{-# 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.Construct.NaturalChoice.Base
import qualified MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp
import qualified MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp
import qualified MAlonzo.Code.Algebra.Morphism
import qualified MAlonzo.Code.Algebra.Structures
import qualified MAlonzo.Code.Data.Bool.Properties
import qualified MAlonzo.Code.Data.Empty
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.Injection
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.Converse
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.Structures
import qualified MAlonzo.Code.Relation.Nullary
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._._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._.Associative
d_Associative_26 :: (Integer -> Integer -> Integer) -> ()
d_Associative_26 :: (Integer -> Integer -> Integer) -> ()
d_Associative_26 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Commutative
d_Commutative_28 :: (Integer -> Integer -> Integer) -> ()
d_Commutative_28 :: (Integer -> Integer -> Integer) -> ()
d_Commutative_28 = (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Identity
d_Identity_40 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Identity_40 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Identity_40 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftIdentity
d_LeftIdentity_52 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftIdentity_52 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftIdentity_52 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.LeftZero
d_LeftZero_56 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftZero_56 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_LeftZero_56 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightIdentity
d_RightIdentity_62 ::
  Integer -> (Integer -> Integer -> Integer) -> ()
d_RightIdentity_62 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightIdentity_62 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.RightZero
d_RightZero_66 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightZero_66 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_RightZero_66 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.Zero
d_Zero_70 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Zero_70 :: Integer -> (Integer -> Integer -> Integer) -> ()
d_Zero_70 = Integer -> (Integer -> Integer -> Integer) -> ()
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup
d_IsAbelianGroup_74 :: p -> p -> p -> ()
d_IsAbelianGroup_74 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsBand
d_IsBand_76 :: p -> ()
d_IsBand_76 p
a0 = ()
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring
d_IsCancellativeCommutativeSemiring_82 :: p -> p -> p -> p -> ()
d_IsCancellativeCommutativeSemiring_82 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsCommutativeMagma
d_IsCommutativeMagma_84 :: p -> ()
d_IsCommutativeMagma_84 p
a0 = ()
-- Data.Nat.Properties._.IsCommutativeMonoid
d_IsCommutativeMonoid_86 :: p -> p -> ()
d_IsCommutativeMonoid_86 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsCommutativeRing
d_IsCommutativeRing_88 :: p -> p -> p -> p -> p -> ()
d_IsCommutativeRing_88 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Nat.Properties._.IsCommutativeSemigroup
d_IsCommutativeSemigroup_90 :: p -> ()
d_IsCommutativeSemigroup_90 p
a0 = ()
-- Data.Nat.Properties._.IsCommutativeSemiring
d_IsCommutativeSemiring_92 :: p -> p -> p -> p -> ()
d_IsCommutativeSemiring_92 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne
d_IsCommutativeSemiringWithoutOne_94 :: p -> p -> p -> ()
d_IsCommutativeSemiringWithoutOne_94 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsGroup
d_IsGroup_98 :: p -> p -> p -> ()
d_IsGroup_98 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid
d_IsIdempotentCommutativeMonoid_100 :: p -> p -> ()
d_IsIdempotentCommutativeMonoid_100 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsMagma
d_IsMagma_104 :: p -> ()
d_IsMagma_104 p
a0 = ()
-- Data.Nat.Properties._.IsMonoid
d_IsMonoid_106 :: p -> p -> ()
d_IsMonoid_106 p
a0 p
a1 = ()
-- Data.Nat.Properties._.IsNearSemiring
d_IsNearSemiring_108 :: p -> p -> p -> ()
d_IsNearSemiring_108 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsRing
d_IsRing_110 :: p -> p -> p -> p -> p -> ()
d_IsRing_110 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Data.Nat.Properties._.IsSelectiveMagma
d_IsSelectiveMagma_112 :: p -> ()
d_IsSelectiveMagma_112 p
a0 = ()
-- Data.Nat.Properties._.IsSemigroup
d_IsSemigroup_114 :: p -> ()
d_IsSemigroup_114 p
a0 = ()
-- Data.Nat.Properties._.IsSemilattice
d_IsSemilattice_116 :: p -> ()
d_IsSemilattice_116 p
a0 = ()
-- Data.Nat.Properties._.IsSemiring
d_IsSemiring_118 :: p -> p -> p -> p -> ()
d_IsSemiring_118 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero
d_IsSemiringWithoutAnnihilatingZero_120 :: p -> p -> p -> p -> ()
d_IsSemiringWithoutAnnihilatingZero_120 p
a0 p
a1 p
a2 p
a3 = ()
-- Data.Nat.Properties._.IsSemiringWithoutOne
d_IsSemiringWithoutOne_122 :: p -> p -> p -> ()
d_IsSemiringWithoutOne_122 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties._.IsAbelianGroup.assoc
d_assoc_128 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_128 :: T_IsAbelianGroup_662
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_128 = T_IsAbelianGroup_662
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.comm
d_comm_130 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_130 :: T_IsAbelianGroup_662 -> Integer -> Integer -> T__'8801'__12
d_comm_130 = T_IsAbelianGroup_662 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.identity
d_identity_132 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_132 :: T_IsAbelianGroup_662 -> T_Σ_14
d_identity_132 T_IsAbelianGroup_662
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
         ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674 (T_IsAbelianGroup_662 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662
v0)))
-- Data.Nat.Properties._.IsAbelianGroup.inverse
d_inverse_138 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_138 :: T_IsAbelianGroup_662 -> T_Σ_14
d_inverse_138 T_IsAbelianGroup_662
v0
  = (T_IsGroup_580 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsGroup_580 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_596
      ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674 (T_IsAbelianGroup_662 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662
v0))
-- Data.Nat.Properties._.IsAbelianGroup.isEquivalence
d_isEquivalence_150 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_150 :: T_IsAbelianGroup_662 -> T_IsEquivalence_26
d_isEquivalence_150 T_IsAbelianGroup_662
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
               ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674 (T_IsAbelianGroup_662 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662
v0)))))
-- Data.Nat.Properties._.IsAbelianGroup.isGroup
d_isGroup_152 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580
d_isGroup_152 :: T_IsAbelianGroup_662 -> T_IsGroup_580
d_isGroup_152 T_IsAbelianGroup_662
v0
  = (T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> T_IsGroup_580
forall a b. a -> b
coe T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674 (T_IsAbelianGroup_662 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662
v0)
-- Data.Nat.Properties._.IsAbelianGroup.isMagma
d_isMagma_154 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_154 :: T_IsAbelianGroup_662 -> T_IsMagma_86
d_isMagma_154 T_IsAbelianGroup_662
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
            ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674 (T_IsAbelianGroup_662 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662
v0))))
-- Data.Nat.Properties._.IsAbelianGroup.isMonoid
d_isMonoid_156 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_156 :: T_IsAbelianGroup_662 -> T_IsMonoid_358
d_isMonoid_156 T_IsAbelianGroup_662
v0
  = (T_IsGroup_580 -> T_IsMonoid_358) -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
      ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674 (T_IsAbelianGroup_662 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662
v0))
-- Data.Nat.Properties._.IsAbelianGroup.isSemigroup
d_isSemigroup_160 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_160 :: T_IsAbelianGroup_662 -> T_IsSemigroup_194
d_isSemigroup_160 T_IsAbelianGroup_662
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
         ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674 (T_IsAbelianGroup_662 -> Any
forall a b. a -> b
coe T_IsAbelianGroup_662
v0)))
-- Data.Nat.Properties._.IsAbelianGroup.⁻¹-cong
d_'8315''185''45'cong_176 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_176 :: T_IsAbelianGroup_662
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_176 = T_IsAbelianGroup_662
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsAbelianGroup.∙-cong
d_'8729''45'cong_178 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662 ->
  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_178 :: T_IsAbelianGroup_662
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_178 = T_IsAbelianGroup_662
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.assoc
d_assoc_186 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_230 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_186 :: T_IsBand_230 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_186 = T_IsBand_230 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.idem
d_idem_188 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_230 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_188 :: T_IsBand_230 -> Integer -> T__'8801'__12
d_idem_188 = T_IsBand_230 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBand.isEquivalence
d_isEquivalence_190 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_230 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_190 :: T_IsBand_230 -> T_IsEquivalence_26
d_isEquivalence_190 T_IsBand_230
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsBand_230 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe T_IsBand_230 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_238 (T_IsBand_230 -> Any
forall a b. a -> b
coe T_IsBand_230
v0)))
-- Data.Nat.Properties._.IsBand.isMagma
d_isMagma_192 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_230 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_192 :: T_IsBand_230 -> T_IsMagma_86
d_isMagma_192 T_IsBand_230
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsBand_230 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe T_IsBand_230 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_238 (T_IsBand_230 -> Any
forall a b. a -> b
coe T_IsBand_230
v0))
-- Data.Nat.Properties._.IsBand.isSemigroup
d_isSemigroup_196 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_230 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_196 :: T_IsBand_230 -> T_IsSemigroup_194
d_isSemigroup_196 T_IsBand_230
v0
  = (T_IsBand_230 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe T_IsBand_230 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_238 (T_IsBand_230 -> Any
forall a b. a -> b
coe T_IsBand_230
v0)
-- Data.Nat.Properties._.IsBand.∙-cong
d_'8729''45'cong_208 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_230 ->
  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_208 :: T_IsBand_230
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_208 = T_IsBand_230
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBoundedLattice.assoc
d_assoc_270 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_270 :: T_IsIdempotentCommutativeMonoid_464
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_270 = T_IsIdempotentCommutativeMonoid_464
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBoundedLattice.comm
d_comm_272 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_272 :: T_IsIdempotentCommutativeMonoid_464
-> Integer -> Integer -> T__'8801'__12
d_comm_272 = T_IsIdempotentCommutativeMonoid_464
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBoundedLattice.idem
d_idem_274 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_274 :: T_IsIdempotentCommutativeMonoid_464 -> Integer -> T__'8801'__12
d_idem_274 = T_IsIdempotentCommutativeMonoid_464 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsBoundedLattice.identity
d_identity_276 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_276 :: T_IsIdempotentCommutativeMonoid_464 -> T_Σ_14
d_identity_276 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474
            (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0)))
-- Data.Nat.Properties._.IsBoundedLattice.isCommutativeMonoid
d_isCommutativeMonoid_284 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_isCommutativeMonoid_284 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
d_isCommutativeMonoid_284 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474 (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0)
-- Data.Nat.Properties._.IsBoundedLattice.isEquivalence
d_isEquivalence_288 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_288 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsEquivalence_26
d_isEquivalence_288 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
               ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474
                  (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0)))))
-- Data.Nat.Properties._.IsBoundedLattice.isMagma
d_isMagma_290 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_290 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsMagma_86
d_isMagma_290 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474
               (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0))))
-- Data.Nat.Properties._.IsBoundedLattice.isMonoid
d_isMonoid_292 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_292 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsMonoid_358
d_isMonoid_292 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
      ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474 (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0))
-- Data.Nat.Properties._.IsBoundedLattice.isSemigroup
d_isSemigroup_296 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_296 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsSemigroup_194
d_isSemigroup_296 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474
            (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0)))
-- Data.Nat.Properties._.IsBoundedLattice.∙-cong
d_'8729''45'cong_308 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  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_308 :: T_IsIdempotentCommutativeMonoid_464
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_308 = T_IsIdempotentCommutativeMonoid_464
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.assoc
d_assoc_316 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_316 :: T_IsCancellativeCommutativeSemiring_1462
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_316 = T_IsCancellativeCommutativeSemiring_1462
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.*-cancelˡ-nonZero
d_'42''45'cancel'737''45'nonZero_318 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  Integer ->
  Integer ->
  Integer ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cancel'737''45'nonZero_318 :: T_IsCancellativeCommutativeSemiring_1462
-> Integer
-> Integer
-> Integer
-> (T__'8801'__12 -> T_'8869'_4)
-> T__'8801'__12
-> T__'8801'__12
d_'42''45'cancel'737''45'nonZero_318 = T_IsCancellativeCommutativeSemiring_1462
-> Integer
-> Integer
-> Integer
-> (T__'8801'__12 -> T_'8869'_4)
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.*-comm
d_'42''45'comm_320 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_320 :: T_IsCancellativeCommutativeSemiring_1462
-> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_320 = T_IsCancellativeCommutativeSemiring_1462
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.∙-cong
d_'8729''45'cong_322 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  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_322 :: T_IsCancellativeCommutativeSemiring_1462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_322 = T_IsCancellativeCommutativeSemiring_1462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.identity
d_identity_328 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_328 :: T_IsCancellativeCommutativeSemiring_1462 -> T_Σ_14
d_identity_328 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
               ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
                  (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0)))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isMagma
d_isMagma_340 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_340 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsMagma_86
d_isMagma_340 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
                  ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
                     (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0))))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.*-isMonoid
d_'42''45'isMonoid_342 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'42''45'isMonoid_342 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsMonoid_358
d_'42''45'isMonoid_342 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
            ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
               (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemigroup
d_isSemigroup_344 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_344 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsSemigroup_194
d_isSemigroup_344 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
               ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
                  (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0)))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.assoc
d_assoc_346 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_346 :: T_IsCancellativeCommutativeSemiring_1462
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_346 = T_IsCancellativeCommutativeSemiring_1462
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.comm
d_comm_348 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_348 :: T_IsCancellativeCommutativeSemiring_1462
-> Integer -> Integer -> T__'8801'__12
d_comm_348 = T_IsCancellativeCommutativeSemiring_1462
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.∙-cong
d_'8729''45'cong_350 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  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_350 :: T_IsCancellativeCommutativeSemiring_1462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_350 = T_IsCancellativeCommutativeSemiring_1462
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.identity
d_identity_356 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_356 :: T_IsCancellativeCommutativeSemiring_1462 -> T_Σ_14
d_identity_356 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
                  ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
                     (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0))))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_364 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_364 :: T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_364 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
            ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
               (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isMagma
d_isMagma_368 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_368 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsMagma_86
d_isMagma_368 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
               ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
                  ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
                     ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
                        T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
                        (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0)))))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isMonoid
d_isMonoid_370 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_370 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsMonoid_358
d_isMonoid_370 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
      ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
               ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
                  (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0)))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemigroup
d_isSemigroup_372 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_372 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsSemigroup_194
d_isSemigroup_372 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
                  ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
                     (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0))))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.distrib
d_distrib_374 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_374 :: T_IsCancellativeCommutativeSemiring_1462 -> T_Σ_14
d_distrib_374 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1162
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
            ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
               T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
               (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isCommutativeSemiring
d_isCommutativeSemiring_380 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344
d_isCommutativeSemiring_380 :: T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
d_isCommutativeSemiring_380 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> T_IsCommutativeSemiring_1344
forall a b. a -> b
coe
      T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
      (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0)
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isEquivalence
d_isEquivalence_384 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_384 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsEquivalence_26
d_isEquivalence_384 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
               ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
                  ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
                     ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
                        T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
                        ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
                           T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
                           (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0))))))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isNearSemiring
d_isNearSemiring_386 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
d_isNearSemiring_386 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1462
-> T_IsNearSemiring_876
d_isNearSemiring_386 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1462
v4 = T_IsCancellativeCommutativeSemiring_1462 -> T_IsNearSemiring_876
du_isNearSemiring_386 T_IsCancellativeCommutativeSemiring_1462
v4
du_isNearSemiring_386 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
du_isNearSemiring_386 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsNearSemiring_876
du_isNearSemiring_386 T_IsCancellativeCommutativeSemiring_1462
v0
  = let v1 :: T_IsCommutativeSemiring_1344
v1
          = T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
              (T_IsCancellativeCommutativeSemiring_1462
-> T_IsCancellativeCommutativeSemiring_1462
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0) in
    Any -> T_IsNearSemiring_876
forall a b. a -> b
coe
      (let v2 :: T_IsSemiring_1238
v2
             = T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> T_IsCommutativeSemiring_1344
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_990
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952) -> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
               (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v2))))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemiring
d_isSemiring_390 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238
d_isSemiring_390 :: T_IsCancellativeCommutativeSemiring_1462 -> T_IsSemiring_1238
d_isSemiring_390 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238)
-> Any -> T_IsSemiring_1238
forall a b. a -> b
coe
      T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
      ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
         (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_392 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142
d_isSemiringWithoutAnnihilatingZero_392 :: T_IsCancellativeCommutativeSemiring_1462
-> T_IsSemiringWithoutAnnihilatingZero_1142
d_isSemiringWithoutAnnihilatingZero_392 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> T_IsSemiringWithoutAnnihilatingZero_1142
forall a b. a -> b
coe
      T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
      ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
         ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
            (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0)))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_394 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_394 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCancellativeCommutativeSemiring_1462
-> T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_394 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCancellativeCommutativeSemiring_1462
v4
  = T_IsCancellativeCommutativeSemiring_1462
-> T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_394 T_IsCancellativeCommutativeSemiring_1462
v4
du_isSemiringWithoutOne_394 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_394 :: T_IsCancellativeCommutativeSemiring_1462
-> T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_394 T_IsCancellativeCommutativeSemiring_1462
v0
  = let v1 :: T_IsCommutativeSemiring_1344
v1
          = T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
              (T_IsCancellativeCommutativeSemiring_1462
-> T_IsCancellativeCommutativeSemiring_1462
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0) in
    Any -> T_IsSemiringWithoutOne_952
forall a b. a -> b
coe
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
         ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v1)))
-- Data.Nat.Properties._.IsCancellativeCommutativeSemiring.zero
d_zero_406 ::
  MAlonzo.Code.Algebra.Structures.T_IsCancellativeCommutativeSemiring_1462 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_406 :: T_IsCancellativeCommutativeSemiring_1462 -> T_Σ_14
d_zero_406 T_IsCancellativeCommutativeSemiring_1462
v0
  = (T_IsSemiring_1238 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiring_1238 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1254
      ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358
         ((T_IsCancellativeCommutativeSemiring_1462
 -> T_IsCommutativeSemiring_1344)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCancellativeCommutativeSemiring_1462
-> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.d_isCommutativeSemiring_1476
            (T_IsCancellativeCommutativeSemiring_1462 -> Any
forall a b. a -> b
coe T_IsCancellativeCommutativeSemiring_1462
v0)))
-- Data.Nat.Properties._.IsCommutativeMagma.comm
d_comm_414 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_122 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_414 :: T_IsCommutativeMagma_122 -> Integer -> Integer -> T__'8801'__12
d_comm_414 = T_IsCommutativeMagma_122 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMagma.isEquivalence
d_isEquivalence_416 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_122 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_416 :: T_IsCommutativeMagma_122 -> T_IsEquivalence_26
d_isEquivalence_416 T_IsCommutativeMagma_122
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsCommutativeMagma_122 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMagma_122 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_130 (T_IsCommutativeMagma_122 -> Any
forall a b. a -> b
coe T_IsCommutativeMagma_122
v0))
-- Data.Nat.Properties._.IsCommutativeMagma.isMagma
d_isMagma_418 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_122 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_418 :: T_IsCommutativeMagma_122 -> T_IsMagma_86
d_isMagma_418 T_IsCommutativeMagma_122
v0
  = (T_IsCommutativeMagma_122 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe T_IsCommutativeMagma_122 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_130 (T_IsCommutativeMagma_122 -> Any
forall a b. a -> b
coe T_IsCommutativeMagma_122
v0)
-- Data.Nat.Properties._.IsCommutativeMagma.∙-cong
d_'8729''45'cong_432 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_122 ->
  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_432 :: T_IsCommutativeMagma_122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_432 = T_IsCommutativeMagma_122
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.assoc
d_assoc_440 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_440 :: T_IsCommutativeMonoid_406
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_440 = T_IsCommutativeMonoid_406
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.comm
d_comm_442 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_442 :: T_IsCommutativeMonoid_406 -> Integer -> Integer -> T__'8801'__12
d_comm_442 = T_IsCommutativeMonoid_406 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeMonoid.identity
d_identity_444 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_444 :: T_IsCommutativeMonoid_406 -> T_Σ_14
d_identity_444 T_IsCommutativeMonoid_406
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416 (T_IsCommutativeMonoid_406 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406
v0))
-- Data.Nat.Properties._.IsCommutativeMonoid.isEquivalence
d_isEquivalence_454 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_454 :: T_IsCommutativeMonoid_406 -> T_IsEquivalence_26
d_isEquivalence_454 T_IsCommutativeMonoid_406
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416 (T_IsCommutativeMonoid_406 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406
v0))))
-- Data.Nat.Properties._.IsCommutativeMonoid.isMagma
d_isMagma_456 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_456 :: T_IsCommutativeMonoid_406 -> T_IsMagma_86
d_isMagma_456 T_IsCommutativeMonoid_406
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416 (T_IsCommutativeMonoid_406 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406
v0)))
-- Data.Nat.Properties._.IsCommutativeMonoid.isMonoid
