Nat.Properties._.IsIdempotentMagma.isPartialEquivalence
d_isPartialEquivalence_1114 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1114 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1114 ~Integer -> Integer -> Integer
v0 T_IsIdempotentMagma_252
v1
  = T_IsIdempotentMagma_252 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1114 T_IsIdempotentMagma_252
v1
du_isPartialEquivalence_1114 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1114 :: T_IsIdempotentMagma_252 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1114 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.Nat.Properties._.IsIdempotentMagma.refl
d_refl_1116 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1116 :: T_IsIdempotentMagma_252 -> Integer -> T__'8801'__12
d_refl_1116 = T_IsIdempotentMagma_252 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMagma.reflexive
d_reflexive_1118 ::
  (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_1118 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1118 = (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMagma.setoid
d_setoid_1120 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1120 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252 -> T_Setoid_46
d_setoid_1120 ~Integer -> Integer -> Integer
v0 T_IsIdempotentMagma_252
v1 = T_IsIdempotentMagma_252 -> T_Setoid_46
du_setoid_1120 T_IsIdempotentMagma_252
v1
du_setoid_1120 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMagma_252 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1120 :: T_IsIdempotentMagma_252 -> T_Setoid_46
du_setoid_1120 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.Nat.Properties._.IsIdempotentMagma.sym
d_sym_1122 ::
  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_1122 :: T_IsIdempotentMagma_252
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1122 = T_IsIdempotentMagma_252
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMagma.trans
d_trans_1124 ::
  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_1124 :: T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1124 = T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMagma.∙-cong
d_'8729''45'cong_1126 ::
  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_1126 :: T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1126 = T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMagma.∙-congʳ
d_'8729''45'cong'691'_1128 ::
  (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'_1128 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1128 = (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMagma.∙-congˡ
d_'8729''45'cong'737'_1130 ::
  (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'_1130 :: (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1130 = (Integer -> Integer -> Integer)
-> T_IsIdempotentMagma_252
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.assoc
d_assoc_1134 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1134 :: T_IsIdempotentMonoid_826
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1134 = T_IsIdempotentMonoid_826
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.idem
d_idem_1136 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_1136 :: T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
d_idem_1136 = T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.identity
d_identity_1138 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1138 :: T_IsIdempotentMonoid_826 -> T_Σ_14
d_identity_1138 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.Nat.Properties._.IsIdempotentMonoid.identityʳ
d_identity'691'_1140 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1140 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
d_identity'691'_1140 = (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.identityˡ
d_identity'737'_1142 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1142 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
d_identity'737'_1142 = (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.isBand
d_isBand_1144 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_1144 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> T_IsBand_526
d_isBand_1144 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.Nat.Properties._.IsIdempotentMonoid.isEquivalence
d_isEquivalence_1146 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1146 :: T_IsIdempotentMonoid_826 -> T_IsEquivalence_28
d_isEquivalence_1146 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.Nat.Properties._.IsIdempotentMonoid.isMagma
d_isMagma_1148 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1148 :: T_IsIdempotentMonoid_826 -> T_IsMagma_178
d_isMagma_1148 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.Nat.Properties._.IsIdempotentMonoid.isMonoid
d_isMonoid_1150 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1150 :: T_IsIdempotentMonoid_826 -> T_IsMonoid_712
d_isMonoid_1150 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.Nat.Properties._.IsIdempotentMonoid.isPartialEquivalence
d_isPartialEquivalence_1152 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1152 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1152 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentMonoid_826
v2
  = T_IsIdempotentMonoid_826 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1152 T_IsIdempotentMonoid_826
v2
du_isPartialEquivalence_1152 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1152 :: T_IsIdempotentMonoid_826 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1152 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.Nat.Properties._.IsIdempotentMonoid.isSemigroup
d_isSemigroup_1154 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1154 :: T_IsIdempotentMonoid_826 -> T_IsSemigroup_488
d_isSemigroup_1154 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.Nat.Properties._.IsIdempotentMonoid.isUnitalMagma
d_isUnitalMagma_1156 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1156 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> T_IsUnitalMagma_666
d_isUnitalMagma_1156 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentMonoid_826
v2 = T_IsIdempotentMonoid_826 -> T_IsUnitalMagma_666
du_isUnitalMagma_1156 T_IsIdempotentMonoid_826
v2
du_isUnitalMagma_1156 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1156 :: T_IsIdempotentMonoid_826 -> T_IsUnitalMagma_666
du_isUnitalMagma_1156 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.Nat.Properties._.IsIdempotentMonoid.refl
d_refl_1158 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1158 :: T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
d_refl_1158 = T_IsIdempotentMonoid_826 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.reflexive
d_reflexive_1160 ::
  (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_1160 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1160 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.setoid
d_setoid_1162 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1162 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsIdempotentMonoid_826 -> T_Setoid_46
d_setoid_1162 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsIdempotentMonoid_826
v2 = T_IsIdempotentMonoid_826 -> T_Setoid_46
du_setoid_1162 T_IsIdempotentMonoid_826
v2
du_setoid_1162 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1162 :: T_IsIdempotentMonoid_826 -> T_Setoid_46
du_setoid_1162 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.Nat.Properties._.IsIdempotentMonoid.sym
d_sym_1164 ::
  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_1164 :: T_IsIdempotentMonoid_826
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1164 = T_IsIdempotentMonoid_826
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.trans
d_trans_1166 ::
  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_1166 :: T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1166 = T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.∙-cong
d_'8729''45'cong_1168 ::
  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_1168 :: T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1168 = T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.∙-congʳ
d_'8729''45'cong'691'_1170 ::
  (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'_1170 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1170 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentMonoid.∙-congˡ
d_'8729''45'cong'737'_1172 ::
  (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'_1172 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1172 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsIdempotentMonoid_826
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.*-assoc
d_'42''45'assoc_1176 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_1176 :: T_IsIdempotentSemiring_1998
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_1176 = T_IsIdempotentSemiring_1998
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.*-cong
d_'42''45'cong_1178 ::
  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_1178 :: T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_1178 = T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.∙-congʳ
d_'8729''45'cong'691'_1180 ::
  (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'_1180 :: (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'_1180 = (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.Nat.Properties._.IsIdempotentSemiring.∙-congˡ
d_'8729''45'cong'737'_1182 ::
  (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'_1182 :: (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'_1182 = (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.Nat.Properties._.IsIdempotentSemiring.*-identity
d_'42''45'identity_1184 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_1184 :: T_IsIdempotentSemiring_1998 -> T_Σ_14
d_'42''45'identity_1184 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.Nat.Properties._.IsIdempotentSemiring.identityʳ
d_identity'691'_1186 ::
  (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'_1186 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_identity'691'_1186 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.identityˡ
d_identity'737'_1188 ::
  (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'_1188 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_identity'737'_1188 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.*-isMagma
d_'42''45'isMagma_1190 ::
  (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_1190 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsMagma_178
d_'42''45'isMagma_1190 ~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_1190 T_IsIdempotentSemiring_1998
v4
du_'42''45'isMagma_1190 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_1190 :: T_IsIdempotentSemiring_1998 -> T_IsMagma_178
du_'42''45'isMagma_1190 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.Nat.Properties._.IsIdempotentSemiring.*-isMonoid
d_'42''45'isMonoid_1192 ::
  (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_1192 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsMonoid_712
d_'42''45'isMonoid_1192 ~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_1192 T_IsIdempotentSemiring_1998
v4
du_'42''45'isMonoid_1192 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_1192 :: T_IsIdempotentSemiring_1998 -> T_IsMonoid_712
du_'42''45'isMonoid_1192 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.Nat.Properties._.IsIdempotentSemiring.*-isSemigroup