d_isMonoid_458 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_458 :: T_IsCommutativeMonoid_406 -> T_IsMonoid_358
d_isMonoid_458 T_IsCommutativeMonoid_406
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416 (T_IsCommutativeMonoid_406 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406
v0)
-- Data.Nat.Properties._.IsCommutativeMonoid.isSemigroup
d_isSemigroup_462 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_462 :: T_IsCommutativeMonoid_406 -> T_IsSemigroup_194
d_isSemigroup_462 T_IsCommutativeMonoid_406
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416 (T_IsCommutativeMonoid_406 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406
v0))
-- Data.Nat.Properties._.IsCommutativeMonoid.∙-cong
d_'8729''45'cong_474 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_474 :: T_IsCommutativeMonoid_406
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_474 = T_IsCommutativeMonoid_406
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.assoc
d_assoc_484 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_484 :: T_IsCommutativeRing_1720
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_484 = T_IsCommutativeRing_1720
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.*-comm
d_'42''45'comm_486 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_486 :: T_IsCommutativeRing_1720 -> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_486 = T_IsCommutativeRing_1720 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.∙-cong
d_'8729''45'cong_488 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  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_488 :: T_IsCommutativeRing_1720
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_488 = T_IsCommutativeRing_1720
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.identity
d_identity_494 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_494 :: T_IsCommutativeRing_1720 -> T_Σ_14
d_identity_494 T_IsCommutativeRing_1720
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsRing_1584 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsRing_1584 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1606
         ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0)))
-- Data.Nat.Properties._.IsCommutativeRing.isMagma
d_isMagma_506 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_506 :: T_IsCommutativeRing_1720 -> T_IsMagma_86
d_isMagma_506 T_IsCommutativeRing_1720
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsRing_1584 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_1584 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1606
            ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))))
-- Data.Nat.Properties._.IsCommutativeRing.*-isMonoid
d_'42''45'isMonoid_508 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'42''45'isMonoid_508 :: T_IsCommutativeRing_1720 -> T_IsMonoid_358
d_'42''45'isMonoid_508 T_IsCommutativeRing_1720
v0
  = (T_IsRing_1584 -> T_IsMonoid_358) -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsRing_1584 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1606
      ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))
-- Data.Nat.Properties._.IsCommutativeRing.isSemigroup
d_isSemigroup_510 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_510 :: T_IsCommutativeRing_1720 -> T_IsSemigroup_194
d_isSemigroup_510 T_IsCommutativeRing_1720
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsRing_1584 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsRing_1584 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1606
         ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0)))
-- Data.Nat.Properties._.IsCommutativeRing.assoc
d_assoc_512 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_512 :: T_IsCommutativeRing_1720
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_512 = T_IsCommutativeRing_1720
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.comm
d_comm_514 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_514 :: T_IsCommutativeRing_1720 -> Integer -> Integer -> T__'8801'__12
d_comm_514 = T_IsCommutativeRing_1720 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.∙-cong
d_'8729''45'cong_516 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  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_516 :: T_IsCommutativeRing_1720
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_516 = T_IsCommutativeRing_1720
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.identity
d_identity_522 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_522 :: T_IsCommutativeRing_1720 -> T_Σ_14
d_identity_522 T_IsCommutativeRing_1720
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
         ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
            T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
            ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
               T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
               ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0)))))
-- Data.Nat.Properties._.IsCommutativeRing.+-isAbelianGroup
d_'43''45'isAbelianGroup_528 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662
d_'43''45'isAbelianGroup_528 :: T_IsCommutativeRing_1720 -> T_IsAbelianGroup_662
d_'43''45'isAbelianGroup_528 T_IsCommutativeRing_1720
v0
  = (T_IsRing_1584 -> T_IsAbelianGroup_662)
-> Any -> T_IsAbelianGroup_662
forall a b. a -> b
coe
      T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
      ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))
-- Data.Nat.Properties._.IsCommutativeRing.isGroup
d_isGroup_536 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580
d_isGroup_536 :: T_IsCommutativeRing_1720 -> T_IsGroup_580
d_isGroup_536 T_IsCommutativeRing_1720
v0
  = (T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> T_IsGroup_580
forall a b. a -> b
coe
      T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
      ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
         T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
         ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0)))
-- Data.Nat.Properties._.IsCommutativeRing.isMagma
d_isMagma_538 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_538 :: T_IsCommutativeRing_1720 -> T_IsMagma_86
d_isMagma_538 T_IsCommutativeRing_1720
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
            ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
               T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
               ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
                  T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
                  ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))))))
-- Data.Nat.Properties._.IsCommutativeRing.isMonoid
d_isMonoid_540 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_540 :: T_IsCommutativeRing_1720 -> T_IsMonoid_358
d_isMonoid_540 T_IsCommutativeRing_1720
v0
  = (T_IsGroup_580 -> T_IsMonoid_358) -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
      ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
         T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
         ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
            ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))))
-- Data.Nat.Properties._.IsCommutativeRing.isSemigroup
d_isSemigroup_542 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_542 :: T_IsCommutativeRing_1720 -> T_IsSemigroup_194
d_isSemigroup_542 T_IsCommutativeRing_1720
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
         ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
            T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
            ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
               T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
               ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0)))))
-- Data.Nat.Properties._.IsCommutativeRing.⁻¹-cong
d_'8315''185''45'cong_544 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_544 :: T_IsCommutativeRing_1720
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_544 = T_IsCommutativeRing_1720
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeRing.inverse
d_inverse_546 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_546 :: T_IsCommutativeRing_1720 -> T_Σ_14
d_inverse_546 T_IsCommutativeRing_1720
v0
  = (T_IsGroup_580 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsGroup_580 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_596
      ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
         T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
         ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
            ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))))
-- Data.Nat.Properties._.IsCommutativeRing.distrib
d_distrib_552 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_552 :: T_IsCommutativeRing_1720 -> T_Σ_14
d_distrib_552 T_IsCommutativeRing_1720
v0
  = (T_IsRing_1584 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsRing_1584 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1608
      ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))
-- Data.Nat.Properties._.IsCommutativeRing.isEquivalence
d_isEquivalence_562 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_562 :: T_IsCommutativeRing_1720 -> T_IsEquivalence_26
d_isEquivalence_562 T_IsCommutativeRing_1720
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
               ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
                  T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
                  ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
                     T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
                     ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0)))))))
-- Data.Nat.Properties._.IsCommutativeRing.isNearSemiring
d_isNearSemiring_564 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
d_isNearSemiring_564 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_1720
-> T_IsNearSemiring_876
d_isNearSemiring_564 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_1720
v5
  = T_IsCommutativeRing_1720 -> T_IsNearSemiring_876
du_isNearSemiring_564 T_IsCommutativeRing_1720
v5
du_isNearSemiring_564 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
du_isNearSemiring_564 :: T_IsCommutativeRing_1720 -> T_IsNearSemiring_876
du_isNearSemiring_564 T_IsCommutativeRing_1720
v0
  = let v1 :: T_IsRing_1584
v1
          = T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> T_IsCommutativeRing_1720
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0) in
    Any -> T_IsNearSemiring_876
forall a b. a -> b
coe
      (let v2 :: t
v2
             = (T_IsRing_1584 -> T_IsSemiring_1238) -> Any -> t
forall a b. a -> b
coe
                 T_IsRing_1584 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.du_isSemiring_1698 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v1) in
       Any -> Any
forall a b. a -> b
coe
         ((T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_990
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952) -> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
               (Any -> Any
forall a b. a -> b
coe Any
forall a. a
v2))))
-- Data.Nat.Properties._.IsCommutativeRing.isRing
d_isRing_568 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584
d_isRing_568 :: T_IsCommutativeRing_1720 -> T_IsRing_1584
d_isRing_568 T_IsCommutativeRing_1720
v0
  = (T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> T_IsRing_1584
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0)
-- Data.Nat.Properties._.IsCommutativeRing.isSemiring
d_isSemiring_570 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238
d_isSemiring_570 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_1720
-> T_IsSemiring_1238
d_isSemiring_570 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_1720
v5 = T_IsCommutativeRing_1720 -> T_IsSemiring_1238
du_isSemiring_570 T_IsCommutativeRing_1720
v5
du_isSemiring_570 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238
du_isSemiring_570 :: T_IsCommutativeRing_1720 -> T_IsSemiring_1238
du_isSemiring_570 T_IsCommutativeRing_1720
v0
  = (T_IsRing_1584 -> T_IsSemiring_1238) -> Any -> T_IsSemiring_1238
forall a b. a -> b
coe
      T_IsRing_1584 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.du_isSemiring_1698
      ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))
-- Data.Nat.Properties._.IsCommutativeRing.isSemiringWithoutOne
d_isSemiringWithoutOne_574 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_574 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeRing_1720
-> T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_574 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsCommutativeRing_1720
v5
  = T_IsCommutativeRing_1720 -> T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_574 T_IsCommutativeRing_1720
v5
du_isSemiringWithoutOne_574 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_574 :: T_IsCommutativeRing_1720 -> T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_574 T_IsCommutativeRing_1720
v0
  = let v1 :: T_IsRing_1584
v1
          = T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> T_IsCommutativeRing_1720
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0) in
    Any -> T_IsSemiringWithoutOne_952
forall a b. a -> b
coe
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
         ((T_IsRing_1584 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsRing_1584 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.du_isSemiring_1698 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v1)))
-- Data.Nat.Properties._.IsCommutativeRing.zero
d_zero_590 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeRing_1720 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_590 :: T_IsCommutativeRing_1720 -> T_Σ_14
d_zero_590 T_IsCommutativeRing_1720
v0
  = (T_IsRing_1584 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsRing_1584 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1610
      ((T_IsCommutativeRing_1720 -> T_IsRing_1584) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720 -> T_IsRing_1584
MAlonzo.Code.Algebra.Structures.d_isRing_1736 (T_IsCommutativeRing_1720 -> Any
forall a b. a -> b
coe T_IsCommutativeRing_1720
v0))
-- Data.Nat.Properties._.IsCommutativeSemigroup.assoc
d_assoc_598 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_598 :: T_IsCommutativeSemigroup_270
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_598 = T_IsCommutativeSemigroup_270
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.comm
d_comm_600 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_600 :: T_IsCommutativeSemigroup_270 -> Integer -> Integer -> T__'8801'__12
d_comm_600 = T_IsCommutativeSemigroup_270 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemigroup.isEquivalence
d_isEquivalence_604 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_604 :: T_IsCommutativeSemigroup_270 -> T_IsEquivalence_26
d_isEquivalence_604 T_IsCommutativeSemigroup_270
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsCommutativeSemigroup_270 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_270 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_278 (T_IsCommutativeSemigroup_270 -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_270
v0)))
-- Data.Nat.Properties._.IsCommutativeSemigroup.isMagma
d_isMagma_606 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_606 :: T_IsCommutativeSemigroup_270 -> T_IsMagma_86
d_isMagma_606 T_IsCommutativeSemigroup_270
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsCommutativeSemigroup_270 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_270 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_278 (T_IsCommutativeSemigroup_270 -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_270
v0))
-- Data.Nat.Properties._.IsCommutativeSemigroup.isSemigroup
d_isSemigroup_610 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_610 :: T_IsCommutativeSemigroup_270 -> T_IsSemigroup_194
d_isSemigroup_610 T_IsCommutativeSemigroup_270
v0
  = (T_IsCommutativeSemigroup_270 -> T_IsSemigroup_194)
-> Any -> T_IsSemigroup_194
forall a b. a -> b
coe T_IsCommutativeSemigroup_270 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_278 (T_IsCommutativeSemigroup_270 -> Any
forall a b. a -> b
coe T_IsCommutativeSemigroup_270
v0)
-- Data.Nat.Properties._.IsCommutativeSemigroup.∙-cong
d_'8729''45'cong_622 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270 ->
  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_622 :: T_IsCommutativeSemigroup_270
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_622 = T_IsCommutativeSemigroup_270
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.assoc
d_assoc_630 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_630 :: T_IsCommutativeSemiring_1344
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_630 = T_IsCommutativeSemiring_1344
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.*-comm
d_'42''45'comm_632 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_632 :: T_IsCommutativeSemiring_1344 -> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_632 = T_IsCommutativeSemiring_1344 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.∙-cong
d_'8729''45'cong_634 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  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_IsCommutativeSemiring_1344
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_634 = T_IsCommutativeSemiring_1344
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.identity
d_identity_640 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_640 :: T_IsCommutativeSemiring_1344 -> T_Σ_14
d_identity_640 T_IsCommutativeSemiring_1344
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0))))
-- Data.Nat.Properties._.IsCommutativeSemiring.isMagma
d_isMagma_652 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_652 :: T_IsCommutativeSemiring_1344 -> T_IsMagma_86
d_isMagma_652 T_IsCommutativeSemiring_1344
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0)))))
-- Data.Nat.Properties._.IsCommutativeSemiring.*-isMonoid
d_'42''45'isMonoid_654 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'42''45'isMonoid_654 :: T_IsCommutativeSemiring_1344 -> T_IsMonoid_358
d_'42''45'isMonoid_654 T_IsCommutativeSemiring_1344
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0)))
-- Data.Nat.Properties._.IsCommutativeSemiring.isSemigroup
d_isSemigroup_656 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_656 :: T_IsCommutativeSemiring_1344 -> T_IsSemigroup_194
d_isSemigroup_656 T_IsCommutativeSemiring_1344
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0))))
-- Data.Nat.Properties._.IsCommutativeSemiring.assoc
d_assoc_658 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_658 :: T_IsCommutativeSemiring_1344
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_658 = T_IsCommutativeSemiring_1344
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.comm
d_comm_660 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_660 :: T_IsCommutativeSemiring_1344 -> Integer -> Integer -> T__'8801'__12
d_comm_660 = T_IsCommutativeSemiring_1344 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.∙-cong
d_'8729''45'cong_662 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  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_662 :: T_IsCommutativeSemiring_1344
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_662 = T_IsCommutativeSemiring_1344
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiring.identity
d_identity_668 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_668 :: T_IsCommutativeSemiring_1344 -> T_Σ_14
d_identity_668 T_IsCommutativeSemiring_1344
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0)))))
-- Data.Nat.Properties._.IsCommutativeSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_676 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_676 :: T_IsCommutativeSemiring_1344 -> T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_676 T_IsCommutativeSemiring_1344
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0)))
-- Data.Nat.Properties._.IsCommutativeSemiring.isMagma
d_isMagma_680 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_680 :: T_IsCommutativeSemiring_1344 -> T_IsMagma_86
d_isMagma_680 T_IsCommutativeSemiring_1344
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
               ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
                  ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
                     T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0))))))
-- Data.Nat.Properties._.IsCommutativeSemiring.isMonoid
d_isMonoid_682 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_682 :: T_IsCommutativeSemiring_1344 -> T_IsMonoid_358
d_isMonoid_682 T_IsCommutativeSemiring_1344
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
      ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0))))
-- Data.Nat.Properties._.IsCommutativeSemiring.isSemigroup
d_isSemigroup_684 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_684 :: T_IsCommutativeSemiring_1344 -> T_IsSemigroup_194
d_isSemigroup_684 T_IsCommutativeSemiring_1344
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0)))))
-- Data.Nat.Properties._.IsCommutativeSemiring.distrib
d_distrib_686 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_686 :: T_IsCommutativeSemiring_1344 -> T_Σ_14
d_distrib_686 T_IsCommutativeSemiring_1344
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1162
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0)))
-- Data.Nat.Properties._.IsCommutativeSemiring.isEquivalence
d_isEquivalence_694 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_694 :: T_IsCommutativeSemiring_1344 -> T_IsEquivalence_26
d_isEquivalence_694 T_IsCommutativeSemiring_1344
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
               ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
                  ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
                     ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe
                        T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0)))))))
-- Data.Nat.Properties._.IsCommutativeSemiring.isNearSemiring
d_isNearSemiring_696 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
d_isNearSemiring_696 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1344
-> T_IsNearSemiring_876
d_isNearSemiring_696 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1344
v4 = T_IsCommutativeSemiring_1344 -> T_IsNearSemiring_876
du_isNearSemiring_696 T_IsCommutativeSemiring_1344
v4
du_isNearSemiring_696 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
du_isNearSemiring_696 :: T_IsCommutativeSemiring_1344 -> T_IsNearSemiring_876
du_isNearSemiring_696 T_IsCommutativeSemiring_1344
v0
  = let v1 :: T_IsSemiring_1238
v1
          = T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> T_IsCommutativeSemiring_1344
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0) in
    Any -> T_IsNearSemiring_876
forall a b. a -> b
coe
      ((T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_990
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
            (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v1)))
-- Data.Nat.Properties._.IsCommutativeSemiring.isSemiring
d_isSemiring_700 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238
d_isSemiring_700 :: T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
d_isSemiring_700 T_IsCommutativeSemiring_1344
v0
  = (T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238)
-> Any -> T_IsSemiring_1238
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0)
-- Data.Nat.Properties._.IsCommutativeSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_702 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142
d_isSemiringWithoutAnnihilatingZero_702 :: T_IsCommutativeSemiring_1344
-> T_IsSemiringWithoutAnnihilatingZero_1142
d_isSemiringWithoutAnnihilatingZero_702 T_IsCommutativeSemiring_1344
v0
  = (T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> T_IsSemiringWithoutAnnihilatingZero_1142
forall a b. a -> b
coe
      T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
      ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0))
-- Data.Nat.Properties._.IsCommutativeSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_704 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_704 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1344
-> T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_704 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsCommutativeSemiring_1344
v4
  = T_IsCommutativeSemiring_1344 -> T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_704 T_IsCommutativeSemiring_1344
v4
du_isSemiringWithoutOne_704 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_704 :: T_IsCommutativeSemiring_1344 -> T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_704 T_IsCommutativeSemiring_1344
v0
  = (T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952)
-> Any -> T_IsSemiringWithoutOne_952
forall a b. a -> b
coe
      T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
      ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0))
-- Data.Nat.Properties._.IsCommutativeSemiring.zero
d_zero_716 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_716 :: T_IsCommutativeSemiring_1344 -> T_Σ_14
d_zero_716 T_IsCommutativeSemiring_1344
v0
  = (T_IsSemiring_1238 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiring_1238 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1254
      ((T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.d_isSemiring_1358 (T_IsCommutativeSemiring_1344 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
v0))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.assoc
d_assoc_724 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_724 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_assoc_724 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-comm
d_'42''45'comm_726 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_726 :: T_IsCommutativeSemiringWithoutOne_1044
-> Integer -> Integer -> T__'8801'__12
d_'42''45'comm_726 = T_IsCommutativeSemiringWithoutOne_1044
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.∙-cong
d_'8729''45'cong_728 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  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_728 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_728 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isMagma
d_isMagma_738 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_738 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> T_IsMagma_86
d_isMagma_738 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1044
v3 = T_IsCommutativeSemiringWithoutOne_1044 -> T_IsMagma_86
du_isMagma_738 T_IsCommutativeSemiringWithoutOne_1044
v3
du_isMagma_738 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
du_isMagma_738 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_IsMagma_86
du_isMagma_738 T_IsCommutativeSemiringWithoutOne_1044
v0
  = let v1 :: T_IsSemiringWithoutOne_952
v1
          = T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
              (T_IsCommutativeSemiringWithoutOne_1044
-> T_IsCommutativeSemiringWithoutOne_1044
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0) in
    Any -> T_IsMagma_86
forall a b. a -> b
coe
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_'42''45'isSemigroup_970
            (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v1)))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-isSemigroup
d_'42''45'isSemigroup_740 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_'42''45'isSemigroup_740 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_IsSemigroup_194
d_'42''45'isSemigroup_740 T_IsCommutativeSemiringWithoutOne_1044
v0
  = (T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194)
-> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_'42''45'isSemigroup_970
      ((T_IsCommutativeSemiringWithoutOne_1044
 -> T_IsSemiringWithoutOne_952)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
         (T_IsCommutativeSemiringWithoutOne_1044 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.assoc
d_assoc_742 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_742 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_assoc_742 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.comm
d_comm_744 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_744 :: T_IsCommutativeSemiringWithoutOne_1044
-> Integer -> Integer -> T__'8801'__12
d_comm_744 = T_IsCommutativeSemiringWithoutOne_1044
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.∙-cong
d_'8729''45'cong_746 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  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_746 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_746 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.identity
d_identity_752 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_752 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> T_Σ_14
d_identity_752 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1044
v3 = T_IsCommutativeSemiringWithoutOne_1044 -> T_Σ_14
du_identity_752 T_IsCommutativeSemiringWithoutOne_1044
v3
du_identity_752 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_identity_752 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_Σ_14
du_identity_752 T_IsCommutativeSemiringWithoutOne_1044
v0
  = let v1 :: T_IsSemiringWithoutOne_952
v1
          = T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
              (T_IsCommutativeSemiringWithoutOne_1044
-> T_IsCommutativeSemiringWithoutOne_1044
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0) in
    Any -> T_Σ_14
forall a b. a -> b
coe
      ((T_IsMonoid_358 -> T_Σ_14) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
               (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v1))))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_760 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_760 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_760 T_IsCommutativeSemiringWithoutOne_1044
v0
  = (T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
      ((T_IsCommutativeSemiringWithoutOne_1044
 -> T_IsSemiringWithoutOne_952)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
         (T_IsCommutativeSemiringWithoutOne_1044 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isMagma
d_isMagma_764 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_764 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> T_IsMagma_86
d_isMagma_764 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1044
v3 = T_IsCommutativeSemiringWithoutOne_1044 -> T_IsMagma_86
du_isMagma_764 T_IsCommutativeSemiringWithoutOne_1044
v3
du_isMagma_764 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