d_'42''45'isSemigroup_1194 ::
  (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_1194 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsSemigroup_488
d_'42''45'isSemigroup_1194 ~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_1194 T_IsIdempotentSemiring_1998
v4
du_'42''45'isSemigroup_1194 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_1194 :: T_IsIdempotentSemiring_1998 -> T_IsSemigroup_488
du_'42''45'isSemigroup_1194 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.Nat.Properties._.IsIdempotentSemiring.assoc
d_assoc_1196 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1196 :: T_IsIdempotentSemiring_1998
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1196 = T_IsIdempotentSemiring_1998
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.comm
d_comm_1198 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1198 :: T_IsIdempotentSemiring_1998 -> Integer -> Integer -> T__'8801'__12
d_comm_1198 = T_IsIdempotentSemiring_1998 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.∙-cong
d_'8729''45'cong_1200 ::
  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_1200 :: T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1200 = T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.∙-congʳ
d_'8729''45'cong'691'_1202 ::
  (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'_1202 :: (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'_1202 = (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.Nat.Properties._.IsIdempotentSemiring.∙-congˡ
d_'8729''45'cong'737'_1204 ::
  (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'_1204 :: (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'_1204 = (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.Nat.Properties._.IsIdempotentSemiring.+-idem
d_'43''45'idem_1206 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'idem_1206 :: T_IsIdempotentSemiring_1998 -> Integer -> T__'8801'__12
d_'43''45'idem_1206 = T_IsIdempotentSemiring_1998 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.identity
d_identity_1208 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1208 :: T_IsIdempotentSemiring_1998 -> T_Σ_14
d_identity_1208 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.Nat.Properties._.IsIdempotentSemiring.identityʳ
d_identity'691'_1210 ::
  (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'_1210 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_identity'691'_1210 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.identityˡ
d_identity'737'_1212 ::
  (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'_1212 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_identity'737'_1212 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.isBand
d_isBand_1214 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
d_isBand_1214 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsBand_526
d_isBand_1214 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4 = T_IsIdempotentSemiring_1998 -> T_IsBand_526
du_isBand_1214 T_IsIdempotentSemiring_1998
v4
du_isBand_1214 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
du_isBand_1214 :: T_IsIdempotentSemiring_1998 -> T_IsBand_526
du_isBand_1214 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.Nat.Properties._.IsIdempotentSemiring.isCommutativeBand
d_isCommutativeBand_1216 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
d_isCommutativeBand_1216 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsCommutativeBand_612
d_isCommutativeBand_1216 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsCommutativeBand_612
du_isCommutativeBand_1216 T_IsIdempotentSemiring_1998
v4
du_isCommutativeBand_1216 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
du_isCommutativeBand_1216 :: T_IsIdempotentSemiring_1998 -> T_IsCommutativeBand_612
du_isCommutativeBand_1216 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.Nat.Properties._.IsIdempotentSemiring.isCommutativeMagma
d_isCommutativeMagma_1218 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_1218 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_1218 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1218 T_IsIdempotentSemiring_1998
v4
du_isCommutativeMagma_1218 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_1218 :: T_IsIdempotentSemiring_1998 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1218 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.Nat.Properties._.IsIdempotentSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_1220 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_1220 :: T_IsIdempotentSemiring_1998 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_1220 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.Nat.Properties._.IsIdempotentSemiring.isCommutativeSemigroup
d_isCommutativeSemigroup_1222 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1222 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1222 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1222 T_IsIdempotentSemiring_1998
v4
du_isCommutativeSemigroup_1222 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1222 :: T_IsIdempotentSemiring_1998 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1222 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.Nat.Properties._.IsIdempotentSemiring.+-isIdempotentCommutativeMonoid
d_'43''45'isIdempotentCommutativeMonoid_1224 ::
  (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_1224 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsIdempotentCommutativeMonoid_884
d_'43''45'isIdempotentCommutativeMonoid_1224 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.Nat.Properties._.IsIdempotentSemiring.isIdempotentMonoid
d_isIdempotentMonoid_1226 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1226 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1226 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1226 T_IsIdempotentSemiring_1998
v4
du_isIdempotentMonoid_1226 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1226 :: T_IsIdempotentSemiring_1998 -> T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1226 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.Nat.Properties._.IsIdempotentSemiring.isMagma
d_isMagma_1228 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1228 :: T_IsIdempotentSemiring_1998 -> T_IsMagma_178
d_isMagma_1228 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.Nat.Properties._.IsIdempotentSemiring.isMonoid
d_isMonoid_1230 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1230 :: T_IsIdempotentSemiring_1998 -> T_IsMonoid_712
d_isMonoid_1230 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.Nat.Properties._.IsIdempotentSemiring.isSemigroup
d_isSemigroup_1232 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1232 :: T_IsIdempotentSemiring_1998 -> T_IsSemigroup_488
d_isSemigroup_1232 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.Nat.Properties._.IsIdempotentSemiring.isUnitalMagma
d_isUnitalMagma_1234 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1234 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsUnitalMagma_666
d_isUnitalMagma_1234 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4 = T_IsIdempotentSemiring_1998 -> T_IsUnitalMagma_666
du_isUnitalMagma_1234 T_IsIdempotentSemiring_1998
v4
du_isUnitalMagma_1234 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1234 :: T_IsIdempotentSemiring_1998 -> T_IsUnitalMagma_666
du_isUnitalMagma_1234 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.Nat.Properties._.IsIdempotentSemiring.distrib
d_distrib_1236 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_1236 :: T_IsIdempotentSemiring_1998 -> T_Σ_14
d_distrib_1236 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.Nat.Properties._.IsIdempotentSemiring.distribʳ
d_distrib'691'_1238 ::
  (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'_1238 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_1238 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.distribˡ
d_distrib'737'_1240 ::
  (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'_1240 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_1240 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.isEquivalence
d_isEquivalence_1242 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1242 :: T_IsIdempotentSemiring_1998 -> T_IsEquivalence_28
d_isEquivalence_1242 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.Nat.Properties._.IsIdempotentSemiring.isNearSemiring
d_isNearSemiring_1244 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_1244 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsNearSemiring_1260
d_isNearSemiring_1244 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsNearSemiring_1260
du_isNearSemiring_1244 T_IsIdempotentSemiring_1998
v4
du_isNearSemiring_1244 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_1244 :: T_IsIdempotentSemiring_1998 -> T_IsNearSemiring_1260
du_isNearSemiring_1244 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.Nat.Properties._.IsIdempotentSemiring.isPartialEquivalence
d_isPartialEquivalence_1246 ::
  (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_1246 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1246 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1246 T_IsIdempotentSemiring_1998
v4
du_isPartialEquivalence_1246 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1246 :: T_IsIdempotentSemiring_1998 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1246 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.Nat.Properties._.IsIdempotentSemiring.isSemiring
d_isSemiring_1248 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_1248 :: T_IsIdempotentSemiring_1998 -> T_IsSemiring_1640
d_isSemiring_1248 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.Nat.Properties._.IsIdempotentSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_1250 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_1250 :: T_IsIdempotentSemiring_1998
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_1250 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.Nat.Properties._.IsIdempotentSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_1252 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_1252 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_1252 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4
  = T_IsIdempotentSemiring_1998 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1252 T_IsIdempotentSemiring_1998
v4
du_isSemiringWithoutOne_1252 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1252 :: T_IsIdempotentSemiring_1998 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1252 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.Nat.Properties._.IsIdempotentSemiring.refl
d_refl_1254 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1254 :: T_IsIdempotentSemiring_1998 -> Integer -> T__'8801'__12
d_refl_1254 = T_IsIdempotentSemiring_1998 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.reflexive
d_reflexive_1256 ::
  (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_1256 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1256 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.setoid
d_setoid_1258 ::
  (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_1258 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> T_Setoid_46
d_setoid_1258 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsIdempotentSemiring_1998
v4 = T_IsIdempotentSemiring_1998 -> T_Setoid_46