du_isMagma_764 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_IsMagma_86
du_isMagma_764 T_IsCommutativeSemiringWithoutOne_1044
v0
  = let v1 :: T_IsSemiringWithoutOne_952
v1
          = T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
              (T_IsCommutativeSemiringWithoutOne_1044
-> T_IsCommutativeSemiringWithoutOne_1044
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0) in
    Any -> T_IsMagma_86
forall a b. a -> b
coe
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
               ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
                  (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v1)))))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isMonoid
d_isMonoid_766 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_766 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_IsMonoid_358
d_isMonoid_766 T_IsCommutativeSemiringWithoutOne_1044
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
      ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
         ((T_IsCommutativeSemiringWithoutOne_1044
 -> T_IsSemiringWithoutOne_952)
-> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
            (T_IsCommutativeSemiringWithoutOne_1044 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0)))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isSemigroup
d_isSemigroup_768 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_768 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemigroup_194
d_isSemigroup_768 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1044
v3 = T_IsCommutativeSemiringWithoutOne_1044 -> T_IsSemigroup_194
du_isSemigroup_768 T_IsCommutativeSemiringWithoutOne_1044
v3
du_isSemigroup_768 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
du_isSemigroup_768 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_IsSemigroup_194
du_isSemigroup_768 T_IsCommutativeSemiringWithoutOne_1044
v0
  = let v1 :: T_IsSemiringWithoutOne_952
v1
          = T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
              (T_IsCommutativeSemiringWithoutOne_1044
-> T_IsCommutativeSemiringWithoutOne_1044
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0) in
    Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
               (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v1))))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.distrib
d_distrib_770 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_770 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_Σ_14
d_distrib_770 T_IsCommutativeSemiringWithoutOne_1044
v0
  = (T_IsSemiringWithoutOne_952 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_972
      ((T_IsCommutativeSemiringWithoutOne_1044
 -> T_IsSemiringWithoutOne_952)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
         (T_IsCommutativeSemiringWithoutOne_1044 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.distribʳ
d_distrib'691'_772 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_772 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_772 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isEquivalence
d_isEquivalence_774 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_774 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> T_IsEquivalence_26
d_isEquivalence_774 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1044
v3 = T_IsCommutativeSemiringWithoutOne_1044 -> T_IsEquivalence_26
du_isEquivalence_774 T_IsCommutativeSemiringWithoutOne_1044
v3
du_isEquivalence_774 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
du_isEquivalence_774 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_IsEquivalence_26
du_isEquivalence_774 T_IsCommutativeSemiringWithoutOne_1044
v0
  = let v1 :: T_IsSemiringWithoutOne_952
v1
          = T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
              (T_IsCommutativeSemiringWithoutOne_1044
-> T_IsCommutativeSemiringWithoutOne_1044
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0) in
    Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      ((T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> Any
forall a b. a -> b
coe
         T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
         ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
            ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
               T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
               ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
                  ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
                     (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v1))))))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isNearSemiring
d_isNearSemiring_776 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
d_isNearSemiring_776 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> T_IsNearSemiring_876
d_isNearSemiring_776 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsCommutativeSemiringWithoutOne_1044
v3 = T_IsCommutativeSemiringWithoutOne_1044 -> T_IsNearSemiring_876
du_isNearSemiring_776 T_IsCommutativeSemiringWithoutOne_1044
v3
du_isNearSemiring_776 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
du_isNearSemiring_776 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_IsNearSemiring_876
du_isNearSemiring_776 T_IsCommutativeSemiringWithoutOne_1044
v0
  = (T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876)
-> Any -> T_IsNearSemiring_876
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_990
      ((T_IsCommutativeSemiringWithoutOne_1044
 -> T_IsSemiringWithoutOne_952)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
         (T_IsCommutativeSemiringWithoutOne_1044 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0))
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isSemiringWithoutOne
d_isSemiringWithoutOne_780 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_780 :: T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_780 T_IsCommutativeSemiringWithoutOne_1044
v0
  = (T_IsCommutativeSemiringWithoutOne_1044
 -> T_IsSemiringWithoutOne_952)
-> Any -> T_IsSemiringWithoutOne_952
forall a b. a -> b
coe
      T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
      (T_IsCommutativeSemiringWithoutOne_1044 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0)
-- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.zero
d_zero_792 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_792 :: T_IsCommutativeSemiringWithoutOne_1044 -> T_Σ_14
d_zero_792 T_IsCommutativeSemiringWithoutOne_1044
v0
  = (T_IsSemiringWithoutOne_952 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_974
      ((T_IsCommutativeSemiringWithoutOne_1044
 -> T_IsSemiringWithoutOne_952)
-> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeSemiringWithoutOne_1044
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1056
         (T_IsCommutativeSemiringWithoutOne_1044 -> Any
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
v0))
-- Data.Nat.Properties._.IsGroup.assoc
d_assoc_848 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_848 :: T_IsGroup_580 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_848 = T_IsGroup_580 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.identity
d_identity_850 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_850 :: T_IsGroup_580 -> T_Σ_14
d_identity_850 T_IsGroup_580
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594 (T_IsGroup_580 -> Any
forall a b. a -> b
coe T_IsGroup_580
v0))
-- Data.Nat.Properties._.IsGroup.inverse
d_inverse_856 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_856 :: T_IsGroup_580 -> T_Σ_14
d_inverse_856 T_IsGroup_580
v0
  = (T_IsGroup_580 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsGroup_580 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_596 (T_IsGroup_580 -> Any
forall a b. a -> b
coe T_IsGroup_580
v0)
-- Data.Nat.Properties._.IsGroup.isEquivalence
d_isEquivalence_862 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_862 :: T_IsGroup_580 -> T_IsEquivalence_26
d_isEquivalence_862 T_IsGroup_580
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594 (T_IsGroup_580 -> Any
forall a b. a -> b
coe T_IsGroup_580
v0))))
-- Data.Nat.Properties._.IsGroup.isMagma
d_isMagma_864 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_864 :: T_IsGroup_580 -> T_IsMagma_86
d_isMagma_864 T_IsGroup_580
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594 (T_IsGroup_580 -> Any
forall a b. a -> b
coe T_IsGroup_580
v0)))
-- Data.Nat.Properties._.IsGroup.isMonoid
d_isMonoid_866 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_866 :: T_IsGroup_580 -> T_IsMonoid_358
d_isMonoid_866 T_IsGroup_580
v0
  = (T_IsGroup_580 -> T_IsMonoid_358) -> Any -> T_IsMonoid_358
forall a b. a -> b
coe T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594 (T_IsGroup_580 -> Any
forall a b. a -> b
coe T_IsGroup_580
v0)
-- Data.Nat.Properties._.IsGroup.isSemigroup
d_isSemigroup_870 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_870 :: T_IsGroup_580 -> T_IsSemigroup_194
d_isSemigroup_870 T_IsGroup_580
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594 (T_IsGroup_580 -> Any
forall a b. a -> b
coe T_IsGroup_580
v0))
-- Data.Nat.Properties._.IsGroup.⁻¹-cong
d_'8315''185''45'cong_886 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_886 :: T_IsGroup_580
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_886 = T_IsGroup_580
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsGroup.∙-cong
d_'8729''45'cong_888 ::
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580 ->
  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_888 :: T_IsGroup_580
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_888 = T_IsGroup_580
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.assoc
d_assoc_896 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_896 :: T_IsIdempotentCommutativeMonoid_464
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_896 = T_IsIdempotentCommutativeMonoid_464
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.comm
d_comm_898 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_898 :: T_IsIdempotentCommutativeMonoid_464
-> Integer -> Integer -> T__'8801'__12
d_comm_898 = T_IsIdempotentCommutativeMonoid_464
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.idem
d_idem_900 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_900 :: T_IsIdempotentCommutativeMonoid_464 -> Integer -> T__'8801'__12
d_idem_900 = T_IsIdempotentCommutativeMonoid_464 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.identity
d_identity_902 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_902 :: T_IsIdempotentCommutativeMonoid_464 -> T_Σ_14
d_identity_902 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474
            (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0)))
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.isCommutativeMonoid
d_isCommutativeMonoid_910 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_isCommutativeMonoid_910 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
d_isCommutativeMonoid_910 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474 (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0)
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.isEquivalence
d_isEquivalence_914 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_914 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsEquivalence_26
d_isEquivalence_914 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
               ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474
                  (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0)))))
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.isMagma
d_isMagma_916 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_916 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsMagma_86
d_isMagma_916 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474
               (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0))))
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.isMonoid
d_isMonoid_918 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_918 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsMonoid_358
d_isMonoid_918 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
      ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
         T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474 (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0))
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.isSemigroup
d_isSemigroup_922 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_922 :: T_IsIdempotentCommutativeMonoid_464 -> T_IsSemigroup_194
d_isSemigroup_922 T_IsIdempotentCommutativeMonoid_464
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsIdempotentCommutativeMonoid_464 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_isCommutativeMonoid_474
            (T_IsIdempotentCommutativeMonoid_464 -> Any
forall a b. a -> b
coe T_IsIdempotentCommutativeMonoid_464
v0)))
-- Data.Nat.Properties._.IsIdempotentCommutativeMonoid.∙-cong
d_'8729''45'cong_934 ::
  MAlonzo.Code.Algebra.Structures.T_IsIdempotentCommutativeMonoid_464 ->
  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_934 :: T_IsIdempotentCommutativeMonoid_464
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_934 = T_IsIdempotentCommutativeMonoid_464
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMagma.isEquivalence
d_isEquivalence_982 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_982 :: T_IsMagma_86 -> T_IsEquivalence_26
d_isEquivalence_982 T_IsMagma_86
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94 (T_IsMagma_86 -> Any
forall a b. a -> b
coe T_IsMagma_86
v0)
-- Data.Nat.Properties._.IsMagma.∙-cong
d_'8729''45'cong_996 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86 ->
  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_996 :: T_IsMagma_86
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_996 = T_IsMagma_86
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.assoc
d_assoc_1004 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1004 :: T_IsMonoid_358 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1004 = T_IsMonoid_358 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsMonoid.identity
d_identity_1006 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1006 :: T_IsMonoid_358 -> T_Σ_14
d_identity_1006 T_IsMonoid_358
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370 (T_IsMonoid_358 -> Any
forall a b. a -> b
coe T_IsMonoid_358
v0)
-- Data.Nat.Properties._.IsMonoid.isEquivalence
d_isEquivalence_1012 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1012 :: T_IsMonoid_358 -> T_IsEquivalence_26
d_isEquivalence_1012 T_IsMonoid_358
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368 (T_IsMonoid_358 -> Any
forall a b. a -> b
coe T_IsMonoid_358
v0)))
-- Data.Nat.Properties._.IsMonoid.isMagma
d_isMagma_1014 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1014 :: T_IsMonoid_358 -> T_IsMagma_86
d_isMagma_1014 T_IsMonoid_358
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368 (T_IsMonoid_358 -> Any
forall a b. a -> b
coe T_IsMonoid_358
v0))
-- Data.Nat.Properties._.IsMonoid.isSemigroup
d_isSemigroup_1018 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1018 :: T_IsMonoid_358 -> T_IsSemigroup_194
d_isSemigroup_1018 T_IsMonoid_358
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368 (T_IsMonoid_358 -> Any
forall a b. a -> b
coe T_IsMonoid_358
v0)
-- Data.Nat.Properties._.IsMonoid.∙-cong
d_'8729''45'cong_1030 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358 ->
  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_1030 :: T_IsMonoid_358
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1030 = T_IsMonoid_358
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsNearSemiring.assoc
d_assoc_1038 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1038 :: T_IsNearSemiring_876
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1038 = T_IsNearSemiring_876
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsNearSemiring.∙-cong
d_'8729''45'cong_1040 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  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_1040 :: T_IsNearSemiring_876
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1040 = T_IsNearSemiring_876
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsNearSemiring.isMagma
d_isMagma_1046 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1046 :: T_IsNearSemiring_876 -> T_IsMagma_86
d_isMagma_1046 T_IsNearSemiring_876
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsNearSemiring_876 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsNearSemiring_876 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_'42''45'isSemigroup_894 (T_IsNearSemiring_876 -> Any
forall a b. a -> b
coe T_IsNearSemiring_876
v0))
-- Data.Nat.Properties._.IsNearSemiring.*-isSemigroup
d_'42''45'isSemigroup_1048 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_'42''45'isSemigroup_1048 :: T_IsNearSemiring_876 -> T_IsSemigroup_194
d_'42''45'isSemigroup_1048 T_IsNearSemiring_876
v0
  = (T_IsNearSemiring_876 -> T_IsSemigroup_194)
-> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsNearSemiring_876 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_'42''45'isSemigroup_894 (T_IsNearSemiring_876 -> Any
forall a b. a -> b
coe T_IsNearSemiring_876
v0)
-- Data.Nat.Properties._.IsNearSemiring.assoc
d_assoc_1050 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1050 :: T_IsNearSemiring_876
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1050 = T_IsNearSemiring_876
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsNearSemiring.∙-cong
d_'8729''45'cong_1052 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1052 :: T_IsNearSemiring_876
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1052 = T_IsNearSemiring_876
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsNearSemiring.identity
d_identity_1058 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1058 :: T_IsNearSemiring_876 -> T_Σ_14
d_identity_1058 T_IsNearSemiring_876
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsNearSemiring_876 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsNearSemiring_876 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'43''45'isMonoid_892 (T_IsNearSemiring_876 -> Any
forall a b. a -> b
coe T_IsNearSemiring_876
v0))
-- Data.Nat.Properties._.IsNearSemiring.isMagma
d_isMagma_1064 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1064 :: T_IsNearSemiring_876 -> T_IsMagma_86
d_isMagma_1064 T_IsNearSemiring_876
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsNearSemiring_876 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsNearSemiring_876 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'43''45'isMonoid_892 (T_IsNearSemiring_876 -> Any
forall a b. a -> b
coe T_IsNearSemiring_876
v0)))
-- Data.Nat.Properties._.IsNearSemiring.+-isMonoid
d_'43''45'isMonoid_1066 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'43''45'isMonoid_1066 :: T_IsNearSemiring_876 -> T_IsMonoid_358
d_'43''45'isMonoid_1066 T_IsNearSemiring_876
v0
  = (T_IsNearSemiring_876 -> T_IsMonoid_358) -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsNearSemiring_876 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'43''45'isMonoid_892 (T_IsNearSemiring_876 -> Any
forall a b. a -> b
coe T_IsNearSemiring_876
v0)
-- Data.Nat.Properties._.IsNearSemiring.isSemigroup
d_isSemigroup_1068 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1068 :: T_IsNearSemiring_876 -> T_IsSemigroup_194
d_isSemigroup_1068 T_IsNearSemiring_876
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsNearSemiring_876 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsNearSemiring_876 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'43''45'isMonoid_892 (T_IsNearSemiring_876 -> Any
forall a b. a -> b
coe T_IsNearSemiring_876
v0))
-- Data.Nat.Properties._.IsNearSemiring.distribʳ
d_distrib'691'_1070 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_1070 :: T_IsNearSemiring_876
-> Integer -> Integer -> Integer -> T__'8801'__12
d_distrib'691'_1070 = T_IsNearSemiring_876
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsNearSemiring.isEquivalence
d_isEquivalence_1072 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1072 :: T_IsNearSemiring_876 -> T_IsEquivalence_26
d_isEquivalence_1072 T_IsNearSemiring_876
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsNearSemiring_876 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsNearSemiring_876 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'43''45'isMonoid_892 (T_IsNearSemiring_876 -> Any
forall a b. a -> b
coe T_IsNearSemiring_876
v0))))
-- Data.Nat.Properties._.IsNearSemiring.zeroˡ
d_zero'737'_1086 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_zero'737'_1086 :: T_IsNearSemiring_876 -> Integer -> T__'8801'__12
d_zero'737'_1086 = T_IsNearSemiring_876 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsRing.assoc
d_assoc_1092 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1092 :: T_IsRing_1584 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1092 = T_IsRing_1584 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsRing.∙-cong
d_'8729''45'cong_1094 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  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_1094 :: T_IsRing_1584
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1094 = T_IsRing_1584
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsRing.identity
d_identity_1100 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1100 :: T_IsRing_1584 -> T_Σ_14
d_identity_1100 T_IsRing_1584
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsRing_1584 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsRing_1584 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1606 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0))
-- Data.Nat.Properties._.IsRing.isMagma
d_isMagma_1106 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1106 :: T_IsRing_1584 -> T_IsMagma_86
d_isMagma_1106 T_IsRing_1584
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsRing_1584 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_1584 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1606 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0)))
-- Data.Nat.Properties._.IsRing.*-isMonoid
d_'42''45'isMonoid_1108 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'42''45'isMonoid_1108 :: T_IsRing_1584 -> T_IsMonoid_358
d_'42''45'isMonoid_1108 T_IsRing_1584
v0
  = (T_IsRing_1584 -> T_IsMonoid_358) -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsRing_1584 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1606 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0)
-- Data.Nat.Properties._.IsRing.isSemigroup
d_isSemigroup_1110 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1110 :: T_IsRing_1584 -> T_IsSemigroup_194
d_isSemigroup_1110 T_IsRing_1584
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsRing_1584 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsRing_1584 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1606 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0))
-- Data.Nat.Properties._.IsRing.assoc
d_assoc_1112 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1112 :: T_IsRing_1584 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1112 = T_IsRing_1584 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsRing.comm
d_comm_1114 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1114 :: T_IsRing_1584 -> Integer -> Integer -> T__'8801'__12
d_comm_1114 = T_IsRing_1584 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsRing.∙-cong
d_'8729''45'cong_1116 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  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_1116 :: T_IsRing_1584
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1116 = T_IsRing_1584
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsRing.identity
d_identity_1122 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1122 :: T_IsRing_1584 -> T_Σ_14
d_identity_1122 T_IsRing_1584
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
         ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
            T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
            ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
               T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
               (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0))))
-- Data.Nat.Properties._.IsRing.+-isAbelianGroup
d_'43''45'isAbelianGroup_1128 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_662
d_'43''45'isAbelianGroup_1128 :: T_IsRing_1584 -> T_IsAbelianGroup_662
d_'43''45'isAbelianGroup_1128 T_IsRing_1584
v0
  = (T_IsRing_1584 -> T_IsAbelianGroup_662)
-> Any -> T_IsAbelianGroup_662
forall a b. a -> b
coe
      T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
      (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0)
-- Data.Nat.Properties._.IsRing.isGroup
d_isGroup_1136 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_580
d_isGroup_1136 :: T_IsRing_1584 -> T_IsGroup_580
d_isGroup_1136 T_IsRing_1584
v0
  = (T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> T_IsGroup_580
forall a b. a -> b
coe
      T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
      ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
         T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
         (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0))
-- Data.Nat.Properties._.IsRing.isMagma
d_isMagma_1138 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1138 :: T_IsRing_1584 -> T_IsMagma_86
d_isMagma_1138 T_IsRing_1584
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
            ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
               T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
               ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
                  T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
                  (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0)))))
-- Data.Nat.Properties._.IsRing.isMonoid
d_isMonoid_1140 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_1140 :: T_IsRing_1584 -> T_IsMonoid_358
d_isMonoid_1140 T_IsRing_1584
v0
  = (T_IsGroup_580 -> T_IsMonoid_358) -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
      ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
         T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
         ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
            (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0)))
-- Data.Nat.Properties._.IsRing.isSemigroup
d_isSemigroup_1142 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1142 :: T_IsRing_1584 -> T_IsSemigroup_194
d_isSemigroup_1142 T_IsRing_1584
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
         ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
            T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
            ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
               T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
               (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0))))
-- Data.Nat.Properties._.IsRing.⁻¹-cong
d_'8315''185''45'cong_1144 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8315''185''45'cong_1144 :: T_IsRing_1584
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8315''185''45'cong_1144 = T_IsRing_1584
-> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsRing.inverse
d_inverse_1146 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_1146 :: T_IsRing_1584 -> T_Σ_14
d_inverse_1146 T_IsRing_1584
v0
  = (T_IsGroup_580 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsGroup_580 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_596
      ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
         T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
         ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
            T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
            (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0)))
-- Data.Nat.Properties._.IsRing.distrib
d_distrib_1152 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_1152 :: T_IsRing_1584 -> T_Σ_14
d_distrib_1152 T_IsRing_1584
v0
  = (T_IsRing_1584 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsRing_1584 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1608 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0)
-- Data.Nat.Properties._.IsRing.isEquivalence
d_isEquivalence_1158 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1158 :: T_IsRing_1584 -> T_IsEquivalence_26
d_isEquivalence_1158 T_IsRing_1584
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsGroup_580 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsGroup_580 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_594
               ((T_IsAbelianGroup_662 -> T_IsGroup_580) -> Any -> Any
forall a b. a -> b
coe
                  T_IsAbelianGroup_662 -> T_IsGroup_580
MAlonzo.Code.Algebra.Structures.d_isGroup_674
                  ((T_IsRing_1584 -> T_IsAbelianGroup_662) -> Any -> Any
forall a b. a -> b
coe
                     T_IsRing_1584 -> T_IsAbelianGroup_662
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_1604
                     (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0))))))
-- Data.Nat.Properties._.IsRing.isNearSemiring
d_isNearSemiring_1160 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
d_isNearSemiring_1160 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsRing_1584
-> T_IsNearSemiring_876
d_isNearSemiring_1160 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsRing_1584
v5
  = T_IsRing_1584 -> T_IsNearSemiring_876
du_isNearSemiring_1160 T_IsRing_1584
v5
du_isNearSemiring_1160 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
du_isNearSemiring_1160 :: T_IsRing_1584 -> T_IsNearSemiring_876
du_isNearSemiring_1160 T_IsRing_1584
v0
  = let v1 :: t
v1
          = (T_IsRing_1584 -> T_IsSemiring_1238) -> Any -> t
forall a b. a -> b
coe
              T_IsRing_1584 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.du_isSemiring_1698 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0) in
    Any -> T_IsNearSemiring_876
forall a b. a -> b
coe
      ((T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_990
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952) -> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
            (Any -> Any
forall a b. a -> b
coe Any
forall a. a
v1)))
-- Data.Nat.Properties._.IsRing.isSemiring
d_isSemiring_1164 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238
d_isSemiring_1164 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsRing_1584
-> T_IsSemiring_1238
d_isSemiring_1164 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer -> Integer
v2 Integer
v3 Integer
v4 T_IsRing_1584
v5
  = (T_IsRing_1584 -> T_IsSemiring_1238)
-> T_IsRing_1584 -> T_IsSemiring_1238
forall a b. a -> b