du_setoid_1258 T_IsIdempotentSemiring_1998
v4
du_setoid_1258 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1258 :: T_IsIdempotentSemiring_1998 -> T_Setoid_46
du_setoid_1258 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.Nat.Properties._.IsIdempotentSemiring.sym
d_sym_1260 ::
  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_1260 :: T_IsIdempotentSemiring_1998
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1260 = T_IsIdempotentSemiring_1998
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.trans
d_trans_1262 ::
  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_1262 :: T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1262 = T_IsIdempotentSemiring_1998
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.zero
d_zero_1264 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_1264 :: T_IsIdempotentSemiring_1998 -> T_Σ_14
d_zero_1264 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.Nat.Properties._.IsIdempotentSemiring.zeroʳ
d_zero'691'_1266 ::
  (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'_1266 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_zero'691'_1266 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentSemiring.zeroˡ
d_zero'737'_1268 ::
  (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'_1268 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
d_zero'737'_1268 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsIdempotentSemiring_1998
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.inverse
d_inverse_1272 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_1272 :: T_IsInvertibleMagma_958 -> T_Σ_14
d_inverse_1272 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.Nat.Properties._.IsInvertibleMagma.inverseʳ
d_inverse'691'_1274 ::
  (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'_1274 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> T__'8801'__12
d_inverse'691'_1274 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.inverseˡ
d_inverse'737'_1276 ::
  (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'_1276 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> T__'8801'__12
d_inverse'737'_1276 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.isEquivalence
d_isEquivalence_1278 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1278 :: T_IsInvertibleMagma_958 -> T_IsEquivalence_28
d_isEquivalence_1278 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.Nat.Properties._.IsInvertibleMagma.isMagma
d_isMagma_1280 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1280 :: T_IsInvertibleMagma_958 -> T_IsMagma_178
d_isMagma_1280 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.Nat.Properties._.IsInvertibleMagma.isPartialEquivalence
d_isPartialEquivalence_1282 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1282 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1282 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsInvertibleMagma_958
v3
  = T_IsInvertibleMagma_958 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1282 T_IsInvertibleMagma_958
v3
du_isPartialEquivalence_1282 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1282 :: T_IsInvertibleMagma_958 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1282 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.Nat.Properties._.IsInvertibleMagma.refl
d_refl_1284 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1284 :: T_IsInvertibleMagma_958 -> Integer -> T__'8801'__12
d_refl_1284 = T_IsInvertibleMagma_958 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.reflexive
d_reflexive_1286 ::
  (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_1286 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1286 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.setoid
d_setoid_1288 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1288 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> T_Setoid_46
d_setoid_1288 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsInvertibleMagma_958
v3 = T_IsInvertibleMagma_958 -> T_Setoid_46
du_setoid_1288 T_IsInvertibleMagma_958
v3
du_setoid_1288 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1288 :: T_IsInvertibleMagma_958 -> T_Setoid_46
du_setoid_1288 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.Nat.Properties._.IsInvertibleMagma.sym
d_sym_1290 ::
  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_1290 :: T_IsInvertibleMagma_958
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1290 = T_IsInvertibleMagma_958
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.trans
d_trans_1292 ::
  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_1292 :: T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1292 = T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.⁻¹-cong
d_'8315''185''45'cong_1294 ::
  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_1294 :: T_IsInvertibleMagma_958
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_1294 = T_IsInvertibleMagma_958
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.∙-cong
d_'8729''45'cong_1296 ::
  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_1296 :: T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1296 = T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.∙-congʳ
d_'8729''45'cong'691'_1298 ::
  (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'_1298 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1298 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleMagma.∙-congˡ
d_'8729''45'cong'737'_1300 ::
  (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'_1300 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1300 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleMagma_958
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.identity
d_identity_1304 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1304 :: T_IsInvertibleUnitalMagma_1012 -> T_Σ_14
d_identity_1304 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.Nat.Properties._.IsInvertibleUnitalMagma.identityʳ
d_identity'691'_1306 ::
  (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'_1306 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
d_identity'691'_1306 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.identityˡ
d_identity'737'_1308 ::
  (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'_1308 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
d_identity'737'_1308 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.inverse
d_inverse_1310 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_1310 :: T_IsInvertibleUnitalMagma_1012 -> T_Σ_14
d_inverse_1310 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.Nat.Properties._.IsInvertibleUnitalMagma.inverseʳ
d_inverse'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_inverse'691'_1312 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
d_inverse'691'_1312 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.inverseˡ
d_inverse'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_inverse'737'_1314 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
d_inverse'737'_1314 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.isEquivalence
d_isEquivalence_1316 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1316 :: T_IsInvertibleUnitalMagma_1012 -> T_IsEquivalence_28
d_isEquivalence_1316 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.Nat.Properties._.IsInvertibleUnitalMagma.isInvertibleMagma
d_isInvertibleMagma_1318 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_1318 :: T_IsInvertibleUnitalMagma_1012 -> T_IsInvertibleMagma_958
d_isInvertibleMagma_1318 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.Nat.Properties._.IsInvertibleUnitalMagma.isMagma
d_isMagma_1320 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1320 :: T_IsInvertibleUnitalMagma_1012 -> T_IsMagma_178
d_isMagma_1320 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.Nat.Properties._.IsInvertibleUnitalMagma.isPartialEquivalence
d_isPartialEquivalence_1322 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1322 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1322 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsInvertibleUnitalMagma_1012
v3
  = T_IsInvertibleUnitalMagma_1012 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1322 T_IsInvertibleUnitalMagma_1012
v3
du_isPartialEquivalence_1322 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1322 :: T_IsInvertibleUnitalMagma_1012 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1322 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.Nat.Properties._.IsInvertibleUnitalMagma.isUnitalMagma
d_isUnitalMagma_1324 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1324 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> T_IsUnitalMagma_666
d_isUnitalMagma_1324 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.Nat.Properties._.IsInvertibleUnitalMagma.refl
d_refl_1326 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1326 :: T_IsInvertibleUnitalMagma_1012 -> Integer -> T__'8801'__12
d_refl_1326 = T_IsInvertibleUnitalMagma_1012 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.reflexive
d_reflexive_1328 ::
  (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_1328 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1328 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.setoid
d_setoid_1330 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  (Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1330 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> T_Setoid_46
d_setoid_1330 ~Integer -> Integer -> Integer
v0 ~Integer
v1 ~Integer -> Integer
v2 T_IsInvertibleUnitalMagma_1012
v3 = T_IsInvertibleUnitalMagma_1012 -> T_Setoid_46
du_setoid_1330 T_IsInvertibleUnitalMagma_1012
v3
du_setoid_1330 ::
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1330 :: T_IsInvertibleUnitalMagma_1012 -> T_Setoid_46
du_setoid_1330 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.Nat.Properties._.IsInvertibleUnitalMagma.sym
d_sym_1332 ::
  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_1332 :: T_IsInvertibleUnitalMagma_1012
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1332 = T_IsInvertibleUnitalMagma_1012
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.trans
d_trans_1334 ::
  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_1334 :: T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1334 = T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.⁻¹-cong
d_'8315''185''45'cong_1336 ::
  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_1336 :: T_IsInvertibleUnitalMagma_1012
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_1336 = T_IsInvertibleUnitalMagma_1012
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.∙-cong
d_'8729''45'cong_1338 ::
  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_1338 :: T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1338 = T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.∙-congʳ
d_'8729''45'cong'691'_1340 ::
  (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'_1340 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1340 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsInvertibleUnitalMagma.∙-congˡ
d_'8729''45'cong'737'_1342 ::
  (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'_1342 :: (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1342 = (Integer -> Integer -> Integer)
-> Integer
-> (Integer -> Integer)