coe T_IsRing_1584 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.du_isSemiring_1698 T_IsRing_1584
v5
-- Data.Nat.Properties._.IsRing.isSemiringWithoutOne
d_isSemiringWithoutOne_1168 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_1168 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer)
-> Integer
-> Integer
-> T_IsRing_1584
-> T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_1168 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer -> Integer
v2 ~Integer
v3 ~Integer
v4 T_IsRing_1584
v5
  = T_IsRing_1584 -> T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_1168 T_IsRing_1584
v5
du_isSemiringWithoutOne_1168 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_1168 :: T_IsRing_1584 -> T_IsSemiringWithoutOne_952
du_isSemiringWithoutOne_1168 T_IsRing_1584
v0
  = (T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952)
-> Any -> T_IsSemiringWithoutOne_952
forall a b. a -> b
coe
      T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
      ((T_IsRing_1584 -> T_IsSemiring_1238) -> Any -> Any
forall a b. a -> b
coe T_IsRing_1584 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.du_isSemiring_1698 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0))
-- Data.Nat.Properties._.IsRing.zero
d_zero_1184 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_1584 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_1184 :: T_IsRing_1584 -> T_Σ_14
d_zero_1184 T_IsRing_1584
v0
  = (T_IsRing_1584 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsRing_1584 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1610 (T_IsRing_1584 -> Any
forall a b. a -> b
coe T_IsRing_1584
v0)
-- Data.Nat.Properties._.IsSelectiveMagma.isEquivalence
d_isEquivalence_1192 ::
  MAlonzo.Code.Algebra.Structures.T_IsSelectiveMagma_158 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1192 :: T_IsSelectiveMagma_158 -> T_IsEquivalence_26
d_isEquivalence_1192 T_IsSelectiveMagma_158
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSelectiveMagma_158 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe T_IsSelectiveMagma_158 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_166 (T_IsSelectiveMagma_158 -> Any
forall a b. a -> b
coe T_IsSelectiveMagma_158
v0))
-- Data.Nat.Properties._.IsSelectiveMagma.isMagma
d_isMagma_1194 ::
  MAlonzo.Code.Algebra.Structures.T_IsSelectiveMagma_158 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1194 :: T_IsSelectiveMagma_158 -> T_IsMagma_86
d_isMagma_1194 T_IsSelectiveMagma_158
v0
  = (T_IsSelectiveMagma_158 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe T_IsSelectiveMagma_158 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_166 (T_IsSelectiveMagma_158 -> Any
forall a b. a -> b
coe T_IsSelectiveMagma_158
v0)
-- Data.Nat.Properties._.IsSelectiveMagma.sel
d_sel_1202 ::
  MAlonzo.Code.Algebra.Structures.T_IsSelectiveMagma_158 ->
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_sel_1202 :: T_IsSelectiveMagma_158 -> Integer -> Integer -> T__'8846'__30
d_sel_1202 T_IsSelectiveMagma_158
v0
  = (T_IsSelectiveMagma_158 -> Any -> Any -> T__'8846'__30)
-> Any -> Integer -> Integer -> T__'8846'__30
forall a b. a -> b
coe T_IsSelectiveMagma_158 -> Any -> Any -> T__'8846'__30
MAlonzo.Code.Algebra.Structures.d_sel_168 (T_IsSelectiveMagma_158 -> Any
forall a b. a -> b
coe T_IsSelectiveMagma_158
v0)
-- Data.Nat.Properties._.IsSelectiveMagma.∙-cong
d_'8729''45'cong_1210 ::
  MAlonzo.Code.Algebra.Structures.T_IsSelectiveMagma_158 ->
  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_1210 :: T_IsSelectiveMagma_158
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1210 = T_IsSelectiveMagma_158
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemigroup.assoc
d_assoc_1218 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1218 :: T_IsSemigroup_194 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1218 = T_IsSemigroup_194 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemigroup.isEquivalence
d_isEquivalence_1220 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1220 :: T_IsSemigroup_194 -> T_IsEquivalence_26
d_isEquivalence_1220 T_IsSemigroup_194
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202 (T_IsSemigroup_194 -> Any
forall a b. a -> b
coe T_IsSemigroup_194
v0))
-- Data.Nat.Properties._.IsSemigroup.isMagma
d_isMagma_1222 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1222 :: T_IsSemigroup_194 -> T_IsMagma_86
d_isMagma_1222 T_IsSemigroup_194
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202 (T_IsSemigroup_194 -> Any
forall a b. a -> b
coe T_IsSemigroup_194
v0)
-- Data.Nat.Properties._.IsSemigroup.∙-cong
d_'8729''45'cong_1236 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194 ->
  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_1236 :: T_IsSemigroup_194
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1236 = T_IsSemigroup_194
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemilattice.assoc
d_assoc_1244 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1244 :: T_IsSemilattice_312
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1244 = T_IsSemilattice_312
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemilattice.comm
d_comm_1246 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1246 :: T_IsSemilattice_312 -> Integer -> Integer -> T__'8801'__12
d_comm_1246 = T_IsSemilattice_312 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemilattice.idem
d_idem_1248 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312 ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_idem_1248 :: T_IsSemilattice_312 -> Integer -> T__'8801'__12
d_idem_1248 = T_IsSemilattice_312 -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemilattice.isBand
d_isBand_1250 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312 ->
  MAlonzo.Code.Algebra.Structures.T_IsBand_230
d_isBand_1250 :: T_IsSemilattice_312 -> T_IsBand_230
d_isBand_1250 T_IsSemilattice_312
v0
  = (T_IsSemilattice_312 -> T_IsBand_230) -> Any -> T_IsBand_230
forall a b. a -> b
coe T_IsSemilattice_312 -> T_IsBand_230
MAlonzo.Code.Algebra.Structures.d_isBand_320 (T_IsSemilattice_312 -> Any
forall a b. a -> b
coe T_IsSemilattice_312
v0)
-- Data.Nat.Properties._.IsSemilattice.isEquivalence
d_isEquivalence_1252 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1252 :: T_IsSemilattice_312 -> T_IsEquivalence_26
d_isEquivalence_1252 T_IsSemilattice_312
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsBand_230 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsBand_230 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_238
            ((T_IsSemilattice_312 -> T_IsBand_230) -> Any -> Any
forall a b. a -> b
coe T_IsSemilattice_312 -> T_IsBand_230
MAlonzo.Code.Algebra.Structures.d_isBand_320 (T_IsSemilattice_312 -> Any
forall a b. a -> b
coe T_IsSemilattice_312
v0))))
-- Data.Nat.Properties._.IsSemilattice.isMagma
d_isMagma_1254 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1254 :: T_IsSemilattice_312 -> T_IsMagma_86
d_isMagma_1254 T_IsSemilattice_312
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsBand_230 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsBand_230 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_238
         ((T_IsSemilattice_312 -> T_IsBand_230) -> Any -> Any
forall a b. a -> b
coe T_IsSemilattice_312 -> T_IsBand_230
MAlonzo.Code.Algebra.Structures.d_isBand_320 (T_IsSemilattice_312 -> Any
forall a b. a -> b
coe T_IsSemilattice_312
v0)))
-- Data.Nat.Properties._.IsSemilattice.isSemigroup
d_isSemigroup_1258 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1258 :: T_IsSemilattice_312 -> T_IsSemigroup_194
d_isSemigroup_1258 T_IsSemilattice_312
v0
  = (T_IsBand_230 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsBand_230 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_238
      ((T_IsSemilattice_312 -> T_IsBand_230) -> Any -> Any
forall a b. a -> b
coe T_IsSemilattice_312 -> T_IsBand_230
MAlonzo.Code.Algebra.Structures.d_isBand_320 (T_IsSemilattice_312 -> Any
forall a b. a -> b
coe T_IsSemilattice_312
v0))
-- Data.Nat.Properties._.IsSemilattice.∙-cong
d_'8729''45'cong_1270 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312 ->
  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_1270 :: T_IsSemilattice_312
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1270 = T_IsSemilattice_312
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiring.assoc
d_assoc_1278 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1278 :: T_IsSemiring_1238 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1278 = T_IsSemiring_1238 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiring.∙-cong
d_'8729''45'cong_1280 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  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_1280 :: T_IsSemiring_1238
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1280 = T_IsSemiring_1238
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiring.identity
d_identity_1286 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1286 :: T_IsSemiring_1238 -> T_Σ_14
d_identity_1286 T_IsSemiring_1238
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0)))
-- Data.Nat.Properties._.IsSemiring.isMagma
d_isMagma_1292 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1292 :: T_IsSemiring_1238 -> T_IsMagma_86
d_isMagma_1292 T_IsSemiring_1238
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0))))
-- Data.Nat.Properties._.IsSemiring.*-isMonoid
d_'42''45'isMonoid_1294 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'42''45'isMonoid_1294 :: T_IsSemiring_1238 -> T_IsMonoid_358
d_'42''45'isMonoid_1294 T_IsSemiring_1238
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0))
-- Data.Nat.Properties._.IsSemiring.isSemigroup
d_isSemigroup_1296 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1296 :: T_IsSemiring_1238 -> T_IsSemigroup_194
d_isSemigroup_1296 T_IsSemiring_1238
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0)))
-- Data.Nat.Properties._.IsSemiring.assoc
d_assoc_1298 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1298 :: T_IsSemiring_1238 -> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1298 = T_IsSemiring_1238 -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiring.comm
d_comm_1300 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1300 :: T_IsSemiring_1238 -> Integer -> Integer -> T__'8801'__12
d_comm_1300 = T_IsSemiring_1238 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiring.∙-cong
d_'8729''45'cong_1302 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8729''45'cong_1302 :: T_IsSemiring_1238
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1302 = T_IsSemiring_1238
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiring.identity
d_identity_1308 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1308 :: T_IsSemiring_1238 -> T_Σ_14
d_identity_1308 T_IsSemiring_1238
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0))))
-- Data.Nat.Properties._.IsSemiring.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_1316 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_1316 :: T_IsSemiring_1238 -> T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_1316 T_IsSemiring_1238
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0))
-- Data.Nat.Properties._.IsSemiring.isMagma
d_isMagma_1320 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1320 :: T_IsSemiring_1238 -> T_IsMagma_86
d_isMagma_1320 T_IsSemiring_1238
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
               ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
                  (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0)))))
-- Data.Nat.Properties._.IsSemiring.isMonoid
d_isMonoid_1322 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_1322 :: T_IsSemiring_1238 -> T_IsMonoid_358
d_isMonoid_1322 T_IsSemiring_1238
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
      ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
         ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
            (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0)))
-- Data.Nat.Properties._.IsSemiring.isSemigroup
d_isSemigroup_1324 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1324 :: T_IsSemiring_1238 -> T_IsSemigroup_194
d_isSemigroup_1324 T_IsSemiring_1238
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
            ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
               (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0))))
-- Data.Nat.Properties._.IsSemiring.distrib
d_distrib_1326 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_1326 :: T_IsSemiring_1238 -> T_Σ_14
d_distrib_1326 T_IsSemiring_1238
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1162
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
         (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0))
-- Data.Nat.Properties._.IsSemiring.isEquivalence
d_isEquivalence_1332 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1332 :: T_IsSemiring_1238 -> T_IsEquivalence_26
d_isEquivalence_1332 T_IsSemiring_1238
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
               ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
                  ((T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any
forall a b. a -> b
coe
                     T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
                     (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0))))))
-- Data.Nat.Properties._.IsSemiring.isNearSemiring
d_isNearSemiring_1334 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
d_isNearSemiring_1334 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsSemiring_1238
-> T_IsNearSemiring_876
d_isNearSemiring_1334 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 ~Integer
v3 T_IsSemiring_1238
v4
  = T_IsSemiring_1238 -> T_IsNearSemiring_876
du_isNearSemiring_1334 T_IsSemiring_1238
v4
du_isNearSemiring_1334 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
du_isNearSemiring_1334 :: T_IsSemiring_1238 -> T_IsNearSemiring_876
du_isNearSemiring_1334 T_IsSemiring_1238
v0
  = (T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876)
-> Any -> T_IsNearSemiring_876
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_990
      ((T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326
         (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0))
-- Data.Nat.Properties._.IsSemiring.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_1338 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142
d_isSemiringWithoutAnnihilatingZero_1338 :: T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
d_isSemiringWithoutAnnihilatingZero_1338 T_IsSemiring_1238
v0
  = (T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> T_IsSemiringWithoutAnnihilatingZero_1142
forall a b. a -> b
coe
      T_IsSemiring_1238 -> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1252
      (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0)
-- Data.Nat.Properties._.IsSemiring.isSemiringWithoutOne
d_isSemiringWithoutOne_1340 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_1340 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsSemiring_1238
-> T_IsSemiringWithoutOne_952
d_isSemiringWithoutOne_1340 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer
v2 Integer
v3 T_IsSemiring_1238
v4
  = (T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952)
-> T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
forall a b. a -> b
coe
      T_IsSemiring_1238 -> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.du_isSemiringWithoutOne_1326 T_IsSemiring_1238
v4
-- Data.Nat.Properties._.IsSemiring.zero
d_zero_1352 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_1352 :: T_IsSemiring_1238 -> T_Σ_14
d_zero_1352 T_IsSemiring_1238
v0
  = (T_IsSemiring_1238 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsSemiring_1238 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1254 (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
v0)
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.assoc
d_assoc_1360 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1360 :: T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1360 = T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.∙-cong
d_'8729''45'cong_1362 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  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_1362 :: T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1362 = T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.identity
d_identity_1368 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1368 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14
d_identity_1368 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160 (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0))
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.isMagma
d_isMagma_1374 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1374 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMagma_86
d_isMagma_1374 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160 (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0)))
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.*-isMonoid
d_'42''45'isMonoid_1376 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'42''45'isMonoid_1376 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
d_'42''45'isMonoid_1376 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160 (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0)
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.isSemigroup
d_isSemigroup_1378 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1378 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsSemigroup_194
d_isSemigroup_1378 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_'42''45'isMonoid_1160 (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0))
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.assoc
d_assoc_1380 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1380 :: T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer -> Integer -> Integer -> T__'8801'__12
d_assoc_1380 = T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.comm
d_comm_1382 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1382 :: T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer -> Integer -> T__'8801'__12
d_comm_1382 = T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.∙-cong
d_'8729''45'cong_1384 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  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_1384 :: T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1384 = T_IsSemiringWithoutAnnihilatingZero_1142
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.identity
d_identity_1390 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1390 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14
d_identity_1390 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
            (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0)))
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_1398 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_1398 :: T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_1398 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
      (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0)
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.isMagma
d_isMagma_1402 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1402 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMagma_86
d_isMagma_1402 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
               (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0))))
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.isMonoid
d_isMonoid_1404 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_1404 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsMonoid_358
d_isMonoid_1404 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
      ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
         (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0))
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.isSemigroup
d_isSemigroup_1406 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1406 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsSemigroup_194
d_isSemigroup_1406 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
            (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0)))
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.distrib
d_distrib_1408 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_1408 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14
d_distrib_1408 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14)
-> Any -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1162 (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0)
-- Data.Nat.Properties._.IsSemiringWithoutAnnihilatingZero.isEquivalence
d_isEquivalence_1414 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1142 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1414 :: T_IsSemiringWithoutAnnihilatingZero_1142 -> T_IsEquivalence_26
d_isEquivalence_1414 T_IsSemiringWithoutAnnihilatingZero_1142
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
               ((T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutAnnihilatingZero_1142
-> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1158
                  (T_IsSemiringWithoutAnnihilatingZero_1142 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1142
v0)))))
-- Data.Nat.Properties._.IsSemiringWithoutOne.assoc
d_assoc_1430 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1430 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_assoc_1430 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutOne.∙-cong
d_'8729''45'cong_1432 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  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_1432 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1432 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutOne.isMagma
d_isMagma_1438 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1438 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> T_IsMagma_86
d_isMagma_1438 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsSemiringWithoutOne_952
v3 = T_IsSemiringWithoutOne_952 -> T_IsMagma_86
du_isMagma_1438 T_IsSemiringWithoutOne_952
v3
du_isMagma_1438 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
du_isMagma_1438 :: T_IsSemiringWithoutOne_952 -> T_IsMagma_86
du_isMagma_1438 T_IsSemiringWithoutOne_952
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_'42''45'isSemigroup_970 (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0))
-- Data.Nat.Properties._.IsSemiringWithoutOne.*-isSemigroup
d_'42''45'isSemigroup_1440 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_'42''45'isSemigroup_1440 :: T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194
d_'42''45'isSemigroup_1440 T_IsSemiringWithoutOne_952
v0
  = (T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194)
-> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_'42''45'isSemigroup_970 (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0)
-- Data.Nat.Properties._.IsSemiringWithoutOne.assoc
d_assoc_1442 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_assoc_1442 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_assoc_1442 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutOne.comm
d_comm_1444 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_comm_1444 :: T_IsSemiringWithoutOne_952 -> Integer -> Integer -> T__'8801'__12
d_comm_1444 = T_IsSemiringWithoutOne_952 -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutOne.∙-cong
d_'8729''45'cong_1446 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  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_1446 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
d_'8729''45'cong_1446 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutOne.identity
d_identity_1452 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1452 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> T_Σ_14
d_identity_1452 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsSemiringWithoutOne_952
v3 = T_IsSemiringWithoutOne_952 -> T_Σ_14
du_identity_1452 T_IsSemiringWithoutOne_952
v3
du_identity_1452 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_identity_1452 :: T_IsSemiringWithoutOne_952 -> T_Σ_14
du_identity_1452 T_IsSemiringWithoutOne_952
v0
  = (T_IsMonoid_358 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_370
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
            (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0)))
-- Data.Nat.Properties._.IsSemiringWithoutOne.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_1460 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_1460 :: T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
d_'43''45'isCommutativeMonoid_1460 T_IsSemiringWithoutOne_952
v0
  = (T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
      (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0)
-- Data.Nat.Properties._.IsSemiringWithoutOne.isMagma
d_isMagma_1464 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_isMagma_1464 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> T_IsMagma_86
d_isMagma_1464 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsSemiringWithoutOne_952
v3 = T_IsSemiringWithoutOne_952 -> T_IsMagma_86
du_isMagma_1464 T_IsSemiringWithoutOne_952
v3
du_isMagma_1464 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
du_isMagma_1464 :: T_IsSemiringWithoutOne_952 -> T_IsMagma_86
du_isMagma_1464 T_IsSemiringWithoutOne_952
v0
  = (T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
      ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
         T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
         ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
            T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
            ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
               T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
               (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0))))
-- Data.Nat.Properties._.IsSemiringWithoutOne.isMonoid
d_isMonoid_1466 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_isMonoid_1466 :: T_IsSemiringWithoutOne_952 -> T_IsMonoid_358
d_isMonoid_1466 T_IsSemiringWithoutOne_952
v0
  = (T_IsCommutativeMonoid_406 -> T_IsMonoid_358)
-> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
      ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
         T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
         (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0))
-- Data.Nat.Properties._.IsSemiringWithoutOne.isSemigroup
d_isSemigroup_1468 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_isSemigroup_1468 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> T_IsSemigroup_194
d_isSemigroup_1468 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsSemiringWithoutOne_952
v3 = T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194
du_isSemigroup_1468 T_IsSemiringWithoutOne_952
v3
du_isSemigroup_1468 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
du_isSemigroup_1468 :: T_IsSemiringWithoutOne_952 -> T_IsSemigroup_194
du_isSemigroup_1468 T_IsSemiringWithoutOne_952
v0
  = (T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
      ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
         ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
            T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
            (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0)))
-- Data.Nat.Properties._.IsSemiringWithoutOne.distrib
d_distrib_1470 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_1470 :: T_IsSemiringWithoutOne_952 -> T_Σ_14
d_distrib_1470 T_IsSemiringWithoutOne_952
v0
  = (T_IsSemiringWithoutOne_952 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutOne_952 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_972 (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0)
-- Data.Nat.Properties._.IsSemiringWithoutOne.distribʳ
d_distrib'691'_1472 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691'_1472 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
d_distrib'691'_1472 = (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> Integer
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._.IsSemiringWithoutOne.isEquivalence
d_isEquivalence_1474 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_1474 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> T_IsEquivalence_26
d_isEquivalence_1474 ~Integer -> Integer -> Integer
v0 ~Integer -> Integer -> Integer
v1 ~Integer
v2 T_IsSemiringWithoutOne_952
v3 = T_IsSemiringWithoutOne_952 -> T_IsEquivalence_26
du_isEquivalence_1474 T_IsSemiringWithoutOne_952
v3
du_isEquivalence_1474 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
du_isEquivalence_1474 :: T_IsSemiringWithoutOne_952 -> T_IsEquivalence_26
du_isEquivalence_1474 T_IsSemiringWithoutOne_952
v0
  = (T_IsMagma_86 -> T_IsEquivalence_26) -> Any -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_86 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_94
      ((T_IsSemigroup_194 -> T_IsMagma_86) -> Any -> Any
forall a b. a -> b
coe
         T_IsSemigroup_194 -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.d_isMagma_202
         ((T_IsMonoid_358 -> T_IsSemigroup_194) -> Any -> Any
forall a b. a -> b
coe
            T_IsMonoid_358 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.d_isSemigroup_368
            ((T_IsCommutativeMonoid_406 -> T_IsMonoid_358) -> Any -> Any
forall a b. a -> b
coe
               T_IsCommutativeMonoid_406 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.d_isMonoid_416
               ((T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406)
-> Any -> Any
forall a b. a -> b
coe
                  T_IsSemiringWithoutOne_952 -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_968
                  (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0)))))
-- Data.Nat.Properties._.IsSemiringWithoutOne.isNearSemiring
d_isNearSemiring_1476 ::
  (Integer -> Integer -> Integer) ->
  (Integer -> Integer -> Integer) ->
  Integer ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_876
d_isNearSemiring_1476 :: (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsSemiringWithoutOne_952
-> T_IsNearSemiring_876
d_isNearSemiring_1476 Integer -> Integer -> Integer
v0 Integer -> Integer -> Integer
v1 Integer
v2 T_IsSemiringWithoutOne_952
v3
  = (T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876)
-> T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876
forall a b. a -> b
coe T_IsSemiringWithoutOne_952 -> T_IsNearSemiring_876
MAlonzo.Code.Algebra.Structures.du_isNearSemiring_990 T_IsSemiringWithoutOne_952
v3
-- Data.Nat.Properties._.IsSemiringWithoutOne.zero
d_zero_1490 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_1490 :: T_IsSemiringWithoutOne_952 -> T_Σ_14
d_zero_1490 T_IsSemiringWithoutOne_952
v0