-> T_IsInvertibleUnitalMagma_1012
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.*-assoc
d_'42''45'assoc_1346 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_1346 :: T_IsKleeneAlgebra_2122
-> Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_1346 = T_IsKleeneAlgebra_2122
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.*-cong
d_'42''45'cong_1348 ::
  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_1348 :: T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cong_1348 = T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.∙-congʳ
d_'8729''45'cong'691'_1350 ::
  (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'_1350 :: (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'_1350 = (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.Nat.Properties._.IsKleeneAlgebra.∙-congˡ
d_'8729''45'cong'737'_1352 ::
  (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'_1352 :: (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'_1352 = (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.Nat.Properties._.IsKleeneAlgebra.*-identity
d_'42''45'identity_1354 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_1354 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_'42''45'identity_1354 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.Nat.Properties._.IsKleeneAlgebra.identityʳ
d_identity'691'_1356 ::
  (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'_1356 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_identity'691'_1356 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.identityˡ
d_identity'737'_1358 ::
  (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'_1358 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_identity'737'_1358 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.*-isMagma
d_'42''45'isMagma_1360 ::
  (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_1360 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsMagma_178
d_'42''45'isMagma_1360 ~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_1360 T_IsKleeneAlgebra_2122
v5
du_'42''45'isMagma_1360 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
du_'42''45'isMagma_1360 :: T_IsKleeneAlgebra_2122 -> T_IsMagma_178
du_'42''45'isMagma_1360 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.Nat.Properties._.IsKleeneAlgebra.*-isMonoid
d_'42''45'isMonoid_1362 ::
  (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_1362 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsMonoid_712
d_'42''45'isMonoid_1362 ~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_1362 T_IsKleeneAlgebra_2122
v5
du_'42''45'isMonoid_1362 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
du_'42''45'isMonoid_1362 :: T_IsKleeneAlgebra_2122 -> T_IsMonoid_712
du_'42''45'isMonoid_1362 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.Nat.Properties._.IsKleeneAlgebra.*-isSemigroup
d_'42''45'isSemigroup_1364 ::
  (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_1364 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsSemigroup_488
d_'42''45'isSemigroup_1364 ~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_1364 T_IsKleeneAlgebra_2122
v5
du_'42''45'isSemigroup_1364 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
du_'42''45'isSemigroup_1364 :: T_IsKleeneAlgebra_2122 -> T_IsSemigroup_488
du_'42''45'isSemigroup_1364 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.Nat.Properties._.IsKleeneAlgebra.assoc
d_assoc_1366 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1366 :: T_IsKleeneAlgebra_2122
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1366 = T_IsKleeneAlgebra_2122
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.comm
d_comm_1368 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1368 :: T_IsKleeneAlgebra_2122 -> Integer -> Integer -> T__'8801'__12
d_comm_1368 = T_IsKleeneAlgebra_2122 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.∙-cong
d_'8729''45'cong_1370 ::
  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_1370 :: T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1370 = T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.∙-congʳ
d_'8729''45'cong'691'_1372 ::
  (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'_1372 :: (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'_1372 = (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.Nat.Properties._.IsKleeneAlgebra.∙-congˡ
d_'8729''45'cong'737'_1374 ::
  (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'_1374 :: (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'_1374 = (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.Nat.Properties._.IsKleeneAlgebra.+-idem
d_'43''45'idem_1376 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'idem_1376 :: T_IsKleeneAlgebra_2122 -> Integer -> T__'8801'__12
d_'43''45'idem_1376 = T_IsKleeneAlgebra_2122 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.identity
d_identity_1378 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1378 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_identity_1378 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.Nat.Properties._.IsKleeneAlgebra.identityʳ
d_identity'691'_1380 ::
  (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'_1380 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_identity'691'_1380 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.identityˡ
d_identity'737'_1382 ::
  (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'_1382 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_identity'737'_1382 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.isBand
d_isBand_1384 ::
  (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_1384 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsBand_526
d_isBand_1384 ~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_1384 T_IsKleeneAlgebra_2122
v5
du_isBand_1384 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_526
du_isBand_1384 :: T_IsKleeneAlgebra_2122 -> T_IsBand_526
du_isBand_1384 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.Nat.Properties._.IsKleeneAlgebra.isCommutativeBand
d_isCommutativeBand_1386 ::
  (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_1386 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsCommutativeBand_612
d_isCommutativeBand_1386 ~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_1386 T_IsKleeneAlgebra_2122
v5
du_isCommutativeBand_1386 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_612
du_isCommutativeBand_1386 :: T_IsKleeneAlgebra_2122 -> T_IsCommutativeBand_612
du_isCommutativeBand_1386 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.Nat.Properties._.IsKleeneAlgebra.isCommutativeMagma
d_isCommutativeMagma_1388 ::
  (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_1388 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_1388 ~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_1388 T_IsKleeneAlgebra_2122
v5
du_isCommutativeMagma_1388 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_1388 :: T_IsKleeneAlgebra_2122 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_1388 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.Nat.Properties._.IsKleeneAlgebra.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_1390 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_1390 :: T_IsKleeneAlgebra_2122 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_1390 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.Nat.Properties._.IsKleeneAlgebra.isCommutativeSemigroup
d_isCommutativeSemigroup_1392 ::
  (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_1392 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_1392 ~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_1392 T_IsKleeneAlgebra_2122
v5
du_isCommutativeSemigroup_1392 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_1392 :: T_IsKleeneAlgebra_2122 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_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_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.Nat.Properties._.IsKleeneAlgebra.+-isIdempotentCommutativeMonoid
d_'43''45'isIdempotentCommutativeMonoid_1394 ::
  (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_1394 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsIdempotentCommutativeMonoid_884
d_'43''45'isIdempotentCommutativeMonoid_1394 ~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_1394 T_IsKleeneAlgebra_2122
v5
du_'43''45'isIdempotentCommutativeMonoid_1394 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_884
du_'43''45'isIdempotentCommutativeMonoid_1394 :: T_IsKleeneAlgebra_2122 -> T_IsIdempotentCommutativeMonoid_884
du_'43''45'isIdempotentCommutativeMonoid_1394 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.Nat.Properties._.IsKleeneAlgebra.isIdempotentMonoid
d_isIdempotentMonoid_1396 ::
  (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_1396 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsIdempotentMonoid_826
d_isIdempotentMonoid_1396 ~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_1396 T_IsKleeneAlgebra_2122
v5
du_isIdempotentMonoid_1396 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1396 :: T_IsKleeneAlgebra_2122 -> T_IsIdempotentMonoid_826
du_isIdempotentMonoid_1396 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.Nat.Properties._.IsKleeneAlgebra.isMagma
d_isMagma_1398 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1398 :: T_IsKleeneAlgebra_2122 -> T_IsMagma_178
d_isMagma_1398 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.Nat.Properties._.IsKleeneAlgebra.isMonoid
d_isMonoid_1400 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_1400 :: T_IsKleeneAlgebra_2122 -> T_IsMonoid_712
d_isMonoid_1400 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.Nat.Properties._.IsKleeneAlgebra.isSemigroup
d_isSemigroup_1402 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1402 :: T_IsKleeneAlgebra_2122 -> T_IsSemigroup_488
d_isSemigroup_1402 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.Nat.Properties._.IsKleeneAlgebra.isUnitalMagma
d_isUnitalMagma_1404 ::
  (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_1404 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsUnitalMagma_666
d_isUnitalMagma_1404 ~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_1404 T_IsKleeneAlgebra_2122
v5
du_isUnitalMagma_1404 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1404 :: T_IsKleeneAlgebra_2122 -> T_IsUnitalMagma_666
du_isUnitalMagma_1404 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.Nat.Properties._.IsKleeneAlgebra.distrib
d_distrib_1406 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_1406 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_distrib_1406 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.Nat.Properties._.IsKleeneAlgebra.distribʳ
d_distrib'691'_1408 ::
  (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'_1408 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_1408 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.distribˡ
d_distrib'737'_1410 ::
  (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'_1410 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'737'_1410 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.isEquivalence
d_isEquivalence_1412 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1412 :: T_IsKleeneAlgebra_2122 -> T_IsEquivalence_28
d_isEquivalence_1412 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.Nat.Properties._.IsKleeneAlgebra.isIdempotentSemiring
d_isIdempotentSemiring_1414 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentSemiring_1998
d_isIdempotentSemiring_1414 :: T_IsKleeneAlgebra_2122 -> T_IsIdempotentSemiring_1998
d_isIdempotentSemiring_1414 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.Nat.Properties._.IsKleeneAlgebra.isNearSemiring
d_isNearSemiring_1416 ::
  (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_1416 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsNearSemiring_1260
d_isNearSemiring_1416 ~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_1416 T_IsKleeneAlgebra_2122
v5
du_isNearSemiring_1416 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_1416 :: T_IsKleeneAlgebra_2122 -> T_IsNearSemiring_1260
du_isNearSemiring_1416 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.Nat.Properties._.IsKleeneAlgebra.isPartialEquivalence
d_isPartialEquivalence_1418 ::
  (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_1418 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1418 ~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_1418 T_IsKleeneAlgebra_2122
v5
du_isPartialEquivalence_1418 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1418 :: T_IsKleeneAlgebra_2122 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1418 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.Nat.Properties._.IsKleeneAlgebra.isSemiring
d_isSemiring_1420 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_1420 :: T_IsKleeneAlgebra_2122 -> T_IsSemiring_1640
d_isSemiring_1420 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.Nat.Properties._.IsKleeneAlgebra.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_1422 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_1422 :: T_IsKleeneAlgebra_2122 -> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_1422 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.Nat.Properties._.IsKleeneAlgebra.isSemiringWithoutOne
d_isSemiringWithoutOne_1424 ::
  (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_1424 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_1424 ~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_1424 T_IsKleeneAlgebra_2122
v5
du_isSemiringWithoutOne_1424 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_1424 :: T_IsKleeneAlgebra_2122 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_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_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.Nat.Properties._.IsKleeneAlgebra.refl
d_refl_1426 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1426 :: T_IsKleeneAlgebra_2122 -> Integer -> T__'8801'__12
d_refl_1426 = T_IsKleeneAlgebra_2122 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.reflexive
d_reflexive_1428 ::
  (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_1428 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1428 = (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.Nat.Properties._.IsKleeneAlgebra.setoid
d_setoid_1430 ::
  (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_1430 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> T_Setoid_46
d_setoid_1430 ~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_1430 T_IsKleeneAlgebra_2122
v5
du_setoid_1430 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1430 :: T_IsKleeneAlgebra_2122 -> T_Setoid_46
du_setoid_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_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.Nat.Properties._.IsKleeneAlgebra.starDestructive
d_starDestructive_1432 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_starDestructive_1432 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_starDestructive_1432 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.Nat.Properties._.IsKleeneAlgebra.starDestructiveʳ
d_starDestructive'691'_1434 ::
  (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'_1434 :: (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'_1434 = (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.Nat.Properties._.IsKleeneAlgebra.starDestructiveˡ
d_starDestructive'737'_1436 ::
  (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'_1436 :: (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'_1436 = (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.Nat.Properties._.IsKleeneAlgebra.starExpansive
d_starExpansive_1438 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_starExpansive_1438 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_starExpansive_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_starExpansive_2142 (T_IsKleeneAlgebra_2122 -> Any
forall a b. a -> b
coe T_IsKleeneAlgebra_2122
v0)
-- Data.Nat.Properties._.IsKleeneAlgebra.starExpansiveʳ
d_starExpansive'691'_1440 ::
  (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'_1440 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_starExpansive'691'_1440 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.starExpansiveˡ
d_starExpansive'737'_1442 ::
  (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'_1442 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_starExpansive'737'_1442 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.sym
d_sym_1444 ::
  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_1444 :: T_IsKleeneAlgebra_2122
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1444 = T_IsKleeneAlgebra_2122
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.trans
d_trans_1446 ::
  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_1446 :: T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1446 = T_IsKleeneAlgebra_2122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.zero
d_zero_1448 ::
  MAlonzo.Code.Algebra.Structures.T_IsKleeneAlgebra_2122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_1448 :: T_IsKleeneAlgebra_2122 -> T_Σ_14
d_zero_1448 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.Nat.Properties._.IsKleeneAlgebra.zeroʳ
d_zero'691'_1450 ::
  (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'_1450 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_zero'691'_1450 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsKleeneAlgebra.zeroˡ
d_zero'737'_1452 ::
  (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'_1452 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
d_zero'737'_1452 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsKleeneAlgebra_2122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.//-cong
d_'47''47''45'cong_1456 ::
  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_1456 :: T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong_1456 = T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.//-congʳ
d_'47''47''45'cong'691'_1458 ::
  (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'_1458 :: (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'_1458 = (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.Nat.Properties._.IsLeftBolLoop.//-congˡ
d_'47''47''45'cong'737'_1460 ::
  (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'_1460 :: (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'_1460 = (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.Nat.Properties._.IsLeftBolLoop.\\-cong
d_'92''92''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_'92''92''45'cong_1462 :: T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong_1462 = T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.\\-congʳ
d_'92''92''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_'92''92''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_'92''92''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.Nat.Properties._.IsLeftBolLoop.\\-congˡ
d_'92''92''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_'92''92''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_'92''92''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.Nat.Properties._.IsLeftBolLoop.identity
d_identity_1468 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1468 :: T_IsLeftBolLoop_3202 -> T_Σ_14
d_identity_1468 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.Nat.Properties._.IsLeftBolLoop.identityʳ
d_identity'691'_1470 ::
  (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'_1470 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> T__'8801'__12
d_identity'691'_1470 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.identityˡ
d_identity'737'_1472 ::
  (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'_1472 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> T__'8801'__12
d_identity'737'_1472 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.isEquivalence
d_isEquivalence_1474 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1474 :: T_IsLeftBolLoop_3202 -> T_IsEquivalence_28
d_isEquivalence_1474 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.Nat.Properties._.IsLeftBolLoop.isLoop
d_isLoop_1476 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122
d_isLoop_1476 :: T_IsLeftBolLoop_3202 -> T_IsLoop_3122
d_isLoop_1476 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.Nat.Properties._.IsLeftBolLoop.isMagma
d_isMagma_1478 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1478 :: T_IsLeftBolLoop_3202 -> T_IsMagma_178
d_isMagma_1478 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.Nat.Properties._.IsLeftBolLoop.isPartialEquivalence
d_isPartialEquivalence_1480 ::
  (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_1480 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1480 ~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_1480 T_IsLeftBolLoop_3202
v4
du_isPartialEquivalence_1480 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1480 :: T_IsLeftBolLoop_3202 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1480 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.Nat.Properties._.IsLeftBolLoop.isQuasigroup
d_isQuasigroup_1482 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Algebra.Structures.T_IsQuasigroup_3038
d_isQuasigroup_1482 :: T_IsLeftBolLoop_3202 -> T_IsQuasigroup_3038
d_isQuasigroup_1482 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.Nat.Properties._.IsLeftBolLoop.leftBol
d_leftBol_1484 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftBol_1484 :: T_IsLeftBolLoop_3202
-> Integer -> Integer -> Integer -> T__'8801'__12
d_leftBol_1484 = T_IsLeftBolLoop_3202
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.leftDivides
d_leftDivides_1486 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_leftDivides_1486 :: T_IsLeftBolLoop_3202 -> T_Σ_14
d_leftDivides_1486 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.Nat.Properties._.IsLeftBolLoop.leftDividesʳ
d_leftDivides'691'_1488 ::
  (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'_1488 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'691'_1488 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.leftDividesˡ
d_leftDivides'737'_1490 ::
  (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'_1490 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'737'_1490 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.refl
d_refl_1492 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1492 :: T_IsLeftBolLoop_3202 -> Integer -> T__'8801'__12
d_refl_1492 = T_IsLeftBolLoop_3202 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.reflexive
d_reflexive_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 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1494 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1494 = (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.Nat.Properties._.IsLeftBolLoop.rightDivides