  = (T_IsSemiringWithoutOne_952 -> T_Σ_14) -> Any -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutOne_952 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_974 (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
v0)
-- Data.Nat.Properties.suc-injective
d_suc'45'injective_1500 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_suc'45'injective_1500 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_suc'45'injective_1500 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≡ᵇ⇒≡
d_'8801''7495''8658''8801'_1506 ::
  Integer ->
  Integer ->
  AgdaAny -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8801''7495''8658''8801'_1506 :: Integer -> Integer -> Any -> T__'8801'__12
d_'8801''7495''8658''8801'_1506 = Integer -> Integer -> Any -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≡⇒≡ᵇ
d_'8801''8658''8801''7495'_1518 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_'8801''8658''8801''7495'_1518 :: Integer -> Integer -> T__'8801'__12 -> Any
d_'8801''8658''8801''7495'_1518 Integer
v0 ~Integer
v1 ~T__'8801'__12
v2
  = Integer -> Any
du_'8801''8658''8801''7495'_1518 Integer
v0
du_'8801''8658''8801''7495'_1518 :: Integer -> AgdaAny
du_'8801''8658''8801''7495'_1518 :: Integer -> Any
du_'8801''8658''8801''7495'_1518 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> () -> Any
forall a b. a -> b
coe ()
MAlonzo.Code.Agda.Builtin.Unit.C_tt_8
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> Any
forall a b. a -> b
coe ((Integer -> Any) -> Any -> Any
forall a b. a -> b
coe Integer -> Any
du_'8801''8658''8801''7495'_1518 (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties._≟_
d__'8799'__1528 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8799'__1528 :: Integer -> Integer -> T_Dec_32
d__'8799'__1528 Integer
v0 Integer
v1
  = ((Any -> Any) -> T_Dec_32 -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe
      (Any -> Any) -> T_Dec_32 -> T_Dec_32
MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
      Any
forall a. a
erased
      ((Bool -> T_Dec_32) -> Any -> Any
forall a b. a -> b
coe
         Bool -> T_Dec_32
MAlonzo.Code.Data.Bool.Properties.d_T'63'_2210
         ((Integer -> Integer -> Bool) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Bool
eqInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.≡-irrelevant
d_'8801''45'irrelevant_1534 ::
  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_'8801''45'irrelevant_1534 :: Integer
-> Integer -> T__'8801'__12 -> T__'8801'__12 -> T__'8801'__12
d_'8801''45'irrelevant_1534 = Integer
-> Integer -> T__'8801'__12 -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≟-diag
d_'8799''45'diag_1542 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8799''45'diag_1542 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8799''45'diag_1542 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≡-isDecEquivalence
d_'8801''45'isDecEquivalence_1544 ::
  MAlonzo.Code.Relation.Binary.Structures.T_IsDecEquivalence_44
d_'8801''45'isDecEquivalence_1544 :: T_IsDecEquivalence_44
d_'8801''45'isDecEquivalence_1544
  = (T_IsEquivalence_26
 -> (Any -> Any -> T_Dec_32) -> T_IsDecEquivalence_44)
-> Any -> Any -> T_IsDecEquivalence_44
forall a b. a -> b
coe
      T_IsEquivalence_26
-> (Any -> Any -> T_Dec_32) -> T_IsDecEquivalence_44
MAlonzo.Code.Relation.Binary.Structures.C_IsDecEquivalence'46'constructor_3075
      (T_IsEquivalence_26 -> Any
forall a b. a -> b
coe
         T_IsEquivalence_26
MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_242)
      ((Integer -> Integer -> T_Dec_32) -> Any
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8799'__1528)
-- Data.Nat.Properties.≡-decSetoid
d_'8801''45'decSetoid_1546 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_DecSetoid_84
d_'8801''45'decSetoid_1546 :: T_DecSetoid_84
d_'8801''45'decSetoid_1546
  = (T_IsDecEquivalence_44 -> T_DecSetoid_84)
-> T_IsDecEquivalence_44 -> T_DecSetoid_84
forall a b. a -> b
coe
      T_IsDecEquivalence_44 -> T_DecSetoid_84
MAlonzo.Code.Relation.Binary.Bundles.C_DecSetoid'46'constructor_1385
      T_IsDecEquivalence_44
d_'8801''45'isDecEquivalence_1544
-- Data.Nat.Properties.0≢1+n
d_0'8802'1'43'n_1550 ::
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_0'8802'1'43'n_1550 :: Integer -> T__'8801'__12 -> T_'8869'_4
d_0'8802'1'43'n_1550 = Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.1+n≢0
d_1'43'n'8802'0_1554 ::
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_1'43'n'8802'0_1554 :: Integer -> T__'8801'__12 -> T_'8869'_4
d_1'43'n'8802'0_1554 = Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.1+n≢n
d_1'43'n'8802'n_1558 ::
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_1'43'n'8802'n_1558 :: Integer -> T__'8801'__12 -> T_'8869'_4
d_1'43'n'8802'n_1558 = Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.<ᵇ⇒<
d_'60''7495''8658''60'_1566 ::
  Integer ->
  Integer -> AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'60''7495''8658''60'_1566 :: Integer -> Integer -> Any -> T__'8804'__18
d_'60''7495''8658''60'_1566 Integer
v0 ~Integer
v1 ~Any
v2
  = Integer -> T__'8804'__18
du_'60''7495''8658''60'_1566 Integer
v0
du_'60''7495''8658''60'_1566 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'60''7495''8658''60'_1566 :: Integer -> T__'8804'__18
du_'60''7495''8658''60'_1566 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
             T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
             (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'60''7495''8658''60'_1566 (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.<⇒<ᵇ
d_'60''8658''60''7495'_1582 ::
  Integer ->
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18 -> AgdaAny
d_'60''8658''60''7495'_1582 :: Integer -> Integer -> T__'8804'__18 -> Any
d_'60''8658''60''7495'_1582 ~Integer
v0 ~Integer
v1 T__'8804'__18
v2
  = T__'8804'__18 -> Any
du_'60''8658''60''7495'_1582 T__'8804'__18
v2
du_'60''8658''60''7495'_1582 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 -> AgdaAny
du_'60''8658''60''7495'_1582 :: T__'8804'__18 -> Any
du_'60''8658''60''7495'_1582 T__'8804'__18
v0
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v3
        -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v3 of
             T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
               -> () -> Any
forall a b. a -> b
coe ()
MAlonzo.Code.Agda.Builtin.Unit.C_tt_8
             MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6
               -> (T__'8804'__18 -> Any) -> Any -> Any
forall a b. a -> b
coe
                    T__'8804'__18 -> Any
du_'60''8658''60''7495'_1582
                    ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6)
             T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError
      T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.<ᵇ-reflects-<
d_'60''7495''45'reflects'45''60'_1590 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Reflects_14
d_'60''7495''45'reflects'45''60'_1590 :: Integer -> Integer -> T_Reflects_14
d_'60''7495''45'reflects'45''60'_1590 Integer
v0 Integer
v1
  = (Bool -> (Any -> Any) -> T_Reflects_14)
-> Any -> (Any -> Any) -> T_Reflects_14
forall a b. a -> b
coe
      Bool -> (Any -> Any) -> T_Reflects_14
MAlonzo.Code.Relation.Nullary.Reflects.du_fromEquivalence_28
      ((Integer -> Integer -> Bool) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Bool
ltInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      (\ Any
v2 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'60''7495''8658''60'_1566 (Integer -> Any
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.≤ᵇ⇒≤
d_'8804''7495''8658''8804'_1600 ::
  Integer ->
  Integer -> AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''7495''8658''8804'_1600 :: Integer -> Integer -> Any -> T__'8804'__18
d_'8804''7495''8658''8804'_1600 Integer
v0 ~Integer
v1 ~Any
v2
  = Integer -> T__'8804'__18
du_'8804''7495''8658''8804'_1600 Integer
v0
du_'8804''7495''8658''8804'_1600 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''7495''8658''8804'_1600 :: Integer -> T__'8804'__18
du_'8804''7495''8658''8804'_1600 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'60''7495''8658''60'_1566 (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties.≤⇒≤ᵇ
d_'8804''8658''8804''7495'_1616 ::
  Integer ->
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18 -> AgdaAny
d_'8804''8658''8804''7495'_1616 :: Integer -> Integer -> T__'8804'__18 -> Any
d_'8804''8658''8804''7495'_1616 ~Integer
v0 ~Integer
v1 T__'8804'__18
v2
  = T__'8804'__18 -> Any
du_'8804''8658''8804''7495'_1616 T__'8804'__18
v2
du_'8804''8658''8804''7495'_1616 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 -> AgdaAny
du_'8804''8658''8804''7495'_1616 :: T__'8804'__18 -> Any
du_'8804''8658''8804''7495'_1616 T__'8804'__18
v0
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0 of
      T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
        -> () -> Any
forall a b. a -> b
coe ()
MAlonzo.Code.Agda.Builtin.Unit.C_tt_8
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v3
        -> (T__'8804'__18 -> Any) -> Any -> Any
forall a b. a -> b
coe
             T__'8804'__18 -> Any
du_'60''8658''60''7495'_1582
             ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v3)
      T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.≤ᵇ-reflects-≤
d_'8804''7495''45'reflects'45''8804'_1624 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Reflects_14
d_'8804''7495''45'reflects'45''8804'_1624 :: Integer -> Integer -> T_Reflects_14
d_'8804''7495''45'reflects'45''8804'_1624 Integer
v0 Integer
v1
  = (Bool -> (Any -> Any) -> T_Reflects_14)
-> Any -> (Any -> Any) -> T_Reflects_14
forall a b. a -> b
coe
      Bool -> (Any -> Any) -> T_Reflects_14
MAlonzo.Code.Relation.Nullary.Reflects.du_fromEquivalence_28
      ((Integer -> Integer -> Bool) -> Any -> Any -> Any
forall a b. a -> b
coe
         Integer -> Integer -> Bool
MAlonzo.Code.Data.Nat.Base.d__'8804''7495'__10 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      (\ Any
v2 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'8804''7495''8658''8804'_1600 (Integer -> Any
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.≤-reflexive
d_'8804''45'reflexive_1630 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''45'reflexive_1630 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_'8804''45'reflexive_1630 Integer
v0 ~Integer
v1 ~T__'8801'__12
v2
  = Integer -> T__'8804'__18
du_'8804''45'reflexive_1630 Integer
v0
du_'8804''45'reflexive_1630 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''45'reflexive_1630 :: Integer -> T__'8804'__18
du_'8804''45'reflexive_1630 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'8804''45'reflexive_1630 (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.≤-refl
d_'8804''45'refl_1634 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''45'refl_1634 :: Integer -> T__'8804'__18
d_'8804''45'refl_1634 Integer
v0 = (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'8804''45'reflexive_1630 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.≤-antisym
d_'8804''45'antisym_1636 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8804''45'antisym_1636 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
d_'8804''45'antisym_1636 = Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≤-trans
d_'8804''45'trans_1642 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''45'trans_1642 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8804''45'trans_1642 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3 T__'8804'__18
v4
  = T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 T__'8804'__18
v3 T__'8804'__18
v4
du_'8804''45'trans_1642 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''45'trans_1642 :: T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 T__'8804'__18
v0 T__'8804'__18
v1
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0 of
      T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
        -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v4
        -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
             MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v7
               -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
                    T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                    ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v4) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v7))
             T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.≤-total
d_'8804''45'total_1648 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'8804''45'total_1648 :: Integer -> Integer -> T__'8846'__30
d_'8804''45'total_1648 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Any -> T__'8846'__30) -> Any -> T__'8846'__30
forall a b. a -> b
coe
             Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
             (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8846'__30
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe
                       Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                       (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       (let v4 :: T__'8846'__30
v4 = Integer -> Integer -> T__'8846'__30
d_'8804''45'total_1648 (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe Integer
v3) in
                        Any -> Any
forall a b. a -> b
coe
                          (case T__'8846'__30 -> T__'8846'__30
forall a b. a -> b
coe T__'8846'__30
v4 of
                             MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
v5
                               -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe
                                    Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
                                    ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 Any
v5)
                             MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
v5
                               -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe
                                    Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                                    ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 Any
v5)
                             T__'8846'__30
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)))
-- Data.Nat.Properties.≤-irrelevant
d_'8804''45'irrelevant_1670 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8804''45'irrelevant_1670 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
d_'8804''45'irrelevant_1670 = Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._≤?_
d__'8804''63'__1676 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8804''63'__1676 :: Integer -> Integer -> T_Dec_32
d__'8804''63'__1676 Integer
v0 Integer
v1
  = ((Any -> Any) -> T_Dec_32 -> T_Dec_32)
-> (Any -> Any) -> Any -> T_Dec_32
forall a b. a -> b
coe
      (Any -> Any) -> T_Dec_32 -> T_Dec_32
MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
      (\ Any
v2 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'8804''7495''8658''8804'_1600 (Integer -> Any
forall a b. a -> b
coe Integer
v0))
      ((Bool -> T_Dec_32) -> Any -> Any
forall a b. a -> b
coe
         Bool -> T_Dec_32
MAlonzo.Code.Data.Bool.Properties.d_T'63'_2210
         ((Integer -> Integer -> Bool) -> Any -> Any -> Any
forall a b. a -> b
coe
            Integer -> Integer -> Bool
MAlonzo.Code.Data.Nat.Base.d__'8804''7495'__10 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties._≥?_
d__'8805''63'__1682 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8805''63'__1682 :: Integer -> Integer -> T_Dec_32
d__'8805''63'__1682 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8804''63'__1676 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.≤-isPreorder
d_'8804''45'isPreorder_1684 ::
  MAlonzo.Code.Relation.Binary.Structures.T_IsPreorder_70
d_'8804''45'isPreorder_1684 :: T_IsPreorder_70
d_'8804''45'isPreorder_1684
  = (T_IsEquivalence_26
 -> (Any -> Any -> Any -> Any)
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> T_IsPreorder_70)
-> Any
-> (Any -> Any -> Any -> Any)
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> T_IsPreorder_70
forall a b. a -> b
coe
      T_IsEquivalence_26
-> (Any -> Any -> Any -> Any)
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> T_IsPreorder_70
MAlonzo.Code.Relation.Binary.Structures.C_IsPreorder'46'constructor_3993
      (T_IsEquivalence_26 -> Any
forall a b. a -> b
coe
         T_IsEquivalence_26
MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_242)
      (\ Any
v0 Any
v1 Any
v2 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'8804''45'reflexive_1630 Any
v0)
      (\ Any
v0 Any
v1 Any
v2 Any
v3 Any
v4 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 Any
v3 Any
v4)
-- Data.Nat.Properties.≤-isTotalPreorder
d_'8804''45'isTotalPreorder_1686 ::
  MAlonzo.Code.Relation.Binary.Structures.T_IsTotalPreorder_118
d_'8804''45'isTotalPreorder_1686 :: T_IsTotalPreorder_118
d_'8804''45'isTotalPreorder_1686
  = (T_IsPreorder_70
 -> (Any -> Any -> T__'8846'__30) -> T_IsTotalPreorder_118)
-> Any -> Any -> T_IsTotalPreorder_118
forall a b. a -> b
coe
      T_IsPreorder_70
-> (Any -> Any -> T__'8846'__30) -> T_IsTotalPreorder_118
MAlonzo.Code.Relation.Binary.Structures.C_IsTotalPreorder'46'constructor_7939
      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684) ((Integer -> Integer -> T__'8846'__30) -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8846'__30
d_'8804''45'total_1648)
-- Data.Nat.Properties.≤-isPartialOrder
d_'8804''45'isPartialOrder_1688 ::
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialOrder_162
d_'8804''45'isPartialOrder_1688 :: T_IsPartialOrder_162
d_'8804''45'isPartialOrder_1688
  = (T_IsPreorder_70
 -> (Any -> Any -> Any -> Any -> Any) -> T_IsPartialOrder_162)
-> Any -> Any -> T_IsPartialOrder_162
forall a b. a -> b
coe
      T_IsPreorder_70
-> (Any -> Any -> Any -> Any -> Any) -> T_IsPartialOrder_162
MAlonzo.Code.Relation.Binary.Structures.C_IsPartialOrder'46'constructor_9297
      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684) Any
forall a. a
erased
-- Data.Nat.Properties.≤-isTotalOrder
d_'8804''45'isTotalOrder_1690 ::
  MAlonzo.Code.Relation.Binary.Structures.T_IsTotalOrder_384
d_'8804''45'isTotalOrder_1690 :: T_IsTotalOrder_384
d_'8804''45'isTotalOrder_1690
  = (T_IsPartialOrder_162
 -> (Any -> Any -> T__'8846'__30) -> T_IsTotalOrder_384)
-> Any -> Any -> T_IsTotalOrder_384
forall a b. a -> b
coe
      T_IsPartialOrder_162
-> (Any -> Any -> T__'8846'__30) -> T_IsTotalOrder_384
MAlonzo.Code.Relation.Binary.Structures.C_IsTotalOrder'46'constructor_19815
      (T_IsPartialOrder_162 -> Any
forall a b. a -> b
coe T_IsPartialOrder_162
d_'8804''45'isPartialOrder_1688) ((Integer -> Integer -> T__'8846'__30) -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8846'__30
d_'8804''45'total_1648)
-- Data.Nat.Properties.≤-isDecTotalOrder
d_'8804''45'isDecTotalOrder_1692 ::
  MAlonzo.Code.Relation.Binary.Structures.T_IsDecTotalOrder_434
d_'8804''45'isDecTotalOrder_1692 :: T_IsDecTotalOrder_434
d_'8804''45'isDecTotalOrder_1692
  = (T_IsTotalOrder_384
 -> (Any -> Any -> T_Dec_32)
 -> (Any -> Any -> T_Dec_32)
 -> T_IsDecTotalOrder_434)
-> Any -> Any -> Any -> T_IsDecTotalOrder_434
forall a b. a -> b
coe
      T_IsTotalOrder_384
-> (Any -> Any -> T_Dec_32)
-> (Any -> Any -> T_Dec_32)
-> T_IsDecTotalOrder_434
MAlonzo.Code.Relation.Binary.Structures.C_IsDecTotalOrder'46'constructor_21785
      (T_IsTotalOrder_384 -> Any
forall a b. a -> b
coe T_IsTotalOrder_384
d_'8804''45'isTotalOrder_1690) ((Integer -> Integer -> T_Dec_32) -> Any
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8799'__1528)
      ((Integer -> Integer -> T_Dec_32) -> Any
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8804''63'__1676)
-- Data.Nat.Properties.≤-preorder
d_'8804''45'preorder_1694 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_Preorder_132
d_'8804''45'preorder_1694 :: T_Preorder_132
d_'8804''45'preorder_1694
  = (T_IsPreorder_70 -> T_Preorder_132)
-> T_IsPreorder_70 -> T_Preorder_132
forall a b. a -> b
coe
      T_IsPreorder_70 -> T_Preorder_132
MAlonzo.Code.Relation.Binary.Bundles.C_Preorder'46'constructor_2269
      T_IsPreorder_70
d_'8804''45'isPreorder_1684
-- Data.Nat.Properties.≤-totalPreorder
d_'8804''45'totalPreorder_1696 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 :: T_TotalPreorder_204
d_'8804''45'totalPreorder_1696
  = (T_IsTotalPreorder_118 -> T_TotalPreorder_204)
-> T_IsTotalPreorder_118 -> T_TotalPreorder_204
forall a b. a -> b
coe
      T_IsTotalPreorder_118 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Bundles.C_TotalPreorder'46'constructor_3669
      T_IsTotalPreorder_118
d_'8804''45'isTotalPreorder_1686
-- Data.Nat.Properties.≤-poset
d_'8804''45'poset_1698 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_Poset_282
d_'8804''45'poset_1698 :: T_Poset_282
d_'8804''45'poset_1698
  = (T_IsPartialOrder_162 -> T_Poset_282)
-> T_IsPartialOrder_162 -> T_Poset_282
forall a b. a -> b
coe
      T_IsPartialOrder_162 -> T_Poset_282
MAlonzo.Code.Relation.Binary.Bundles.C_Poset'46'constructor_5219
      T_IsPartialOrder_162
d_'8804''45'isPartialOrder_1688
-- Data.Nat.Properties.≤-totalOrder
d_'8804''45'totalOrder_1700 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_TotalOrder_652
d_'8804''45'totalOrder_1700 :: T_TotalOrder_652
d_'8804''45'totalOrder_1700
  = (T_IsTotalOrder_384 -> T_TotalOrder_652)
-> T_IsTotalOrder_384 -> T_TotalOrder_652
forall a b. a -> b
coe
      T_IsTotalOrder_384 -> T_TotalOrder_652
MAlonzo.Code.Relation.Binary.Bundles.C_TotalOrder'46'constructor_12489
      T_IsTotalOrder_384
d_'8804''45'isTotalOrder_1690
-- Data.Nat.Properties.≤-decTotalOrder
d_'8804''45'decTotalOrder_1702 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_DecTotalOrder_740
d_'8804''45'decTotalOrder_1702 :: T_DecTotalOrder_740
d_'8804''45'decTotalOrder_1702
  = (T_IsDecTotalOrder_434 -> T_DecTotalOrder_740)
-> T_IsDecTotalOrder_434 -> T_DecTotalOrder_740
forall a b. a -> b
coe
      T_IsDecTotalOrder_434 -> T_DecTotalOrder_740
MAlonzo.Code.Relation.Binary.Bundles.C_DecTotalOrder'46'constructor_14337
      T_IsDecTotalOrder_434
d_'8804''45'isDecTotalOrder_1692
-- Data.Nat.Properties.s≤s-injective
d_s'8804's'45'injective_1712 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_s'8804's'45'injective_1712 :: Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8801'__12
-> T__'8801'__12
d_s'8804's'45'injective_1712 = Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≤-step
d_'8804''45'step_1718 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''45'step_1718 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8804''45'step_1718 ~Integer
v0 ~Integer
v1 T__'8804'__18
v2 = T__'8804'__18 -> T__'8804'__18
du_'8804''45'step_1718 T__'8804'__18
v2
du_'8804''45'step_1718 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''45'step_1718 :: T__'8804'__18 -> T__'8804'__18
du_'8804''45'step_1718 T__'8804'__18
v0 = T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0
-- Data.Nat.Properties.n≤1+n
d_n'8804'1'43'n_1724 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8804'1'43'n_1724 :: Integer -> T__'8804'__18
d_n'8804'1'43'n_1724 Integer
v0 = (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.1+n≰n
d_1'43'n'8816'n_1728 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_1'43'n'8816'n_1728 :: Integer -> T__'8804'__18 -> T_'8869'_4
d_1'43'n'8816'n_1728 = Integer -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.n≤0⇒n≡0
d_n'8804'0'8658'n'8801'0_1734 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_n'8804'0'8658'n'8801'0_1734 :: Integer -> T__'8804'__18 -> T__'8801'__12
d_n'8804'0'8658'n'8801'0_1734 = Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.<⇒≤
d_'60''8658''8804'_1736 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'60''8658''8804'_1736 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'60''8658''8804'_1736 ~Integer
v0 Integer
v1 T__'8804'__18
v2 = Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 Integer
v1 T__'8804'__18
v2
du_'60''8658''8804'_1736 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'60''8658''8804'_1736 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 Integer
v0 T__'8804'__18
v1
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v4
        -> let v5 :: Integer
v5 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v4)
                ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v5)))
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.<⇒≢
d_'60''8658''8802'_1740 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'60''8658''8802'_1740 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
d_'60''8658''8802'_1740 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.>⇒≢
d_'62''8658''8802'_1744 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'62''8658''8802'_1744 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
d_'62''8658''8802'_1744 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.≤⇒≯
d_'8804''8658''8815'_1746 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'8804''8658''8815'_1746 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T_'8869'_4
d_'8804''8658''8815'_1746 = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.<⇒≱
d_'60''8658''8817'_1752 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'60''8658''8817'_1752 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T_'8869'_4
d_'60''8658''8817'_1752 = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.<⇒≯
d_'60''8658''8815'_1758 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'60''8658''8815'_1758 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T_'8869'_4
d_'60''8658''8815'_1758 = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.≰⇒≮
d_'8816''8658''8814'_1764 ::
  Integer ->
  Integer ->
  (MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'8816''8658''8814'_1764 :: Integer
-> Integer
-> (T__'8804'__18 -> T_'8869'_4)
-> T__'8804'__18
-> T_'8869'_4
d_'8816''8658''8814'_1764 = Integer
-> Integer
-> (T__'8804'__18 -> T_'8869'_4)
-> T__'8804'__18
-> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.≰⇒>
d_'8816''8658''62'_1770 ::
  Integer ->
  Integer ->
  (MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8816''8658''62'_1770 :: Integer
-> Integer -> (T__'8804'__18 -> T_'8869'_4) -> T__'8804'__18
d_'8816''8658''62'_1770 Integer
v0 Integer
v1 ~T__'8804'__18 -> T_'8869'_4
v2 = Integer -> Integer -> T__'8804'__18
du_'8816''8658''62'_1770 Integer
v0 Integer
v1
du_'8816''8658''62'_1770 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8816''8658''62'_1770 :: Integer -> Integer -> T__'8804'__18
du_'8816''8658''62'_1770 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> Any -> T__'8804'__18
forall a b. a -> b
coe
             Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> (T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                       (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
du_'8816''8658''62'_1770 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3))))
-- Data.Nat.Properties.≰⇒≥
d_'8816''8658''8805'_1782 ::
  Integer ->
  Integer ->
  (MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8816''8658''8805'_1782 :: Integer
-> Integer -> (T__'8804'__18 -> T_'8869'_4) -> T__'8804'__18
d_'8816''8658''8805'_1782 Integer
v0 Integer
v1 ~T__'8804'__18 -> T_'8869'_4
v2
  = Integer -> Integer -> T__'8804'__18
du_'8816''8658''8805'_1782 Integer
v0 Integer
v1
du_'8816''8658''8805'_1782 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8816''8658''8805'_1782 :: Integer -> Integer -> T__'8804'__18
du_'8816''8658''8805'_1782 Integer
v0 Integer
v1
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
      ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
du_'8816''8658''62'_1770 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties.≮⇒≥
d_'8814''8658''8805'_1784 ::
  Integer ->
  Integer ->
  (MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8814''8658''8805'_1784 :: Integer
-> Integer -> (T__'8804'__18 -> T_'8869'_4) -> T__'8804'__18
d_'8814''8658''8805'_1784 Integer
v0 Integer
v1 ~T__'8804'__18 -> T_'8869'_4
v2
  = Integer -> Integer -> T__'8804'__18
du_'8814''8658''8805'_1784 Integer
v0 Integer
v1
du_'8814''8658''8805'_1784 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8814''8658''8805'_1784 :: Integer -> Integer -> T__'8804'__18
du_'8814''8658''8805'_1784 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
                Integer
0 -> Any -> Any
forall a b. a -> b
coe
                       Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
du_'8814''8658''8805'_1784 (Integer -> Any
forall a b. a -> b
coe Integer
v3) (Integer -> Any
forall a b. a -> b
coe Integer
v2))))
-- Data.Nat.Properties.≤∧≢⇒<
d_'8804''8743''8802''8658''60'_1800 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''8743''8802''8658''60'_1800 :: Integer
-> Integer
-> T__'8804'__18
-> (T__'8801'__12 -> T_'8869'_4)
-> T__'8804'__18
d_'8804''8743''8802''8658''60'_1800 ~Integer
v0 Integer
v1 T__'8804'__18
v2 ~T__'8801'__12 -> T_'8869'_4
v3
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804''8743''8802''8658''60'_1800 Integer
v1 T__'8804'__18
v2
du_'8804''8743''8802''8658''60'_1800 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''8743''8802''8658''60'_1800 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804''8743''8802''8658''60'_1800 Integer
v0 T__'8804'__18
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Any -> Any -> Any) -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
             Any -> Any -> Any
forall a b. a -> b -> b
seq (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)
             (Any -> Any
forall a b. a -> b
coe
                Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24)
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
                T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
                  -> (T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                       (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
                MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5
                  -> (T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                       ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804''8743''8802''8658''60'_1800 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5))
                T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.≤∧≮⇒≡
d_'8804''8743''8814''8658''8801'_1818 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  (MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8804''8743''8814''8658''8801'_1818 :: Integer
-> Integer
-> T__'8804'__18
-> (T__'8804'__18 -> T_'8869'_4)
-> T__'8801'__12
d_'8804''8743''8814''8658''8801'_1818 = Integer
-> Integer
-> T__'8804'__18
-> (T__'8804'__18 -> T_'8869'_4)
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≤-<-connex