d_rightDivides_1496 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_rightDivides_1496 :: T_IsLeftBolLoop_3202 -> T_Σ_14
d_rightDivides_1496 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.Nat.Properties._.IsLeftBolLoop.rightDividesʳ
d_rightDivides'691'_1498 ::
  (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'_1498 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'691'_1498 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.rightDividesˡ
d_rightDivides'737'_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
d_rightDivides'737'_1500 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'737'_1500 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.setoid
d_setoid_1502 ::
  (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_1502 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLeftBolLoop_3202
-> T_Setoid_46
d_setoid_1502 ~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_1502 T_IsLeftBolLoop_3202
v4
du_setoid_1502 ::
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1502 :: T_IsLeftBolLoop_3202 -> T_Setoid_46
du_setoid_1502 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.Nat.Properties._.IsLeftBolLoop.sym
d_sym_1504 ::
  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_1504 :: T_IsLeftBolLoop_3202
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1504 = T_IsLeftBolLoop_3202
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.trans
d_trans_1506 ::
  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_1506 :: T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1506 = T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.∙-cong
d_'8729''45'cong_1508 ::
  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_1508 :: T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1508 = T_IsLeftBolLoop_3202
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLeftBolLoop.∙-congʳ
d_'8729''45'cong'691'_1510 ::
  (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'_1510 :: (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'_1510 = (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.Nat.Properties._.IsLeftBolLoop.∙-congˡ
d_'8729''45'cong'737'_1512 ::
  (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'_1512 :: (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'_1512 = (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.Nat.Properties._.IsLoop.//-cong
d_'47''47''45'cong_1516 ::
  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_1516 :: T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong_1516 = T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.//-congʳ
d_'47''47''45'cong'691'_1518 ::
  (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'_1518 :: (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'_1518 = (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.Nat.Properties._.IsLoop.//-congˡ
d_'47''47''45'cong'737'_1520 ::
  (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'_1520 :: (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'_1520 = (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.Nat.Properties._.IsLoop.\\-cong
d_'92''92''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_'92''92''45'cong_1522 :: T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong_1522 = T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.\\-congʳ
d_'92''92''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_'92''92''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_'92''92''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.Nat.Properties._.IsLoop.\\-congˡ
d_'92''92''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_'92''92''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_'92''92''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.Nat.Properties._.IsLoop.identity
d_identity_1528 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1528 :: T_IsLoop_3122 -> T_Σ_14
d_identity_1528 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.Nat.Properties._.IsLoop.identityʳ
d_identity'691'_1530 ::
  (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'_1530 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> T__'8801'__12
d_identity'691'_1530 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.identityˡ
d_identity'737'_1532 ::
  (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'_1532 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> T__'8801'__12
d_identity'737'_1532 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.isEquivalence
d_isEquivalence_1534 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1534 :: T_IsLoop_3122 -> T_IsEquivalence_28
d_isEquivalence_1534 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.Nat.Properties._.IsLoop.isMagma
d_isMagma_1536 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1536 :: T_IsLoop_3122 -> T_IsMagma_178
d_isMagma_1536 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.Nat.Properties._.IsLoop.isPartialEquivalence
d_isPartialEquivalence_1538 ::
  (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_1538 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1538 ~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_1538 T_IsLoop_3122
v4
du_isPartialEquivalence_1538 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1538 :: T_IsLoop_3122 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1538 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.Nat.Properties._.IsLoop.isQuasigroup
d_isQuasigroup_1540 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Algebra.Structures.T_IsQuasigroup_3038
d_isQuasigroup_1540 :: T_IsLoop_3122 -> T_IsQuasigroup_3038
d_isQuasigroup_1540 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.Nat.Properties._.IsLoop.leftDivides
d_leftDivides_1542 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_leftDivides_1542 :: T_IsLoop_3122 -> T_Σ_14
d_leftDivides_1542 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.Nat.Properties._.IsLoop.leftDividesʳ
d_leftDivides'691'_1544 ::
  (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'_1544 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'691'_1544 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.leftDividesˡ
d_leftDivides'737'_1546 ::
  (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'_1546 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'737'_1546 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.refl
d_refl_1548 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1548 :: T_IsLoop_3122 -> Integer -> T__'8801'__12
d_refl_1548 = T_IsLoop_3122 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.reflexive
d_reflexive_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 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1550 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1550 = (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.Nat.Properties._.IsLoop.rightDivides
d_rightDivides_1552 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_rightDivides_1552 :: T_IsLoop_3122 -> T_Σ_14
d_rightDivides_1552 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.Nat.Properties._.IsLoop.rightDividesʳ
d_rightDivides'691'_1554 ::
  (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'_1554 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'691'_1554 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.rightDividesˡ
d_rightDivides'737'_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
d_rightDivides'737'_1556 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'737'_1556 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.setoid
d_setoid_1558 ::
  (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_1558 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsLoop_3122
-> T_Setoid_46
d_setoid_1558 ~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_1558 T_IsLoop_3122
v4
du_setoid_1558 ::
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1558 :: T_IsLoop_3122 -> T_Setoid_46
du_setoid_1558 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.Nat.Properties._.IsLoop.sym
d_sym_1560 ::
  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_1560 :: T_IsLoop_3122
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1560 = T_IsLoop_3122
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.trans
d_trans_1562 ::
  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_1562 :: T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1562 = T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.∙-cong
d_'8729''45'cong_1564 ::
  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_1564 :: T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1564 = T_IsLoop_3122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsLoop.∙-congʳ
d_'8729''45'cong'691'_1566 ::
  (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'_1566 :: (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'_1566 = (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.Nat.Properties._.IsLoop.∙-congˡ
d_'8729''45'cong'737'_1568 ::
  (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'_1568 :: (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'_1568 = (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.Nat.Properties._.IsMagma.isEquivalence
d_isEquivalence_1572 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1572 :: T_IsMagma_178 -> T_IsEquivalence_28
d_isEquivalence_1572 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.Nat.Properties._.IsMagma.isPartialEquivalence
d_isPartialEquivalence_1574 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1574 :: (Integer -> Integer -> Integer)
-> T_IsMagma_178 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1574 ~Integer -> Integer -> Integer
v0 T_IsMagma_178
v1
  = T_IsMagma_178 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1574 T_IsMagma_178
v1
du_isPartialEquivalence_1574 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1574 :: T_IsMagma_178 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1574 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.Nat.Properties._.IsMagma.refl
d_refl_1576 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1576 :: T_IsMagma_178 -> Integer -> T__'8801'__12
d_refl_1576 = T_IsMagma_178 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMagma.reflexive
d_reflexive_1578 ::
  (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_1578 :: (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1578 = (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMagma.setoid
d_setoid_1580 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1580 :: (Integer -> Integer -> Integer) -> T_IsMagma_178 -> T_Setoid_46
d_setoid_1580 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.Nat.Properties._.IsMagma.sym
d_sym_1582 ::
  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_1582 :: T_IsMagma_178
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1582 = T_IsMagma_178
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMagma.trans
d_trans_1584 ::
  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_1584 :: T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1584 = T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMagma.∙-cong
d_'8729''45'cong_1586 ::
  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_1586 :: T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1586 = T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMagma.∙-congʳ
d_'8729''45'cong'691'_1588 ::
  (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'_1588 :: (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1588 = (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMagma.∙-congˡ
d_'8729''45'cong'737'_1590 ::
  (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'_1590 :: (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1590 = (Integer -> Integer -> Integer)
-> T_IsMagma_178
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMedialMagma.isEquivalence
d_isEquivalence_1594 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1594 :: T_IsMedialMagma_370 -> T_IsEquivalence_28
d_isEquivalence_1594 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.Nat.Properties._.IsMedialMagma.isMagma
d_isMagma_1596 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1596 :: T_IsMedialMagma_370 -> T_IsMagma_178
d_isMagma_1596 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.Nat.Properties._.IsMedialMagma.isPartialEquivalence
d_isPartialEquivalence_1598 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1598 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1598 ~Integer -> Integer -> Integer
v0 T_IsMedialMagma_370
v1
  = T_IsMedialMagma_370 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1598 T_IsMedialMagma_370
v1
du_isPartialEquivalence_1598 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1598 :: T_IsMedialMagma_370 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1598 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.Nat.Properties._.IsMedialMagma.medial
d_medial_1600 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_medial_1600 :: T_IsMedialMagma_370
-> Integer -> Integer -> Integer -> Integer -> T__'8801'__12
d_medial_1600 = T_IsMedialMagma_370
-> Integer -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMedialMagma.refl
d_refl_1602 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1602 :: T_IsMedialMagma_370 -> Integer -> T__'8801'__12
d_refl_1602 = T_IsMedialMagma_370 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMedialMagma.reflexive
d_reflexive_1604 ::
  (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_1604 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1604 = (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMedialMagma.setoid
d_setoid_1606 ::
  (Integer -> Integer -> Integer) ->
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1606 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370 -> T_Setoid_46
d_setoid_1606 ~Integer -> Integer -> Integer
v0 T_IsMedialMagma_370
v1 = T_IsMedialMagma_370 -> T_Setoid_46
du_setoid_1606 T_IsMedialMagma_370
v1
du_setoid_1606 ::
  MAlonzo.Code.Algebra.Structures.T_IsMedialMagma_370 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1606 :: T_IsMedialMagma_370 -> T_Setoid_46
du_setoid_1606 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.Nat.Properties._.IsMedialMagma.sym
d_sym_1608 ::
  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_1608 :: T_IsMedialMagma_370
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1608 = T_IsMedialMagma_370
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMedialMagma.trans
d_trans_1610 ::
  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_1610 :: T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1610 = T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMedialMagma.∙-cong
d_'8729''45'cong_1612 ::
  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_1612 :: T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1612 = T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMedialMagma.∙-congʳ
d_'8729''45'cong'691'_1614 ::
  (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'_1614 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1614 = (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMedialMagma.∙-congˡ
d_'8729''45'cong'737'_1616 ::
  (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'_1616 :: (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1616 = (Integer -> Integer -> Integer)
-> T_IsMedialMagma_370
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.//-cong
d_'47''47''45'cong_1620 ::
  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_1620 :: T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong_1620 = T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.//-congʳ
d_'47''47''45'cong'691'_1622 ::
  (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'_1622 :: (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'_1622 = (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.Nat.Properties._.IsMiddleBolLoop.//-congˡ
d_'47''47''45'cong'737'_1624 ::
  (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'_1624 :: (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'_1624 = (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.Nat.Properties._.IsMiddleBolLoop.\\-cong
d_'92''92''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_'92''92''45'cong_1626 :: T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong_1626 = T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.\\-congʳ
d_'92''92''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_'92''92''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_'92''92''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.Nat.Properties._.IsMiddleBolLoop.\\-congˡ
d_'92''92''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_'92''92''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_'92''92''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.Nat.Properties._.IsMiddleBolLoop.identity
d_identity_1632 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1632 :: T_IsMiddleBolLoop_3462 -> T_Σ_14
d_identity_1632 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.Nat.Properties._.IsMiddleBolLoop.identityʳ
d_identity'691'_1634 ::
  (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'_1634 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> T__'8801'__12
d_identity'691'_1634 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.identityˡ
d_identity'737'_1636 ::
  (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'_1636 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> T__'8801'__12
d_identity'737'_1636 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.isEquivalence
d_isEquivalence_1638 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1638 :: T_IsMiddleBolLoop_3462 -> T_IsEquivalence_28
d_isEquivalence_1638 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.Nat.Properties._.IsMiddleBolLoop.isLoop
d_isLoop_1640 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122
d_isLoop_1640 :: T_IsMiddleBolLoop_3462 -> T_IsLoop_3122
d_isLoop_1640 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.Nat.Properties._.IsMiddleBolLoop.isMagma
d_isMagma_1642 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1642 :: T_IsMiddleBolLoop_3462 -> T_IsMagma_178
d_isMagma_1642 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.Nat.Properties._.IsMiddleBolLoop.isPartialEquivalence
d_isPartialEquivalence_1644 ::
  (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_1644 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1644 ~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_1644 T_IsMiddleBolLoop_3462
v4
du_isPartialEquivalence_1644 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1644 :: T_IsMiddleBolLoop_3462 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1644 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.Nat.Properties._.IsMiddleBolLoop.isQuasigroup
d_isQuasigroup_1646 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Algebra.Structures.T_IsQuasigroup_3038
d_isQuasigroup_1646 :: T_IsMiddleBolLoop_3462 -> T_IsQuasigroup_3038
d_isQuasigroup_1646 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.Nat.Properties._.IsMiddleBolLoop.leftDivides
d_leftDivides_1648 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_leftDivides_1648 :: T_IsMiddleBolLoop_3462 -> T_Σ_14
d_leftDivides_1648 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.Nat.Properties._.IsMiddleBolLoop.leftDividesʳ
d_leftDivides'691'_1650 ::
  (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'_1650 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'691'_1650 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.leftDividesˡ
d_leftDivides'737'_1652 ::
  (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'_1652 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'737'_1652 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.middleBol
d_middleBol_1654 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_middleBol_1654 :: T_IsMiddleBolLoop_3462
-> Integer -> Integer -> Integer -> T__'8801'__12
d_middleBol_1654 = T_IsMiddleBolLoop_3462
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.refl
d_refl_1656 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1656 :: T_IsMiddleBolLoop_3462 -> Integer -> T__'8801'__12
d_refl_1656 = T_IsMiddleBolLoop_3462 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.reflexive
d_reflexive_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 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1658 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1658 = (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.Nat.Properties._.IsMiddleBolLoop.rightDivides
d_rightDivides_1660 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_rightDivides_1660 :: T_IsMiddleBolLoop_3462 -> T_Σ_14
d_rightDivides_1660 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.Nat.Properties._.IsMiddleBolLoop.rightDividesʳ
d_rightDivides'691'_1662 ::
  (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'_1662 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'691'_1662 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.rightDividesˡ
d_rightDivides'737'_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
d_rightDivides'737'_1664 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
d_rightDivides'737'_1664 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.setoid
d_setoid_1666 ::
  (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_1666 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMiddleBolLoop_3462
-> T_Setoid_46
d_setoid_1666 ~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_1666 T_IsMiddleBolLoop_3462
v4
du_setoid_1666 ::
  MAlonzo.Code.Algebra.Structures.T_IsMiddleBolLoop_3462 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1666 :: T_IsMiddleBolLoop_3462 -> T_Setoid_46
du_setoid_1666 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.Nat.Properties._.IsMiddleBolLoop.sym
d_sym_1668 ::
  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_1668 :: T_IsMiddleBolLoop_3462
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1668 = T_IsMiddleBolLoop_3462
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.trans
d_trans_1670 ::
  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_1670 :: T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1670 = T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.∙-cong
d_'8729''45'cong_1672 ::
  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_1672 :: T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1672 = T_IsMiddleBolLoop_3462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMiddleBolLoop.∙-congʳ
d_'8729''45'cong'691'_1674 ::
  (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'_1674 :: (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'_1674 = (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.Nat.Properties._.IsMiddleBolLoop.∙-congˡ