d_'8804''45''60''45'connex_1824 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'8804''45''60''45'connex_1824 :: Integer -> Integer -> T__'8846'__30
d_'8804''45''60''45'connex_1824 Integer
v0 Integer
v1
  = let v2 :: T_Dec_32
v2 = Integer -> Integer -> T_Dec_32
d__'8804''63'__1676 (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe Integer
v1) in
    Any -> T__'8846'__30
forall a b. a -> b
coe
      (case T_Dec_32 -> T_Dec_32
forall a b. a -> b
coe T_Dec_32
v2 of
         MAlonzo.Code.Relation.Nullary.C__because__46 Bool
v3 T_Reflects_14
v4
           -> if Bool -> Bool
forall a b. a -> b
coe Bool
v3
                then case T_Reflects_14 -> T_Reflects_14
forall a b. a -> b
coe T_Reflects_14
v4 of
                       MAlonzo.Code.Relation.Nullary.C_of'696'_22 Any
v5
                         -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 (Any -> Any
forall a b. a -> b
coe Any
v5)
                       T_Reflects_14
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError
                else (Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
                       Any -> Any -> Any
forall a b. a -> b -> b
seq (T_Reflects_14 -> Any
forall a b. a -> b
coe T_Reflects_14
v4)
                       ((Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe
                          Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
du_'8816''8658''62'_1770 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
         T_Dec_32
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.≥->-connex
d_'8805''45''62''45'connex_1846 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'8805''45''62''45'connex_1846 :: Integer -> Integer -> T__'8846'__30
d_'8805''45''62''45'connex_1846 Integer
v0 Integer
v1
  = (Integer -> Integer -> T__'8846'__30)
-> Any -> Any -> T__'8846'__30
forall a b. a -> b
coe Integer -> Integer -> T__'8846'__30
d_'8804''45''60''45'connex_1824 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.<-≤-connex
d_'60''45''8804''45'connex_1848 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'60''45''8804''45'connex_1848 :: Integer -> Integer -> T__'8846'__30
d_'60''45''8804''45'connex_1848
  = ((Any -> Any -> T__'8846'__30) -> Any -> Any -> T__'8846'__30)
-> Any -> Integer -> Integer -> T__'8846'__30
forall a b. a -> b
coe
      (Any -> Any -> T__'8846'__30) -> Any -> Any -> T__'8846'__30
MAlonzo.Code.Relation.Binary.Consequences.du_flip'45'Connex_788
      ((Integer -> Integer -> T__'8846'__30) -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8846'__30
d_'8804''45''60''45'connex_1824)
-- Data.Nat.Properties.>-≥-connex
d_'62''45''8805''45'connex_1850 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'62''45''8805''45'connex_1850 :: Integer -> Integer -> T__'8846'__30
d_'62''45''8805''45'connex_1850
  = ((Any -> Any -> T__'8846'__30) -> Any -> Any -> T__'8846'__30)
-> Any -> Integer -> Integer -> T__'8846'__30
forall a b. a -> b
coe
      (Any -> Any -> T__'8846'__30) -> Any -> Any -> T__'8846'__30
MAlonzo.Code.Relation.Binary.Consequences.du_flip'45'Connex_788
      ((Integer -> Integer -> T__'8846'__30) -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8846'__30
d_'8805''45''62''45'connex_1846)
-- Data.Nat.Properties.<-irrefl
d_'60''45'irrefl_1852 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'60''45'irrefl_1852 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18 -> T_'8869'_4
d_'60''45'irrefl_1852 = Integer -> Integer -> T__'8801'__12 -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.<-asym
d_'60''45'asym_1856 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'60''45'asym_1856 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T_'8869'_4
d_'60''45'asym_1856 = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.<-trans
d_'60''45'trans_1862 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'60''45'trans_1862 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'60''45'trans_1862 ~Integer
v0 Integer
v1 ~Integer
v2 T__'8804'__18
v3 T__'8804'__18
v4
  = Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans_1862 Integer
v1 T__'8804'__18
v3 T__'8804'__18
v4
du_'60''45'trans_1862 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'60''45'trans_1862 :: Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans_1862 Integer
v0 T__'8804'__18
v1 T__'8804'__18
v2
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5
        -> let v6 :: Integer
v6 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v2 of
                MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9
                  -> (T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                       ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5)
                          ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                             T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_n'8804'1'43'n_1724 (Integer -> Any
forall a b. a -> b
coe Integer
v6))
                             (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v9)))
                T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.<-transʳ
d_'60''45'trans'691'_1868 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'60''45'trans'691'_1868 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'60''45'trans'691'_1868 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3 T__'8804'__18
v4
  = T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 T__'8804'__18
v3 T__'8804'__18
v4
du_'60''45'trans'691'_1868 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'60''45'trans'691'_1868 :: T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 T__'8804'__18
v0 T__'8804'__18
v1
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v4
        -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
             T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
             ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v4))
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.<-transˡ
d_'60''45'trans'737'_1874 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'60''45'trans'737'_1874 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'60''45'trans'737'_1874 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3 T__'8804'__18
v4
  = T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'737'_1874 T__'8804'__18
v3 T__'8804'__18
v4
du_'60''45'trans'737'_1874 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'60''45'trans'737'_1874 :: T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'737'_1874 T__'8804'__18
v0 T__'8804'__18
v1
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v4
        -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
             MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v7
               -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
                    T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                    ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v4) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v7))
             T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.<-cmp
d_'60''45'cmp_1880 ::
  Integer ->
  Integer -> MAlonzo.Code.Relation.Binary.Definitions.T_Tri_136
d_'60''45'cmp_1880 :: Integer -> Integer -> T_Tri_136
d_'60''45'cmp_1880 Integer
v0 Integer
v1
  = let v2 :: T_Dec_32
v2 = Integer -> Integer -> T_Dec_32
d__'8799'__1528 (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe Integer
v1) in
    Any -> T_Tri_136
forall a b. a -> b
coe
      (let v3 :: T_Dec_32
v3
             = Bool -> T_Dec_32
MAlonzo.Code.Data.Bool.Properties.d_T'63'_2210
                 ((Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe Integer -> Integer -> Bool
ltInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)) in
       Any -> Any
forall a b. a -> b
coe
         (case T_Dec_32 -> T_Dec_32
forall a b. a -> b
coe T_Dec_32
v2 of
            MAlonzo.Code.Relation.Nullary.C__because__46 Bool
v4 T_Reflects_14
v5
              -> if Bool -> Bool
forall a b. a -> b
coe Bool
v4
                   then case T_Reflects_14 -> T_Reflects_14
forall a b. a -> b
coe T_Reflects_14
v5 of
                          MAlonzo.Code.Relation.Nullary.C_of'696'_22 Any
v6
                            -> (Any -> T_Tri_136) -> Any -> Any
forall a b. a -> b
coe Any -> T_Tri_136
MAlonzo.Code.Relation.Binary.Definitions.C_tri'8776'_158 Any
v6
                          T_Reflects_14
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError
                   else (Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
                          Any -> Any -> Any
forall a b. a -> b -> b
seq (T_Reflects_14 -> Any
forall a b. a -> b
coe T_Reflects_14
v5)
                          (case T_Dec_32 -> T_Dec_32
forall a b. a -> b
coe T_Dec_32
v3 of
                             MAlonzo.Code.Relation.Nullary.C__because__46 Bool
v6 T_Reflects_14
v7
                               -> if Bool -> Bool
forall a b. a -> b
coe Bool
v6
                                    then (Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
                                           Any -> Any -> Any
forall a b. a -> b -> b
seq (T_Reflects_14 -> Any
forall a b. a -> b
coe T_Reflects_14
v7)
                                           ((Any -> T_Tri_136) -> Any -> Any
forall a b. a -> b
coe
                                              Any -> T_Tri_136
MAlonzo.Code.Relation.Binary.Definitions.C_tri'60'_150
                                              ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'60''7495''8658''60'_1566 (Integer -> Any
forall a b. a -> b
coe Integer
v0)))
                                    else (Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
                                           Any -> Any -> Any
forall a b. a -> b -> b
seq (T_Reflects_14 -> Any
forall a b. a -> b
coe T_Reflects_14
v7)
                                           ((Any -> T_Tri_136) -> Any -> Any
forall a b. a -> b
coe
                                              Any -> T_Tri_136
MAlonzo.Code.Relation.Binary.Definitions.C_tri'62'_166
                                              ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe
                                                 Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804''8743''8802''8658''60'_1800 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                                                 ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe
                                                    Integer -> Integer -> T__'8804'__18
du_'8814''8658''8805'_1784 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))))
                             T_Dec_32
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
            T_Dec_32
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError))
-- Data.Nat.Properties._<?_
d__'60''63'__1912 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'60''63'__1912 :: Integer -> Integer -> T_Dec_32
d__'60''63'__1912 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe
      Integer -> Integer -> T_Dec_32
d__'8804''63'__1676 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0))
      (Integer -> Any
forall a b. a -> b
coe Integer
v1)
-- Data.Nat.Properties._>?_
d__'62''63'__1918 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'62''63'__1918 :: Integer -> Integer -> T_Dec_32
d__'62''63'__1918 Integer
v0 Integer
v1 = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'60''63'__1912 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.<-irrelevant
d_'60''45'irrelevant_1920 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'60''45'irrelevant_1920 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
d_'60''45'irrelevant_1920 = Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.<-resp₂-≡
d_'60''45'resp'8322''45''8801'_1922 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'60''45'resp'8322''45''8801'_1922 :: T_Σ_14
d_'60''45'resp'8322''45''8801'_1922
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe
      Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
      ((Any -> Any -> Any -> Any -> Any -> Any) -> Any
forall a b. a -> b
coe (\ Any
v0 Any
v1 Any
v2 Any
v3 Any
v4 -> Any
v4)) ((Any -> Any -> Any -> Any -> Any -> Any) -> Any
forall a b. a -> b
coe (\ Any
v0 Any
v1 Any
v2 Any
v3 Any
v4 -> Any
v4))
-- Data.Nat.Properties.<-isStrictPartialOrder
d_'60''45'isStrictPartialOrder_1928 ::
  MAlonzo.Code.Relation.Binary.Structures.T_IsStrictPartialOrder_266
d_'60''45'isStrictPartialOrder_1928 :: T_IsStrictPartialOrder_266
d_'60''45'isStrictPartialOrder_1928
  = (T_IsEquivalence_26
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> T_Σ_14
 -> T_IsStrictPartialOrder_266)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> T_Σ_14
-> T_IsStrictPartialOrder_266
forall a b. a -> b
coe
      T_IsEquivalence_26
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> T_Σ_14
-> T_IsStrictPartialOrder_266
MAlonzo.Code.Relation.Binary.Structures.C_IsStrictPartialOrder'46'constructor_13145
      (T_IsEquivalence_26 -> Any
forall a b. a -> b
coe
         T_IsEquivalence_26
MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_242)
      (\ Any
v0 Any
v1 Any
v2 Any
v3 Any
v4 -> (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans_1862 Any
v1 Any
v3 Any
v4)
      T_Σ_14
d_'60''45'resp'8322''45''8801'_1922
-- Data.Nat.Properties.<-isStrictTotalOrder
d_'60''45'isStrictTotalOrder_1930 ::
  MAlonzo.Code.Relation.Binary.Structures.T_IsStrictTotalOrder_502
d_'60''45'isStrictTotalOrder_1930 :: T_IsStrictTotalOrder_502
d_'60''45'isStrictTotalOrder_1930
  = (T_IsEquivalence_26
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> (Any -> Any -> T_Tri_136)
 -> T_IsStrictTotalOrder_502)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> T_IsStrictTotalOrder_502
forall a b. a -> b
coe
      T_IsEquivalence_26
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> (Any -> Any -> T_Tri_136)
-> T_IsStrictTotalOrder_502
MAlonzo.Code.Relation.Binary.Structures.C_IsStrictTotalOrder'46'constructor_23999
      (T_IsEquivalence_26 -> Any
forall a b. a -> b
coe
         T_IsEquivalence_26
MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_242)
      (\ Any
v0 Any
v1 Any
v2 Any
v3 Any
v4 -> (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans_1862 Any
v1 Any
v3 Any
v4)
      ((Integer -> Integer -> T_Tri_136) -> Any
forall a b. a -> b
coe Integer -> Integer -> T_Tri_136
d_'60''45'cmp_1880)
-- Data.Nat.Properties.<-strictPartialOrder
d_'60''45'strictPartialOrder_1932 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_StrictPartialOrder_472
d_'60''45'strictPartialOrder_1932 :: T_StrictPartialOrder_472
d_'60''45'strictPartialOrder_1932
  = (T_IsStrictPartialOrder_266 -> T_StrictPartialOrder_472)
-> T_IsStrictPartialOrder_266 -> T_StrictPartialOrder_472
forall a b. a -> b
coe
      T_IsStrictPartialOrder_266 -> T_StrictPartialOrder_472
MAlonzo.Code.Relation.Binary.Bundles.C_StrictPartialOrder'46'constructor_8957
      T_IsStrictPartialOrder_266
d_'60''45'isStrictPartialOrder_1928
-- Data.Nat.Properties.<-strictTotalOrder
d_'60''45'strictTotalOrder_1934 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_StrictTotalOrder_864
d_'60''45'strictTotalOrder_1934 :: T_StrictTotalOrder_864
d_'60''45'strictTotalOrder_1934
  = (T_IsStrictTotalOrder_502 -> T_StrictTotalOrder_864)
-> T_IsStrictTotalOrder_502 -> T_StrictTotalOrder_864
forall a b. a -> b
coe
      T_IsStrictTotalOrder_502 -> T_StrictTotalOrder_864
MAlonzo.Code.Relation.Binary.Bundles.C_StrictTotalOrder'46'constructor_16739
      T_IsStrictTotalOrder_502
d_'60''45'isStrictTotalOrder_1930
-- Data.Nat.Properties.n≮n
d_n'8814'n_1938 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_n'8814'n_1938 :: Integer -> T__'8804'__18 -> T_'8869'_4
d_n'8814'n_1938 = Integer -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.0<1+n
d_0'60'1'43'n_1944 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_0'60'1'43'n_1944 :: Integer -> T__'8804'__18
d_0'60'1'43'n_1944 ~Integer
v0 = T__'8804'__18
du_0'60'1'43'n_1944
du_0'60'1'43'n_1944 :: MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_0'60'1'43'n_1944 :: T__'8804'__18
du_0'60'1'43'n_1944
  = (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
      T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
      (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
-- Data.Nat.Properties.n<1+n
d_n'60'1'43'n_1948 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'60'1'43'n_1948 :: Integer -> T__'8804'__18
d_n'60'1'43'n_1948 Integer
v0
  = (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> T__'8804'__18
d_'8804''45'refl_1634 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.n<1⇒n≡0
d_n'60'1'8658'n'8801'0_1954 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_n'60'1'8658'n'8801'0_1954 :: Integer -> T__'8804'__18 -> T__'8801'__12
d_n'60'1'8658'n'8801'0_1954 = Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.n≢0⇒n>0
d_n'8802'0'8658'n'62'0_1960 ::
  Integer ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8802'0'8658'n'62'0_1960 :: Integer -> (T__'8801'__12 -> T_'8869'_4) -> T__'8804'__18
d_n'8802'0'8658'n'62'0_1960 Integer
v0 ~T__'8801'__12 -> T_'8869'_4
v1
  = Integer -> T__'8804'__18
du_n'8802'0'8658'n'62'0_1960 Integer
v0
du_n'8802'0'8658'n'62'0_1960 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_n'8802'0'8658'n'62'0_1960 :: Integer -> T__'8804'__18
du_n'8802'0'8658'n'62'0_1960 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> Any -> T__'8804'__18
forall a b. a -> b
coe
             Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24
      Integer
_ -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
du_0'60'1'43'n_1944
-- Data.Nat.Properties.m<n⇒0<n
d_m'60'n'8658'0'60'n_1970 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8658'0'60'n_1970 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8658'0'60'n_1970 ~Integer
v0 ~Integer
v1 = T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'0'60'n_1970
du_m'60'n'8658'0'60'n_1970 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'60'n'8658'0'60'n_1970 :: T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'0'60'n_1970
  = (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
du_0'60'1'43'n_1944)
-- Data.Nat.Properties.m<n⇒n≢0
d_m'60'n'8658'n'8802'0_1976 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'60'n'8658'n'8802'0_1976 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
d_m'60'n'8658'n'8802'0_1976 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.m<n⇒m≤1+n
d_m'60'n'8658'm'8804'1'43'n_1984 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8658'm'8804'1'43'n_1984 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8658'm'8804'1'43'n_1984 ~Integer
v0 Integer
v1 T__'8804'__18
v2
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'm'8804'1'43'n_1984 Integer
v1 T__'8804'__18
v2
du_m'60'n'8658'm'8804'1'43'n_1984 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'60'n'8658'm'8804'1'43'n_1984 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'm'8804'1'43'n_1984 Integer
v0 T__'8804'__18
v1
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)
-- Data.Nat.Properties.∀[m≤n⇒m≢o]⇒n<o
d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 ::
  Integer ->
  Integer ->
  (Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 :: Integer
-> Integer
-> (Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4)
-> T__'8804'__18
d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 Integer
v0 Integer
v1 ~Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
v2
  = Integer -> Integer -> T__'8804'__18
du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 Integer
v0 Integer
v1
du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 :: Integer -> Integer -> T__'8804'__18
du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
      Integer
0 -> Any -> T__'8804'__18
forall a b. a -> b
coe
             Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
                Integer
0 -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
du_0'60'1'43'n_1944
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe
                             Integer -> Integer -> T__'8804'__18
du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 (Integer -> Any
forall a b. a -> b
coe Integer
v3)
                             (Integer -> Any
forall a b. a -> b
coe Integer
v2))))
-- Data.Nat.Properties._.rec
d_rec_2010 ::
  Integer ->
  Integer ->
  (Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_rec_2010 :: Integer
-> Integer
-> (Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4)
-> Integer
-> T__'8804'__18
-> T__'8801'__12
-> T_'8869'_4
d_rec_2010 = Integer
-> Integer
-> (Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4)
-> Integer
-> T__'8804'__18
-> T__'8801'__12
-> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.∀[m<n⇒m≢o]⇒n≤o
d_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 ::
  Integer ->
  Integer ->
  (Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 :: Integer
-> Integer
-> (Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4)
-> T__'8804'__18
d_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 Integer
v0 Integer
v1 ~Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
v2
  = Integer -> Integer -> T__'8804'__18
du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 Integer
v0 Integer
v1
du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 :: Integer -> Integer -> T__'8804'__18
du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> Any -> Any
forall a b. a -> b
coe
                       Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe
                             Integer -> Integer -> T__'8804'__18
du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 (Integer -> Any
forall a b. a -> b
coe Integer
v2)
                             (Integer -> Any
forall a b. a -> b
coe Integer
v3))))
-- Data.Nat.Properties._.rec
d_rec_2040 ::
  Integer ->
  Integer ->
  (Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_rec_2040 :: Integer
-> Integer
-> (Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4)
-> Integer
-> T__'8804'__18
-> T__'8801'__12
-> T_'8869'_4
d_rec_2040 = Integer
-> Integer
-> (Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4)
-> Integer
-> T__'8804'__18
-> T__'8801'__12
-> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.≤-Reasoning._._IsRelatedTo_
d__IsRelatedTo__2048 :: p -> p -> ()
d__IsRelatedTo__2048 p
a0 p
a1 = ()
-- Data.Nat.Properties.≤-Reasoning._._∎
d__'8718'_2050 ::
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70
d__'8718'_2050 :: Integer -> T__IsRelatedTo__70
d__'8718'_2050
  = (T_IsPreorder_70 -> Any -> T__IsRelatedTo__70)
-> Any -> Integer -> T__IsRelatedTo__70
forall a b. a -> b
coe
      T_IsPreorder_70 -> Any -> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du__'8718'_346
      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
-- Data.Nat.Properties.≤-Reasoning._._≡⟨⟩_
d__'8801''10216''10217'__2052 ::
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70
d__'8801''10216''10217'__2052 :: T__IsRelatedTo__70 -> T__IsRelatedTo__70
d__'8801''10216''10217'__2052 T__IsRelatedTo__70
v0 = T__IsRelatedTo__70 -> T__IsRelatedTo__70
forall a b. a -> b
coe T__IsRelatedTo__70
v0
-- Data.Nat.Properties.≤-Reasoning._.IsEquality
d_IsEquality_2054 :: p -> p -> p -> ()
d_IsEquality_2054 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties.≤-Reasoning._.IsEquality?
d_IsEquality'63'_2056 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Relation.Nullary.T_Dec_32
d_IsEquality'63'_2056 :: Integer -> Integer -> T__IsRelatedTo__70 -> T_Dec_32
d_IsEquality'63'_2056 Integer
v0 Integer
v1 T__IsRelatedTo__70
v2
  = (T__IsRelatedTo__70 -> T_Dec_32) -> T__IsRelatedTo__70 -> T_Dec_32
forall a b. a -> b
coe
      T__IsRelatedTo__70 -> T_Dec_32
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_IsEquality'63'_142
      T__IsRelatedTo__70
v2
-- Data.Nat.Properties.≤-Reasoning._.IsStrict
d_IsStrict_2058 :: p -> p -> p -> ()
d_IsStrict_2058 p
a0 p
a1 p
a2 = ()
-- Data.Nat.Properties.≤-Reasoning._.IsStrict?
d_IsStrict'63'_2060 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Relation.Nullary.T_Dec_32
d_IsStrict'63'_2060 :: Integer -> Integer -> T__IsRelatedTo__70 -> T_Dec_32
d_IsStrict'63'_2060 Integer
v0 Integer
v1 T__IsRelatedTo__70
v2
  = (T__IsRelatedTo__70 -> T_Dec_32) -> T__IsRelatedTo__70 -> T_Dec_32
forall a b. a -> b
coe
      T__IsRelatedTo__70 -> T_Dec_32
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_IsStrict'63'_108
      T__IsRelatedTo__70
v2
-- Data.Nat.Properties.≤-Reasoning._.begin_
d_begin__2062 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_begin__2062 :: Integer -> Integer -> T__IsRelatedTo__70 -> T__'8804'__18
d_begin__2062
  = (T_IsPreorder_70
 -> (Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any)
-> Any
-> (Any -> Any -> Any -> Any)
-> Integer
-> Integer
-> T__IsRelatedTo__70
-> T__'8804'__18
forall a b. a -> b
coe
      T_IsPreorder_70
-> (Any -> Any -> Any -> Any)
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin__160
      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
      (\ Any
v0 Any
v1 Any
v2 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 Any
v1 Any
v2)
-- Data.Nat.Properties.≤-Reasoning._.begin-equality_
d_begin'45'equality__2064 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  AgdaAny -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_begin'45'equality__2064 :: Integer -> Integer -> T__IsRelatedTo__70 -> Any -> T__'8801'__12
d_begin'45'equality__2064 = Integer -> Integer -> T__IsRelatedTo__70 -> Any -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≤-Reasoning._.begin-strict_
d_begin'45'strict__2066 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_begin'45'strict__2066 :: Integer -> Integer -> T__IsRelatedTo__70 -> Any -> T__'8804'__18
d_begin'45'strict__2066 Integer
v0 Integer
v1 T__IsRelatedTo__70
v2 Any
v3
  = (T__IsRelatedTo__70 -> Any) -> T__IsRelatedTo__70 -> T__'8804'__18
forall a b. a -> b
coe
      T__IsRelatedTo__70 -> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin'45'strict__176
      T__IsRelatedTo__70
v2
-- Data.Nat.Properties.≤-Reasoning._.extractEquality
d_extractEquality_2070 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T_IsEquality_126 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_extractEquality_2070 :: Integer
-> Integer
-> T__IsRelatedTo__70
-> T_IsEquality_126
-> T__'8801'__12
d_extractEquality_2070 = Integer
-> Integer
-> T__IsRelatedTo__70
-> T_IsEquality_126
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≤-Reasoning._.extractStrict
d_extractStrict_2072 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T_IsStrict_92 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_extractStrict_2072 :: Integer
-> Integer -> T__IsRelatedTo__70 -> T_IsStrict_92 -> T__'8804'__18
d_extractStrict_2072 Integer
v0 Integer
v1 T__IsRelatedTo__70
v2 T_IsStrict_92
v3
  = (T__IsRelatedTo__70 -> T_IsStrict_92 -> Any)
-> T__IsRelatedTo__70 -> T_IsStrict_92 -> T__'8804'__18
forall a b. a -> b
coe
      T__IsRelatedTo__70 -> T_IsStrict_92 -> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_extractStrict_118
      T__IsRelatedTo__70
v2 T_IsStrict_92
v3
-- Data.Nat.Properties.≤-Reasoning._.step-<
d_step'45''60'_2080 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70
d_step'45''60'_2080 :: Integer
-> Integer
-> Integer
-> T__IsRelatedTo__70
-> T__'8804'__18
-> T__IsRelatedTo__70
d_step'45''60'_2080
  = ((Any -> Any -> Any -> Any -> Any -> Any)
 -> T_Σ_14
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Integer
-> Integer
-> Integer
-> T__IsRelatedTo__70
-> T__'8804'__18
-> T__IsRelatedTo__70
forall a b. a -> b
coe
      (Any -> Any -> Any -> Any -> Any -> Any)
-> T_Σ_14
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''60'_202
      (\ Any
v0 Any
v1 Any
v2 Any
v3 Any
v4 -> (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans_1862 Any
v1 Any
v3 Any
v4)
      (T_Σ_14 -> Any
forall a b. a -> b
coe
         T_Σ_14
MAlonzo.Code.Relation.Binary.PropositionalEquality.Core.du_resp'8322'_144)
      (\ Any
v0 Any
v1 Any
v2 Any
v3 Any
v4 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'737'_1874 Any
v3 Any
v4)
-- Data.Nat.Properties.≤-Reasoning._.step-≡
d_step'45''8801'_2082 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70
d_step'45''8801'_2082 :: Integer
-> Integer
-> Integer
-> T__IsRelatedTo__70
-> T__'8801'__12
-> T__IsRelatedTo__70
d_step'45''8801'_2082 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__IsRelatedTo__70
v3 ~T__'8801'__12
v4
  = T__IsRelatedTo__70 -> T__IsRelatedTo__70
du_step'45''8801'_2082 T__IsRelatedTo__70
v3
du_step'45''8801'_2082 ::
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70
du_step'45''8801'_2082 :: T__IsRelatedTo__70 -> T__IsRelatedTo__70
du_step'45''8801'_2082 T__IsRelatedTo__70
v0 = T__IsRelatedTo__70 -> T__IsRelatedTo__70
forall a b. a -> b
coe T__IsRelatedTo__70
v0
-- Data.Nat.Properties.≤-Reasoning._.step-≡˘
d_step'45''8801''728'_2084 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70
d_step'45''8801''728'_2084 :: Integer
-> Integer
-> Integer
-> T__IsRelatedTo__70
-> T__'8801'__12
-> T__IsRelatedTo__70
d_step'45''8801''728'_2084 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__IsRelatedTo__70
v3 ~T__'8801'__12
v4
  = T__IsRelatedTo__70 -> T__IsRelatedTo__70
du_step'45''8801''728'_2084 T__IsRelatedTo__70
v3
du_step'45''8801''728'_2084 ::
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70
du_step'45''8801''728'_2084 :: T__IsRelatedTo__70 -> T__IsRelatedTo__70
du_step'45''8801''728'_2084 T__IsRelatedTo__70
v0 = T__IsRelatedTo__70 -> T__IsRelatedTo__70
forall a b. a -> b
coe T__IsRelatedTo__70
v0
-- Data.Nat.Properties.≤-Reasoning._.step-≤
d_step'45''8804'_2086 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__70
d_step'45''8804'_2086 :: Integer
-> Integer
-> Integer
-> T__IsRelatedTo__70
-> T__'8804'__18
-> T__IsRelatedTo__70
d_step'45''8804'_2086
  = (T_IsPreorder_70
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Integer
-> Integer
-> Integer
-> T__IsRelatedTo__70
-> T__'8804'__18
-> T__IsRelatedTo__70
forall a b. a -> b
coe
      T_IsPreorder_70
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''8804'_228
      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
      (\ Any
v0 Any
v1 Any
v2 Any
v3 Any
v4 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 Any
v3 Any
v4)
-- Data.Nat.Properties.+-suc
d_'43''45'suc_2110 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'suc_2110 :: Integer -> Integer -> T__'8801'__12
d_'43''45'suc_2110 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-assoc
d_'43''45'assoc_2118 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'assoc_2118 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'43''45'assoc_2118 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-identityˡ
d_'43''45'identity'737'_2126 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'identity'737'_2126 :: Integer -> T__'8801'__12
d_'43''45'identity'737'_2126 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-identityʳ
d_'43''45'identity'691'_2128 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'identity'691'_2128 :: Integer -> T__'8801'__12
d_'43''45'identity'691'_2128 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-identity
d_'43''45'identity_2132 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'43''45'identity_2132 :: T_Σ_14
d_'43''45'identity_2132
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.+-comm
d_'43''45'comm_2134 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'comm_2134 :: Integer -> Integer -> T__'8801'__12
d_'43''45'comm_2134 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-cancelˡ-≡
d_'43''45'cancel'737''45''8801'_2142 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'cancel'737''45''8801'_2142 :: Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'43''45'cancel'737''45''8801'_2142 = Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-cancelʳ-≡
d_'43''45'cancel'691''45''8801'_2150 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'cancel'691''45''8801'_2150 :: Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'43''45'cancel'691''45''8801'_2150 = Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-cancel-≡
d_'43''45'cancel'45''8801'_2152 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'43''45'cancel'45''8801'_2152 :: T_Σ_14
d_'43''45'cancel'45''8801'_2152
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.+-isMagma
d_'43''45'isMagma_2154 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_'43''45'isMagma_2154 :: T_IsMagma_86
d_'43''45'isMagma_2154
  = (T_IsEquivalence_26
 -> (Any -> Any -> Any -> Any -> Any -> Any -> Any) -> T_IsMagma_86)
-> Any -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsEquivalence_26
-> (Any -> Any -> Any -> Any -> Any -> Any -> Any) -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.C_IsMagma'46'constructor_553
      (T_IsEquivalence_26 -> Any
forall a b. a -> b
coe
         T_IsEquivalence_26
MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_242)
      Any
forall a. a
erased
-- Data.Nat.Properties.+-isSemigroup
d_'43''45'isSemigroup_2156 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_'43''45'isSemigroup_2156 :: T_IsSemigroup_194
d_'43''45'isSemigroup_2156
  = (T_IsMagma_86 -> (Any -> Any -> Any -> Any) -> T_IsSemigroup_194)
-> Any -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMagma_86 -> (Any -> Any -> Any -> Any) -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.C_IsSemigroup'46'constructor_4001
      (T_IsMagma_86 -> Any
forall a b. a -> b
coe T_IsMagma_86
d_'43''45'isMagma_2154) Any
forall a. a
erased
-- Data.Nat.Properties.+-isCommutativeSemigroup
d_'43''45'isCommutativeSemigroup_2158 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270
d_'43''45'isCommutativeSemigroup_2158 :: T_IsCommutativeSemigroup_270
d_'43''45'isCommutativeSemigroup_2158
  = (T_IsSemigroup_194
 -> (Any -> Any -> Any) -> T_IsCommutativeSemigroup_270)
-> Any -> Any -> T_IsCommutativeSemigroup_270
forall a b. a -> b
coe
      T_IsSemigroup_194
-> (Any -> Any -> Any) -> T_IsCommutativeSemigroup_270
MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemigroup'46'constructor_5673
      (T_IsSemigroup_194 -> Any
forall a b. a -> b
coe T_IsSemigroup_194
d_'43''45'isSemigroup_2156) Any
forall a. a
erased
-- Data.Nat.Properties.+-0-isMonoid
d_'43''45'0'45'isMonoid_2160 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'43''45'0'45'isMonoid_2160 :: T_IsMonoid_358
d_'43''45'0'45'isMonoid_2160
  = (T_IsSemigroup_194 -> T_Σ_14 -> T_IsMonoid_358)
-> Any -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_Σ_14 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_7687
      (T_IsSemigroup_194 -> Any
forall a b. a -> b
coe T_IsSemigroup_194
d_'43''45'isSemigroup_2156) (T_Σ_14 -> Any
forall a b. a -> b
coe T_Σ_14
d_'43''45'identity_2132)
-- Data.Nat.Properties.+-0-isCommutativeMonoid
d_'43''45'0'45'isCommutativeMonoid_2162 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'43''45'0'45'isCommutativeMonoid_2162 :: T_IsCommutativeMonoid_406
d_'43''45'0'45'isCommutativeMonoid_2162
  = (T_IsMonoid_358
 -> (Any -> Any -> Any) -> T_IsCommutativeMonoid_406)
-> Any -> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsMonoid_358 -> (Any -> Any -> Any) -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_9361
      (T_IsMonoid_358 -> Any
forall a b. a -> b
coe T_IsMonoid_358
d_'43''45'0'45'isMonoid_2160) Any
forall a. a
erased
-- Data.Nat.Properties.+-rawMagma
d_'43''45'rawMagma_2164 ::
  MAlonzo.Code.Algebra.Bundles.T_RawMagma_8
d_'43''45'rawMagma_2164 :: T_RawMagma_8
d_'43''45'rawMagma_2164
  = ((Any -> Any -> Any) -> T_RawMagma_8)
-> (Integer -> Integer -> Integer) -> T_RawMagma_8
forall a b. a -> b
coe
      (Any -> Any -> Any) -> T_RawMagma_8
MAlonzo.Code.Algebra.Bundles.C_RawMagma'46'constructor_79 Integer -> Integer -> Integer
addInt
-- Data.Nat.Properties.+-0-rawMonoid
d_'43''45'0'45'rawMonoid_2166 ::
  MAlonzo.Code.Algebra.Bundles.T_RawMonoid_474
d_'43''45'0'45'rawMonoid_2166 :: T_RawMonoid_474
d_'43''45'0'45'rawMonoid_2166
  = ((Any -> Any -> Any) -> Any -> T_RawMonoid_474)
-> (Integer -> Integer -> Integer) -> Integer -> T_RawMonoid_474
forall a b. a -> b
coe
      (Any -> Any -> Any) -> Any -> T_RawMonoid_474
MAlonzo.Code.Algebra.Bundles.C_RawMonoid'46'constructor_8313 Integer -> Integer -> Integer
addInt
      (Integer
0 :: Integer)
-- Data.Nat.Properties.+-magma
d_'43''45'magma_2168 :: MAlonzo.Code.Algebra.Bundles.T_Magma_36
d_'43''45'magma_2168 :: T_Magma_36
d_'43''45'magma_2168
  = ((Any -> Any -> Any) -> T_IsMagma_86 -> T_Magma_36)
-> (Integer -> Integer -> Integer) -> T_IsMagma_86 -> T_Magma_36
forall a b. a -> b
coe
      (Any -> Any -> Any) -> T_IsMagma_86 -> T_Magma_36
MAlonzo.Code.Algebra.Bundles.C_Magma'46'constructor_581 Integer -> Integer -> Integer
addInt
      T_IsMagma_86
d_'43''45'isMagma_2154
-- Data.Nat.Properties.+-semigroup
d_'43''45'semigroup_2170 ::
  MAlonzo.Code.Algebra.Bundles.T_Semigroup_206
d_'43''45'semigroup_2170 :: T_Semigroup_206
d_'43''45'semigroup_2170
  = ((Any -> Any -> Any) -> T_IsSemigroup_194 -> T_Semigroup_206)
-> (Integer -> Integer -> Integer)
-> T_IsSemigroup_194
-> T_Semigroup_206
forall a b. a -> b
coe
      (Any -> Any -> Any) -> T_IsSemigroup_194 -> T_Semigroup_206
MAlonzo.Code.Algebra.Bundles.C_Semigroup'46'constructor_3669 Integer -> Integer -> Integer
addInt
      T_IsSemigroup_194
d_'43''45'isSemigroup_2156
-- Data.Nat.Properties.+-commutativeSemigroup
d_'43''45'commutativeSemigroup_2172 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemigroup_332
d_'43''45'commutativeSemigroup_2172 :: T_CommutativeSemigroup_332
d_'43''45'commutativeSemigroup_2172
  = ((Any -> Any -> Any)
 -> T_IsCommutativeSemigroup_270 -> T_CommutativeSemigroup_332)
-> (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_270
-> T_CommutativeSemigroup_332
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> T_IsCommutativeSemigroup_270 -> T_CommutativeSemigroup_332
MAlonzo.Code.Algebra.Bundles.C_CommutativeSemigroup'46'constructor_5887
      Integer -> Integer -> Integer
addInt T_IsCommutativeSemigroup_270
d_'43''45'isCommutativeSemigroup_2158
-- Data.Nat.Properties.+-0-monoid
d_'43''45'0'45'monoid_2174 ::
  MAlonzo.Code.Algebra.Bundles.T_Monoid_506
d_'43''45'0'45'monoid_2174 :: T_Monoid_506
d_'43''45'0'45'monoid_2174
  = ((Any -> Any -> Any) -> Any -> T_IsMonoid_358 -> T_Monoid_506)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_358
-> T_Monoid_506
forall a b. a -> b
coe
      (Any -> Any -> Any) -> Any -> T_IsMonoid_358 -> T_Monoid_506
MAlonzo.Code.Algebra.Bundles.C_Monoid'46'constructor_8851 Integer -> Integer -> Integer
addInt
      (Integer
0 :: Integer) T_IsMonoid_358
d_'43''45'0'45'isMonoid_2160
-- Data.Nat.Properties.+-0-commutativeMonoid
d_'43''45'0'45'commutativeMonoid_2176 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeMonoid_582
d_'43''45'0'45'commutativeMonoid_2176 :: T_CommutativeMonoid_582
d_'43''45'0'45'commutativeMonoid_2176
  = ((Any -> Any -> Any)
 -> Any -> T_IsCommutativeMonoid_406 -> T_CommutativeMonoid_582)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_406
-> T_CommutativeMonoid_582
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> Any -> T_IsCommutativeMonoid_406 -> T_CommutativeMonoid_582
MAlonzo.Code.Algebra.Bundles.C_CommutativeMonoid'46'constructor_10343
      Integer -> Integer -> Integer
addInt (Integer
0 :: Integer) T_IsCommutativeMonoid_406
d_'43''45'0'45'isCommutativeMonoid_2162
-- Data.Nat.Properties.∸-magma
d_'8760''45'magma_2178 :: MAlonzo.Code.Algebra.Bundles.T_Magma_36
d_'8760''45'magma_2178 :: T_Magma_36
d_'8760''45'magma_2178
  = ((Any -> Any -> Any) -> T_Magma_36) -> Any -> T_Magma_36
forall a b. a -> b
coe
      (Any -> Any -> Any) -> T_Magma_36
MAlonzo.Code.Relation.Binary.PropositionalEquality.Algebra.du_magma_20
      ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22)
-- Data.Nat.Properties.m≢1+m+n
d_m'8802'1'43'm'43'n_2184 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'8802'1'43'm'43'n_2184 :: Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
d_m'8802'1'43'm'43'n_2184 = Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.m≢1+n+m
d_m'8802'1'43'n'43'm_2194 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'8802'1'43'n'43'm_2194 :: Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
d_m'8802'1'43'n'43'm_2194 = Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.m+1+n≢m
d_m'43'1'43'n'8802'm_2204 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'43'1'43'n'8802'm_2204 :: Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
d_m'43'1'43'n'8802'm_2204 = Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.m+1+n≢0
d_m'43'1'43'n'8802'0_2212 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'43'1'43'n'8802'0_2212 :: Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
d_m'43'1'43'n'8802'0_2212 = Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.m+n≡0⇒m≡0
d_m'43'n'8801'0'8658'm'8801'0_2226 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'43'n'8801'0'8658'm'8801'0_2226 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_m'43'n'8801'0'8658'm'8801'0_2226 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m+n≡0⇒n≡0
d_m'43'n'8801'0'8658'n'8801'0_2234 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'43'n'8801'0'8658'n'8801'0_2234 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_m'43'n'8801'0'8658'n'8801'0_2234 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-cancelˡ-≤
d_'43''45'cancel'737''45''8804'_2242 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'cancel'737''45''8804'_2242 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'43''45'cancel'737''45''8804'_2242 Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''8804'_2242 Integer
v0 T__'8804'__18
v3
du_'43''45'cancel'737''45''8804'_2242 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'cancel'737''45''8804'_2242 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''8804'_2242 Integer
v0 T__'8804'__18
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
                MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5
                  -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''8804'_2242 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5)
                T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.+-cancelʳ-≤
d_'43''45'cancel'691''45''8804'_2250 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'cancel'691''45''8804'_2250 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'43''45'cancel'691''45''8804'_2250 Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'691''45''8804'_2250 Integer
v0 T__'8804'__18
v3
du_'43''45'cancel'691''45''8804'_2250 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'cancel'691''45''8804'_2250 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'691''45''8804'_2250 Integer
v0 T__'8804'__18
v1
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''8804'_2242 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)
-- Data.Nat.Properties.+-cancel-≤
d_'43''45'cancel'45''8804'_2260 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'43''45'cancel'45''8804'_2260 :: T_Σ_14
d_'43''45'cancel'45''8804'_2260
  = (Any -> Any -> T_Σ_14)
-> (Any -> Any -> Any -> Any -> Any)
-> (Any -> Any -> Any -> Any -> Any)
-> T_Σ_14
forall a b. a -> b
coe
      Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
      (\ Any
v0 Any
v1 Any
v2 Any
v3 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''8804'_2242 Any
v0 Any
v3)
      (\ Any
v0 Any
v1 Any
v2 Any
v3 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'691''45''8804'_2250 Any
v0 Any
v3)
-- Data.Nat.Properties.+-cancelˡ-<
d_'43''45'cancel'737''45''60'_2262 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'cancel'737''45''60'_2262 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'43''45'cancel'737''45''60'_2262 Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''60'_2262 Integer
v0 T__'8804'__18
v3
du_'43''45'cancel'737''45''60'_2262 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'cancel'737''45''60'_2262 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''60'_2262 Integer
v0 T__'8804'__18
v1
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''8804'_2242 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)
-- Data.Nat.Properties.+-cancelʳ-<
d_'43''45'cancel'691''45''60'_2272 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'cancel'691''45''60'_2272 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'43''45'cancel'691''45''60'_2272 Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'691''45''60'_2272 Integer
v0 T__'8804'__18
v3
du_'43''45'cancel'691''45''60'_2272 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'cancel'691''45''60'_2272 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'691''45''60'_2272 Integer
v0 T__'8804'__18
v1
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'691''45''8804'_2250 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)
-- Data.Nat.Properties.+-cancel-<
d_'43''45'cancel'45''60'_2280 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'43''45'cancel'45''60'_2280 :: T_Σ_14
d_'43''45'cancel'45''60'_2280
  = (Any -> Any -> T_Σ_14)
-> (Any -> Any -> Any -> Any -> Any)
-> (Any -> Any -> Any -> Any -> Any)
-> T_Σ_14
forall a b. a -> b
coe
      Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
      (\ Any
v0 Any
v1 Any
v2 Any
v3 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''60'_2262 Any
v0 Any
v3)
      (\ Any
v0 Any
v1 Any
v2 Any
v3 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'691''45''60'_2272 Any
v0 Any
v3)
-- Data.Nat.Properties.≤-stepsˡ
d_'8804''45'steps'737'_2288 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''45'steps'737'_2288 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8804''45'steps'737'_2288 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = T__'8804'__18 -> T__'8804'__18
du_'8804''45'steps'737'_2288 T__'8804'__18
v3
du_'8804''45'steps'737'_2288 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''45'steps'737'_2288 :: T__'8804'__18 -> T__'8804'__18
du_'8804''45'steps'737'_2288 T__'8804'__18
v0 = T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0
-- Data.Nat.Properties.≤-stepsʳ
d_'8804''45'steps'691'_2302 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''45'steps'691'_2302 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8804''45'steps'691'_2302 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = T__'8804'__18 -> T__'8804'__18
du_'8804''45'steps'691'_2302 T__'8804'__18
v3
du_'8804''45'steps'691'_2302 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''45'steps'691'_2302 :: T__'8804'__18 -> T__'8804'__18
du_'8804''45'steps'691'_2302 T__'8804'__18
v0 = T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0
-- Data.Nat.Properties.m≤m+n
d_m'8804'm'43'n_2316 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8804'm'43'n_2316 :: Integer -> Integer -> T__'8804'__18
d_m'8804'm'43'n_2316 Integer
v0 ~Integer
v1 = Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 Integer
v0
du_m'8804'm'43'n_2316 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'8804'm'43'n_2316 :: Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.m≤n+m
d_m'8804'n'43'm_2328 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8804'n'43'm_2328 :: Integer -> Integer -> T__'8804'__18
d_m'8804'n'43'm_2328 Integer
v0 ~Integer
v1 = Integer -> T__'8804'__18
du_m'8804'n'43'm_2328 Integer
v0
du_m'8804'n'43'm_2328 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'8804'n'43'm_2328 :: Integer -> T__'8804'__18
du_m'8804'n'43'm_2328 Integer
v0 = (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.m≤n⇒m<n∨m≡n
d_m'8804'n'8658'm'60'n'8744'm'8801'n_2340 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_m'8804'n'8658'm'60'n'8744'm'8801'n_2340 :: Integer -> Integer -> T__'8804'__18 -> T__'8846'__30
d_m'8804'n'8658'm'60'n'8744'm'8801'n_2340 Integer
v0 Integer
v1 T__'8804'__18
v2
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
             Integer
0 -> (Any -> T__'8846'__30) -> Any -> T__'8846'__30
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
forall a. a
erased
             Integer
_ -> (Any -> T__'8846'__30) -> Any -> T__'8846'__30
forall a b. a -> b
coe
                    Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
du_0'60'1'43'n_1944)
      Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8846'__30
forall a b. a -> b
coe
             (let v4 :: Integer
v4 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
              Any -> Any
forall a b. a -> b
coe
                (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v2 of
                   MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v7
                     -> let v8 :: T__'8846'__30
v8
                              = Integer -> Integer -> T__'8804'__18 -> T__'8846'__30
d_m'8804'n'8658'm'60'n'8744'm'8801'n_2340
                                  (Integer -> Integer
forall a b. a -> b
coe Integer
v3) (Integer -> Integer
forall a b. a -> b
coe Integer
v4) (T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v7) in
                        Any -> Any
forall a b. a -> b
coe
                          (case T__'8846'__30 -> T__'8846'__30
forall a b. a -> b
coe T__'8846'__30
v8 of
                             MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
v9
                               -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe
                                    Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
                                    ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 Any
v9)
                             MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
v9
                               -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
forall a. a
erased
                             T__'8846'__30
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
                   T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError))
-- Data.Nat.Properties.m+n≤o⇒m≤o
d_m'43'n'8804'o'8658'm'8804'o_2376 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'43'n'8804'o'8658'm'8804'o_2376 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'43'n'8804'o'8658'm'8804'o_2376 Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_m'43'n'8804'o'8658'm'8804'o_2376 Integer
v0 T__'8804'__18
v3
du_m'43'n'8804'o'8658'm'8804'o_2376 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'43'n'8804'o'8658'm'8804'o_2376 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_m'43'n'8804'o'8658'm'8804'o_2376 Integer
v0 T__'8804'__18
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
                MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5
                  -> (T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                       ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_m'43'n'8804'o'8658'm'8804'o_2376 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5))
                T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.m+n≤o⇒n≤o
d_m'43'n'8804'o'8658'n'8804'o_2390 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'43'n'8804'o'8658'n'8804'o_2390 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'43'n'8804'o'8658'n'8804'o_2390 Integer
v0 ~Integer
v1 Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'43'n'8804'o'8658'n'8804'o_2390 Integer
v0 Integer
v2 T__'8804'__18
v3
du_m'43'n'8804'o'8658'n'8804'o_2390 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'43'n'8804'o'8658'n'8804'o_2390 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'43'n'8804'o'8658'n'8804'o_2390 Integer
v0 Integer
v1 T__'8804'__18
v2
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v2
      Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'43'n'8804'o'8658'n'8804'o_2390 (Integer -> Any
forall a b. a -> b
coe Integer
v3) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
                ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)))
-- Data.Nat.Properties.+-mono-≤
d_'43''45'mono'45''8804'_2398 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'mono'45''8804'_2398 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'43''45'mono'45''8804'_2398 ~Integer
v0 ~Integer
v1 ~Integer
v2 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
  = Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
du_'43''45'mono'45''8804'_2398 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'mono'45''8804'_2398 :: Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 Integer
v0 T__'8804'__18
v1 T__'8804'__18
v2
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
      T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
        -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
             T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
             ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'n'43'm_2328 (Integer -> Any
forall a b. a -> b
coe Integer
v0))
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5
        -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
             T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
             ((Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2))
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.+-monoˡ-≤
d_'43''45'mono'737''45''8804'_2412 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'mono'737''45''8804'_2412 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'43''45'mono'737''45''8804'_2412 Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'737''45''8804'_2412 Integer
v0 T__'8804'__18
v3
du_'43''45'mono'737''45''8804'_2412 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'mono'737''45''8804'_2412 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'737''45''8804'_2412 Integer
v0 T__'8804'__18
v1
  = (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.+-monoʳ-≤
d_'43''45'mono'691''45''8804'_2422 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'mono'691''45''8804'_2422 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'43''45'mono'691''45''8804'_2422 Integer
v0 ~Integer
v1 Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'691''45''8804'_2422 Integer
v0 Integer
v2 T__'8804'__18
v3
du_'43''45'mono'691''45''8804'_2422 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'mono'691''45''8804'_2422 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'691''45''8804'_2422 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 (Integer -> Any
forall a b. a -> b
coe Integer
v1)
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v0)) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
-- Data.Nat.Properties.+-mono-<-≤
d_'43''45'mono'45''60''45''8804'_2428 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'mono'45''60''45''8804'_2428 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'43''45'mono'45''60''45''8804'_2428 ~Integer
v0 ~Integer
v1 ~Integer
v2 ~Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
  = T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''60''45''8804'_2428 T__'8804'__18
v4 T__'8804'__18
v5
du_'43''45'mono'45''60''45''8804'_2428 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'mono'45''60''45''8804'_2428 :: T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''60''45''8804'_2428 T__'8804'__18
v0 T__'8804'__18
v1
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v4
        -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v4 of
             T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
               -> (T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v1
             MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v7
               -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
                    T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                    ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''60''45''8804'_2428
                       ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v7) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1))
             T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.+-mono-≤-<
d_'43''45'mono'45''8804''45''60'_2438 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'mono'45''8804''45''60'_2438 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'43''45'mono'45''8804''45''60'_2438 ~Integer
v0 ~Integer
v1 ~Integer
v2 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
  = Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804''45''60'_2438 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
du_'43''45'mono'45''8804''45''60'_2438 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'mono'45''8804''45''60'_2438 :: Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804''45''60'_2438 Integer
v0 T__'8804'__18
v1 T__'8804'__18
v2
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
      T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
        -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
             T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
             ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'n'43'm_2328 (Integer -> Any
forall a b. a -> b
coe Integer
v0))
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5
        -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
             T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
             ((Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804''45''60'_2438 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2))
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.+-mono-<
d_'43''45'mono'45''60'_2448 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'mono'45''60'_2448 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'43''45'mono'45''60'_2448 ~Integer
v0 Integer
v1 ~Integer
v2 Integer
v3 T__'8804'__18
v4
  = Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''60'_2448 Integer
v1 Integer
v3 T__'8804'__18
v4
du_'43''45'mono'45''60'_2448 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'mono'45''60'_2448 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''60'_2448 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804''45''60'_2438 (Integer -> Any
forall a b. a -> b
coe Integer
v1)
      ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2))
-- Data.Nat.Properties.+-monoˡ-<
d_'43''45'mono'737''45''60'_2456 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'mono'737''45''60'_2456 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'43''45'mono'737''45''60'_2456 Integer
v0 ~Integer
v1 ~Integer
v2
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'737''45''60'_2456 Integer
v0
du_'43''45'mono'737''45''60'_2456 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'mono'737''45''60'_2456 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'737''45''60'_2456 Integer
v0
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'737''45''8804'_2412 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.+-monoʳ-<
d_'43''45'mono'691''45''60'_2464 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'43''45'mono'691''45''60'_2464 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'43''45'mono'691''45''60'_2464 Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'691''45''60'_2464 Integer
v0 T__'8804'__18
v3
du_'43''45'mono'691''45''60'_2464 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'43''45'mono'691''45''60'_2464 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'691''45''60'_2464 Integer
v0 T__'8804'__18
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'691''45''60'_2464 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)))
-- Data.Nat.Properties.m+1+n≰m
d_m'43'1'43'n'8816'm_2476 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'43'1'43'n'8816'm_2476 :: Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
d_m'43'1'43'n'8816'm_2476 = Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.m<m+n
d_m'60'm'43'n_2486 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'm'43'n_2486 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'm'43'n_2486 Integer
v0 ~Integer
v1 T__'8804'__18
v2 = Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'm'43'n_2486 Integer
v0 T__'8804'__18
v2
du_m'60'm'43'n_2486 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'60'm'43'n_2486 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'm'43'n_2486 Integer
v0 T__'8804'__18
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'm'43'n_2486 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)))
-- Data.Nat.Properties.m<n+m
d_m'60'n'43'm_2498 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'43'm_2498 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'43'm_2498 Integer
v0 ~Integer
v1 T__'8804'__18
v2 = Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'43'm_2498 Integer
v0 T__'8804'__18
v2
du_m'60'n'43'm_2498 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'60'n'43'm_2498 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'43'm_2498 Integer
v0 T__'8804'__18
v1
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'm'43'n_2486 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)
-- Data.Nat.Properties.m+n≮n
d_m'43'n'8814'n_2514 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'43'n'8814'n_2514 :: Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
d_m'43'n'8814'n_2514 = Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.m+n≮m
d_m'43'n'8814'm_2528 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'43'n'8814'm_2528 :: Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
d_m'43'n'8814'm_2528 = Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.*-suc
d_'42''45'suc_2540 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'suc_2540 :: Integer -> Integer -> T__'8801'__12
d_'42''45'suc_2540 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-identityˡ
d_'42''45'identity'737'_2552 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'identity'737'_2552 :: Integer -> T__'8801'__12
d_'42''45'identity'737'_2552 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-identityʳ