d_'8729''45'cong'737'_1676 ::
  (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'_1676 :: (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'_1676 = (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.Nat.Properties._.IsMonoid.assoc
d_assoc_1680 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1680 :: T_IsMonoid_712 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1680 = T_IsMonoid_712 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.identity
d_identity_1682 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1682 :: T_IsMonoid_712 -> T_Σ_14
d_identity_1682 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.Nat.Properties._.IsMonoid.identityʳ
d_identity'691'_1684 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'691'_1684 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> Integer -> T__'8801'__12
d_identity'691'_1684 = (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.identityˡ
d_identity'737'_1686 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identity'737'_1686 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> Integer -> T__'8801'__12
d_identity'737'_1686 = (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.isEquivalence
d_isEquivalence_1688 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1688 :: T_IsMonoid_712 -> T_IsEquivalence_28
d_isEquivalence_1688 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.Nat.Properties._.IsMonoid.isMagma
d_isMagma_1690 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1690 :: T_IsMonoid_712 -> T_IsMagma_178
d_isMagma_1690 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.Nat.Properties._.IsMonoid.isPartialEquivalence
d_isPartialEquivalence_1692 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1692 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> T_IsPartialEquivalence_16
d_isPartialEquivalence_1692 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsMonoid_712
v2
  = T_IsMonoid_712 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1692 T_IsMonoid_712
v2
du_isPartialEquivalence_1692 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1692 :: T_IsMonoid_712 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1692 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.Nat.Properties._.IsMonoid.isSemigroup
d_isSemigroup_1694 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1694 :: T_IsMonoid_712 -> T_IsSemigroup_488
d_isSemigroup_1694 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.Nat.Properties._.IsMonoid.isUnitalMagma
d_isUnitalMagma_1696 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1696 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> T_IsUnitalMagma_666
d_isUnitalMagma_1696 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.Nat.Properties._.IsMonoid.refl
d_refl_1698 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1698 :: T_IsMonoid_712 -> Integer -> T__'8801'__12
d_refl_1698 = T_IsMonoid_712 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.reflexive
d_reflexive_1700 ::
  (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_1700 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1700 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.setoid
d_setoid_1702 ::
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1702 :: (Integer -> Integer -> Integer)
-> Integer -> T_IsMonoid_712 -> T_Setoid_46
d_setoid_1702 ~Integer -> Integer -> Integer
v0 ~Integer
v1 T_IsMonoid_712
v2 = T_IsMonoid_712 -> T_Setoid_46
du_setoid_1702 T_IsMonoid_712
v2
du_setoid_1702 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1702 :: T_IsMonoid_712 -> T_Setoid_46
du_setoid_1702 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.Nat.Properties._.IsMonoid.sym
d_sym_1704 ::
  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_1704 :: T_IsMonoid_712
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_sym_1704 = T_IsMonoid_712
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.trans
d_trans_1706 ::
  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_1706 :: T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_trans_1706 = T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.∙-cong
d_'8729''45'cong_1708 ::
  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_1708 :: T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1708 = T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.∙-congʳ
d_'8729''45'cong'691'_1710 ::
  (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'_1710 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'691'_1710 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.∙-congˡ
d_'8729''45'cong'737'_1712 ::
  (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'_1712 :: (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong'737'_1712 = (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_712
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.//-cong
d_'47''47''45'cong_1716 ::
  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_1716 :: T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'47''47''45'cong_1716 = T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.//-congʳ
d_'47''47''45'cong'691'_1718 ::
  (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'_1718 :: (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'_1718 = (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.Nat.Properties._.IsMoufangLoop.//-congˡ
d_'47''47''45'cong'737'_1720 ::
  (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'_1720 :: (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'_1720 = (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.Nat.Properties._.IsMoufangLoop.\\-cong
d_'92''92''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_'92''92''45'cong_1722 :: T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'92''92''45'cong_1722 = T_IsMoufangLoop_3370
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.\\-congʳ
d_'92''92''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_'92''92''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_'92''92''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.Nat.Properties._.IsMoufangLoop.\\-congˡ
d_'92''92''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_'92''92''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_'92''92''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.Nat.Properties._.IsMoufangLoop.identical
d_identical_1728 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_identical_1728 :: T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
d_identical_1728 = T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.identity
d_identity_1730 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1730 :: T_IsMoufangLoop_3370 -> T_Σ_14
d_identity_1730 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.Nat.Properties._.IsMoufangLoop.identityʳ
d_identity'691'_1732 ::
  (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'_1732 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> T__'8801'__12
d_identity'691'_1732 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.identityˡ
d_identity'737'_1734 ::
  (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'_1734 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> T__'8801'__12
d_identity'737'_1734 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.isEquivalence
d_isEquivalence_1736 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_1736 :: T_IsMoufangLoop_3370 -> T_IsEquivalence_28
d_isEquivalence_1736 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.Nat.Properties._.IsMoufangLoop.isLeftBolLoop
d_isLeftBolLoop_1738 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Algebra.Structures.T_IsLeftBolLoop_3202
d_isLeftBolLoop_1738 :: T_IsMoufangLoop_3370 -> T_IsLeftBolLoop_3202
d_isLeftBolLoop_1738 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.Nat.Properties._.IsMoufangLoop.isLoop
d_isLoop_1740 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Algebra.Structures.T_IsLoop_3122
d_isLoop_1740 :: T_IsMoufangLoop_3370 -> T_IsLoop_3122
d_isLoop_1740 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.Nat.Properties._.IsMoufangLoop.isMagma
d_isMagma_1742 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_1742 :: T_IsMoufangLoop_3370 -> T_IsMagma_178
d_isMagma_1742 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.Nat.Properties._.IsMoufangLoop.isPartialEquivalence
d_isPartialEquivalence_1744 ::
  (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_1744 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1744 ~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_1744 T_IsMoufangLoop_3370
v4
du_isPartialEquivalence_1744 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1744 :: T_IsMoufangLoop_3370 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1744 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.Nat.Properties._.IsMoufangLoop.isQuasigroup
d_isQuasigroup_1746 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Algebra.Structures.T_IsQuasigroup_3038
d_isQuasigroup_1746 :: T_IsMoufangLoop_3370 -> T_IsQuasigroup_3038
d_isQuasigroup_1746 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.Nat.Properties._.IsMoufangLoop.leftBol
d_leftBol_1748 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_leftBol_1748 :: T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
d_leftBol_1748 = T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.leftDivides
d_leftDivides_1750 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_leftDivides_1750 :: T_IsMoufangLoop_3370 -> T_Σ_14
d_leftDivides_1750 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.Nat.Properties._.IsMoufangLoop.leftDividesʳ
d_leftDivides'691'_1752 ::
  (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'_1752 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'691'_1752 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.leftDividesˡ
d_leftDivides'737'_1754 ::
  (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'_1754 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
d_leftDivides'737'_1754 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.refl
d_refl_1756 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_refl_1756 :: T_IsMoufangLoop_3370 -> Integer -> T__'8801'__12
d_refl_1756 = T_IsMoufangLoop_3370 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.reflexive
d_reflexive_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 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_reflexive_1758 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
d_reflexive_1758 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMoufangLoop_3370
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.rightBol
d_rightBol_1760 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_rightBol_1760 :: T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
d_rightBol_1760 = T_IsMoufangLoop_3370
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMoufangLoop.rightDivides
d_rightDivides_1762 ::
  MAlonzo.Code.Algebra.Structures.T_IsMoufangLoop_3370 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_rightDivides_1762 :: T_IsMoufangLoop_3370 -> T_Σ_14
d_rightDivides_1762 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_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_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.Nat.Properties._.IsMoufangLoop.rightDividesʳ