d_'42''45'identity'691'_2556 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'identity'691'_2556 :: Integer -> T__'8801'__12
d_'42''45'identity'691'_2556 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-identity
d_'42''45'identity_2560 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_2560 :: T_Σ_14
d_'42''45'identity_2560
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.*-zeroˡ
d_'42''45'zero'737'_2562 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'zero'737'_2562 :: Integer -> T__'8801'__12
d_'42''45'zero'737'_2562 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-zeroʳ
d_'42''45'zero'691'_2564 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'zero'691'_2564 :: Integer -> T__'8801'__12
d_'42''45'zero'691'_2564 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-zero
d_'42''45'zero_2568 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'zero_2568 :: T_Σ_14
d_'42''45'zero_2568
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.*-comm
d_'42''45'comm_2570 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'comm_2570 :: Integer -> Integer -> T__'8801'__12
d_'42''45'comm_2570 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distribʳ-+
d_'42''45'distrib'691''45''43'_2580 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'691''45''43'_2580 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'691''45''43'_2580 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distribˡ-+
d_'42''45'distrib'737''45''43'_2594 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'737''45''43'_2594 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'737''45''43'_2594 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distrib-+
d_'42''45'distrib'45''43'_2596 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'distrib'45''43'_2596 :: T_Σ_14
d_'42''45'distrib'45''43'_2596
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.*-assoc
d_'42''45'assoc_2598 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'assoc_2598 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'assoc_2598 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-isMagma
d_'42''45'isMagma_2612 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_'42''45'isMagma_2612 :: T_IsMagma_86
d_'42''45'isMagma_2612
  = (T_IsEquivalence_26
 -> (Any -> Any -> Any -> Any -> Any -> Any -> Any) -> T_IsMagma_86)
-> Any -> Any -> T_IsMagma_86
forall a b. a -> b
coe
      T_IsEquivalence_26
-> (Any -> Any -> Any -> Any -> Any -> Any -> Any) -> T_IsMagma_86
MAlonzo.Code.Algebra.Structures.C_IsMagma'46'constructor_553
      (T_IsEquivalence_26 -> Any
forall a b. a -> b
coe
         T_IsEquivalence_26
MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_242)
      Any
forall a. a
erased
-- Data.Nat.Properties.*-isSemigroup
d_'42''45'isSemigroup_2614 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_'42''45'isSemigroup_2614 :: T_IsSemigroup_194
d_'42''45'isSemigroup_2614
  = (T_IsMagma_86 -> (Any -> Any -> Any -> Any) -> T_IsSemigroup_194)
-> Any -> Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      T_IsMagma_86 -> (Any -> Any -> Any -> Any) -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Structures.C_IsSemigroup'46'constructor_4001
      (T_IsMagma_86 -> Any
forall a b. a -> b
coe T_IsMagma_86
d_'42''45'isMagma_2612) Any
forall a. a
erased
-- Data.Nat.Properties.*-isCommutativeSemigroup
d_'42''45'isCommutativeSemigroup_2616 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270
d_'42''45'isCommutativeSemigroup_2616 :: T_IsCommutativeSemigroup_270
d_'42''45'isCommutativeSemigroup_2616
  = (T_IsSemigroup_194
 -> (Any -> Any -> Any) -> T_IsCommutativeSemigroup_270)
-> Any -> Any -> T_IsCommutativeSemigroup_270
forall a b. a -> b
coe
      T_IsSemigroup_194
-> (Any -> Any -> Any) -> T_IsCommutativeSemigroup_270
MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemigroup'46'constructor_5673
      (T_IsSemigroup_194 -> Any
forall a b. a -> b
coe T_IsSemigroup_194
d_'42''45'isSemigroup_2614) Any
forall a. a
erased
-- Data.Nat.Properties.*-1-isMonoid
d_'42''45'1'45'isMonoid_2618 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'42''45'1'45'isMonoid_2618 :: T_IsMonoid_358
d_'42''45'1'45'isMonoid_2618
  = (T_IsSemigroup_194 -> T_Σ_14 -> T_IsMonoid_358)
-> Any -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_Σ_14 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_7687
      (T_IsSemigroup_194 -> Any
forall a b. a -> b
coe T_IsSemigroup_194
d_'42''45'isSemigroup_2614) (T_Σ_14 -> Any
forall a b. a -> b
coe T_Σ_14
d_'42''45'identity_2560)
-- Data.Nat.Properties.*-1-isCommutativeMonoid
d_'42''45'1'45'isCommutativeMonoid_2620 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'42''45'1'45'isCommutativeMonoid_2620 :: T_IsCommutativeMonoid_406
d_'42''45'1'45'isCommutativeMonoid_2620
  = (T_IsMonoid_358
 -> (Any -> Any -> Any) -> T_IsCommutativeMonoid_406)
-> Any -> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsMonoid_358 -> (Any -> Any -> Any) -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_9361
      (T_IsMonoid_358 -> Any
forall a b. a -> b
coe T_IsMonoid_358
d_'42''45'1'45'isMonoid_2618) Any
forall a. a
erased
-- Data.Nat.Properties.+-*-isSemiring
d_'43''45''42''45'isSemiring_2622 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238
d_'43''45''42''45'isSemiring_2622 :: T_IsSemiring_1238
d_'43''45''42''45'isSemiring_2622
  = (T_IsSemiringWithoutAnnihilatingZero_1142
 -> T_Σ_14 -> T_IsSemiring_1238)
-> Any -> Any -> T_IsSemiring_1238
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1142
-> T_Σ_14 -> T_IsSemiring_1238
MAlonzo.Code.Algebra.Structures.C_IsSemiring'46'constructor_37213
      ((T_IsCommutativeMonoid_406
 -> T_IsMonoid_358
 -> T_Σ_14
 -> T_IsSemiringWithoutAnnihilatingZero_1142)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         T_IsCommutativeMonoid_406
-> T_IsMonoid_358
-> T_Σ_14
-> T_IsSemiringWithoutAnnihilatingZero_1142
MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutAnnihilatingZero'46'constructor_33703
         (T_IsCommutativeMonoid_406 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406
d_'43''45'0'45'isCommutativeMonoid_2162)
         (T_IsMonoid_358 -> Any
forall a b. a -> b
coe T_IsMonoid_358
d_'42''45'1'45'isMonoid_2618)
         (T_Σ_14 -> Any
forall a b. a -> b
coe T_Σ_14
d_'42''45'distrib'45''43'_2596))
      (T_Σ_14 -> Any
forall a b. a -> b
coe T_Σ_14
d_'42''45'zero_2568)
-- Data.Nat.Properties.+-*-isCommutativeSemiring
d_'43''45''42''45'isCommutativeSemiring_2624 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344
d_'43''45''42''45'isCommutativeSemiring_2624 :: T_IsCommutativeSemiring_1344
d_'43''45''42''45'isCommutativeSemiring_2624
  = (T_IsSemiring_1238
 -> (Any -> Any -> Any) -> T_IsCommutativeSemiring_1344)
-> Any -> Any -> T_IsCommutativeSemiring_1344
forall a b. a -> b
coe
      T_IsSemiring_1238
-> (Any -> Any -> Any) -> T_IsCommutativeSemiring_1344
MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemiring'46'constructor_40675
      (T_IsSemiring_1238 -> Any
forall a b. a -> b
coe T_IsSemiring_1238
d_'43''45''42''45'isSemiring_2622) Any
forall a. a
erased
-- Data.Nat.Properties.*-rawMagma
d_'42''45'rawMagma_2626 ::
  MAlonzo.Code.Algebra.Bundles.T_RawMagma_8
d_'42''45'rawMagma_2626 :: T_RawMagma_8
d_'42''45'rawMagma_2626
  = ((Any -> Any -> Any) -> T_RawMagma_8)
-> (Integer -> Integer -> Integer) -> T_RawMagma_8
forall a b. a -> b
coe
      (Any -> Any -> Any) -> T_RawMagma_8
MAlonzo.Code.Algebra.Bundles.C_RawMagma'46'constructor_79 Integer -> Integer -> Integer
mulInt
-- Data.Nat.Properties.*-1-rawMonoid
d_'42''45'1'45'rawMonoid_2628 ::
  MAlonzo.Code.Algebra.Bundles.T_RawMonoid_474
d_'42''45'1'45'rawMonoid_2628 :: T_RawMonoid_474
d_'42''45'1'45'rawMonoid_2628
  = ((Any -> Any -> Any) -> Any -> T_RawMonoid_474)
-> (Integer -> Integer -> Integer) -> Integer -> T_RawMonoid_474
forall a b. a -> b
coe
      (Any -> Any -> Any) -> Any -> T_RawMonoid_474
MAlonzo.Code.Algebra.Bundles.C_RawMonoid'46'constructor_8313 Integer -> Integer -> Integer
mulInt
      (Integer
1 :: Integer)
-- Data.Nat.Properties.*-magma
d_'42''45'magma_2630 :: MAlonzo.Code.Algebra.Bundles.T_Magma_36
d_'42''45'magma_2630 :: T_Magma_36
d_'42''45'magma_2630
  = ((Any -> Any -> Any) -> T_IsMagma_86 -> T_Magma_36)
-> (Integer -> Integer -> Integer) -> T_IsMagma_86 -> T_Magma_36
forall a b. a -> b
coe
      (Any -> Any -> Any) -> T_IsMagma_86 -> T_Magma_36
MAlonzo.Code.Algebra.Bundles.C_Magma'46'constructor_581 Integer -> Integer -> Integer
mulInt
      T_IsMagma_86
d_'42''45'isMagma_2612
-- Data.Nat.Properties.*-semigroup
d_'42''45'semigroup_2632 ::
  MAlonzo.Code.Algebra.Bundles.T_Semigroup_206
d_'42''45'semigroup_2632 :: T_Semigroup_206
d_'42''45'semigroup_2632
  = ((Any -> Any -> Any) -> T_IsSemigroup_194 -> T_Semigroup_206)
-> (Integer -> Integer -> Integer)
-> T_IsSemigroup_194
-> T_Semigroup_206
forall a b. a -> b
coe
      (Any -> Any -> Any) -> T_IsSemigroup_194 -> T_Semigroup_206
MAlonzo.Code.Algebra.Bundles.C_Semigroup'46'constructor_3669 Integer -> Integer -> Integer
mulInt
      T_IsSemigroup_194
d_'42''45'isSemigroup_2614
-- Data.Nat.Properties.*-commutativeSemigroup
d_'42''45'commutativeSemigroup_2634 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemigroup_332
d_'42''45'commutativeSemigroup_2634 :: T_CommutativeSemigroup_332
d_'42''45'commutativeSemigroup_2634
  = ((Any -> Any -> Any)
 -> T_IsCommutativeSemigroup_270 -> T_CommutativeSemigroup_332)
-> (Integer -> Integer -> Integer)
-> T_IsCommutativeSemigroup_270
-> T_CommutativeSemigroup_332
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> T_IsCommutativeSemigroup_270 -> T_CommutativeSemigroup_332
MAlonzo.Code.Algebra.Bundles.C_CommutativeSemigroup'46'constructor_5887
      Integer -> Integer -> Integer
mulInt T_IsCommutativeSemigroup_270
d_'42''45'isCommutativeSemigroup_2616
-- Data.Nat.Properties.*-1-monoid
d_'42''45'1'45'monoid_2636 ::
  MAlonzo.Code.Algebra.Bundles.T_Monoid_506
d_'42''45'1'45'monoid_2636 :: T_Monoid_506
d_'42''45'1'45'monoid_2636
  = ((Any -> Any -> Any) -> Any -> T_IsMonoid_358 -> T_Monoid_506)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_358
-> T_Monoid_506
forall a b. a -> b
coe
      (Any -> Any -> Any) -> Any -> T_IsMonoid_358 -> T_Monoid_506
MAlonzo.Code.Algebra.Bundles.C_Monoid'46'constructor_8851 Integer -> Integer -> Integer
mulInt
      (Integer
1 :: Integer) T_IsMonoid_358
d_'42''45'1'45'isMonoid_2618
-- Data.Nat.Properties.*-1-commutativeMonoid
d_'42''45'1'45'commutativeMonoid_2638 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeMonoid_582
d_'42''45'1'45'commutativeMonoid_2638 :: T_CommutativeMonoid_582
d_'42''45'1'45'commutativeMonoid_2638
  = ((Any -> Any -> Any)
 -> Any -> T_IsCommutativeMonoid_406 -> T_CommutativeMonoid_582)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_406
-> T_CommutativeMonoid_582
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> Any -> T_IsCommutativeMonoid_406 -> T_CommutativeMonoid_582
MAlonzo.Code.Algebra.Bundles.C_CommutativeMonoid'46'constructor_10343
      Integer -> Integer -> Integer
mulInt (Integer
1 :: Integer) T_IsCommutativeMonoid_406
d_'42''45'1'45'isCommutativeMonoid_2620
-- Data.Nat.Properties.+-*-semiring
d_'43''45''42''45'semiring_2640 ::
  MAlonzo.Code.Algebra.Bundles.T_Semiring_1932
d_'43''45''42''45'semiring_2640 :: T_Semiring_1932
d_'43''45''42''45'semiring_2640
  = ((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> Any
 -> Any
 -> T_IsSemiring_1238
 -> T_Semiring_1932)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsSemiring_1238
-> T_Semiring_1932
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> Any
-> Any
-> T_IsSemiring_1238
-> T_Semiring_1932
MAlonzo.Code.Algebra.Bundles.C_Semiring'46'constructor_33613 Integer -> Integer -> Integer
addInt
      Integer -> Integer -> Integer
mulInt (Integer
0 :: Integer) (Integer
1 :: Integer)
      T_IsSemiring_1238
d_'43''45''42''45'isSemiring_2622
-- Data.Nat.Properties.+-*-commutativeSemiring
d_'43''45''42''45'commutativeSemiring_2642 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemiring_2094
d_'43''45''42''45'commutativeSemiring_2642 :: T_CommutativeSemiring_2094
d_'43''45''42''45'commutativeSemiring_2642
  = ((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> Any
 -> Any
 -> T_IsCommutativeSemiring_1344
 -> T_CommutativeSemiring_2094)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> Integer
-> T_IsCommutativeSemiring_1344
-> T_CommutativeSemiring_2094
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> Any
-> Any
-> T_IsCommutativeSemiring_1344
-> T_CommutativeSemiring_2094
MAlonzo.Code.Algebra.Bundles.C_CommutativeSemiring'46'constructor_36513
      Integer -> Integer -> Integer
addInt Integer -> Integer -> Integer
mulInt (Integer
0 :: Integer) (Integer
1 :: Integer)
      T_IsCommutativeSemiring_1344
d_'43''45''42''45'isCommutativeSemiring_2624
-- Data.Nat.Properties.*-cancelʳ-≡
d_'42''45'cancel'691''45''8801'_2650 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cancel'691''45''8801'_2650 :: Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'42''45'cancel'691''45''8801'_2650 = Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-cancelˡ-≡
d_'42''45'cancel'737''45''8801'_2668 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'cancel'737''45''8801'_2668 :: Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'42''45'cancel'737''45''8801'_2668 = Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m*n≡0⇒m≡0∨n≡0
d_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_2682 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_2682 :: Integer -> Integer -> T__'8801'__12 -> T__'8846'__30
d_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_2682 Integer
v0 ~Integer
v1 ~T__'8801'__12
v2
  = Integer -> T__'8846'__30
du_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_2682 Integer
v0
du_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_2682 ::
  Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
du_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_2682 :: Integer -> T__'8846'__30
du_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_2682 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Any -> T__'8846'__30) -> Any -> T__'8846'__30
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
forall a. a
erased
      Integer
_ -> (Any -> T__'8846'__30) -> Any -> T__'8846'__30
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
forall a. a
erased
-- Data.Nat.Properties.m*n≡1⇒m≡1
d_m'42'n'8801'1'8658'm'8801'1_2696 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'42'n'8801'1'8658'm'8801'1_2696 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_m'42'n'8801'1'8658'm'8801'1_2696 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m*n≡1⇒n≡1
d_m'42'n'8801'1'8658'n'8801'1_2710 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'42'n'8801'1'8658'n'8801'1_2710 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_m'42'n'8801'1'8658'n'8801'1_2710 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.[m*n]*[o*p]≡[m*o]*[n*p]
d_'91'm'42'n'93''42''91'o'42'p'93''8801''91'm'42'o'93''42''91'n'42'p'93'_2726 ::
  Integer ->
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'91'm'42'n'93''42''91'o'42'p'93''8801''91'm'42'o'93''42''91'n'42'p'93'_2726 :: Integer -> Integer -> Integer -> Integer -> T__'8801'__12
d_'91'm'42'n'93''42''91'o'42'p'93''8801''91'm'42'o'93''42''91'n'42'p'93'_2726
  = Integer -> Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-cancelʳ-≤
d_'42''45'cancel'691''45''8804'_2792 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'cancel'691''45''8804'_2792 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'cancel'691''45''8804'_2792 Integer
v0 ~Integer
v1 ~Integer
v2 ~T__'8804'__18
v3
  = Integer -> T__'8804'__18
du_'42''45'cancel'691''45''8804'_2792 Integer
v0
du_'42''45'cancel'691''45''8804'_2792 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'42''45'cancel'691''45''8804'_2792 :: Integer -> T__'8804'__18
du_'42''45'cancel'691''45''8804'_2792 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'42''45'cancel'691''45''8804'_2792 (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.*-cancelˡ-≤
d_'42''45'cancel'737''45''8804'_2808 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'cancel'737''45''8804'_2808 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'cancel'737''45''8804'_2808 Integer
v0 ~Integer
v1 ~Integer
v2
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'cancel'737''45''8804'_2808 Integer
v0
du_'42''45'cancel'737''45''8804'_2808 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'42''45'cancel'737''45''8804'_2808 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'cancel'737''45''8804'_2808 Integer
v0 T__'8804'__18
v1
  = (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'42''45'cancel'691''45''8804'_2792 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.*-mono-≤
d_'42''45'mono'45''8804'_2824 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'mono'45''8804'_2824 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'42''45'mono'45''8804'_2824 ~Integer
v0 Integer
v1 ~Integer
v2 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
  = Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''8804'_2824 Integer
v1 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
du_'42''45'mono'45''8804'_2824 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'42''45'mono'45''8804'_2824 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''8804'_2824 Integer
v0 Integer
v1 T__'8804'__18
v2 T__'8804'__18
v3
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v2 of
      T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
        -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6
        -> let v7 :: Integer
v7 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v7) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v3)
                ((Integer
 -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                   Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''8804'_2824 (Integer -> Any
forall a b. a -> b
coe Integer
v7) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v6)
                   (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v3)))
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.*-monoˡ-≤
d_'42''45'mono'737''45''8804'_2834 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'mono'737''45''8804'_2834 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'mono'737''45''8804'_2834 Integer
v0 ~Integer
v1 Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'737''45''8804'_2834 Integer
v0 Integer
v2 T__'8804'__18
v3
du_'42''45'mono'737''45''8804'_2834 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'42''45'mono'737''45''8804'_2834 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'737''45''8804'_2834 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (Integer
 -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''8804'_2824 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.*-monoʳ-≤
d_'42''45'mono'691''45''8804'_2844 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'mono'691''45''8804'_2844 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'mono'691''45''8804'_2844 Integer
v0 ~Integer
v1 Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'691''45''8804'_2844 Integer
v0 Integer
v2 T__'8804'__18
v3
du_'42''45'mono'691''45''8804'_2844 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'42''45'mono'691''45''8804'_2844 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'691''45''8804'_2844 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (Integer
 -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''8804'_2824 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v0)) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
-- Data.Nat.Properties.*-mono-<
d_'42''45'mono'45''60'_2850 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'mono'45''60'_2850 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'42''45'mono'45''60'_2850 ~Integer
v0 Integer
v1 ~Integer
v2 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
  = Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''60'_2850 Integer
v1 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
du_'42''45'mono'45''60'_2850 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'42''45'mono'45''60'_2850 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''60'_2850 Integer
v0 Integer
v1 T__'8804'__18
v2 T__'8804'__18
v3
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v2 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6
        -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v6 of
             T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
               -> (Any -> Any -> Any) -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
                    Any -> Any -> Any
forall a b. a -> b -> b
seq (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v3)
                    ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                       (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22))
             MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9
               -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v3 of
                    MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v12
                      -> (Integer
 -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Integer -> Integer -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
                           Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''60'_2448 Integer
v1
                           (Integer -> Integer -> Integer
mulInt ((Integer -> Integer -> Integer) -> Any -> Any -> Integer
forall a b. a -> b
coe Integer -> Integer -> Integer
subInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer))) (Integer -> Integer
forall a b. a -> b
coe Integer
v1))
                           ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v12)
                           ((Integer
 -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                              Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''60'_2850
                              ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
subInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer))) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
                              ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9)
                              ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v12))
                    T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
             T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.*-monoˡ-<
d_'42''45'mono'737''45''60'_2862 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'mono'737''45''60'_2862 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'mono'737''45''60'_2862 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v3 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6
        -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v6 of
             T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
               -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
                    T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                    (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
             MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9
               -> let v10 :: Integer
v10 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                  Any -> T__'8804'__18
forall a b. a -> b
coe
                    ((Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                       Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804''45''60'_2438
                       (((Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any)
-> (Any -> Any -> Any)
-> (Any -> Integer)
-> Integer
-> Integer
-> Any
forall a b. a -> b
coe
                          (Any -> Any -> Any) -> (Any -> Any) -> Any -> Any -> Any
MAlonzo.Code.Function.Base.du_'8739'_'10217''45'__308
                          (\ Any
v11 Any
v12 -> Any
v12)
                          (\ Any
v11 ->
                             Integer -> Integer -> Integer
mulInt (Any -> Integer
forall a b. a -> b
coe Any
v11) ((Integer -> Integer -> Integer) -> Any -> Any -> Integer
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0)))
                          Integer
v10 (Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer))))
                       ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                          Integer -> T__'8804'__18
d_'8804''45'refl_1634 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0)))
                       ((Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                          Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'mono'737''45''60'_2862 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v10)
                          ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
subInt (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)))
                          ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9)))
             T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.*-monoʳ-<
d_'42''45'mono'691''45''60'_2874 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'mono'691''45''60'_2874 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'mono'691''45''60'_2874 Integer
v0 ~Integer
v1 Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'691''45''60'_2874 Integer
v0 Integer
v2 T__'8804'__18
v3
du_'42''45'mono'691''45''60'_2874 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'42''45'mono'691''45''60'_2874 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'691''45''60'_2874 Integer
v0 Integer
v1 T__'8804'__18
v2
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v2 of
             MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5
               -> (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
                    Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer))
                    ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5)
                    (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
             T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
      Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v2 of
                MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6
                  -> (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                       Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                       ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6)
                       ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe
                          Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                          ((Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                             Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'691''45''60'_2874 (Integer -> Any
forall a b. a -> b
coe Integer
v3) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
                             ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6)))
                T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.m≤m*n
d_m'8804'm'42'n_2886 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8804'm'42'n_2886 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'8804'm'42'n_2886 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (T_IsPreorder_70
 -> (Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any)
-> Any
-> (Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__'8804'__18
forall a b. a -> b
coe
      T_IsPreorder_70
-> (Any -> Any -> Any -> Any)
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin__160
      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
      (\ Any
v3 Any
v4 Any
v5 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 Any
v4 Any
v5) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
      ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      ((T_IsPreorder_70
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
forall a b. a -> b
coe
         T_IsPreorder_70
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''8804'_228
         (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
         (\ Any
v3 Any
v4 Any
v5 Any
v6 Any
v7 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 Any
v6 Any
v7)
         ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)))
         ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
         ((T_IsPreorder_70 -> Any -> T__IsRelatedTo__70) -> Any -> Any -> Any
forall a b. a -> b
coe
            T_IsPreorder_70 -> Any -> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du__'8718'_346
            (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
         ((Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
            Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'691''45''8804'_2844 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)))
-- Data.Nat.Properties.m≤n*m
d_m'8804'n'42'm_2898 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8804'n'42'm_2898 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'8804'n'42'm_2898 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (T_IsPreorder_70
 -> (Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any)
-> Any
-> (Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__'8804'__18
forall a b. a -> b
coe
      T_IsPreorder_70
-> (Any -> Any -> Any -> Any)
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin__160
      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
      (\ Any
v3 Any
v4 Any
v5 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 Any
v4 Any
v5) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
      ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0))
      ((T_IsPreorder_70
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
forall a b. a -> b
coe
         T_IsPreorder_70
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''8804'_228
         (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
         (\ Any
v3 Any
v4 Any
v5 Any
v6 Any
v7 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 Any
v6 Any
v7) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
         ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0))
         ((T_IsPreorder_70 -> Any -> T__IsRelatedTo__70) -> Any -> Any -> Any
forall a b. a -> b
coe
            T_IsPreorder_70 -> Any -> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du__'8718'_346
            (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)))
         ((Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'8804'm'42'n_2886 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)))
-- Data.Nat.Properties.m<m*n
d_m'60'm'42'n_2910 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'm'42'n_2910 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
d_m'60'm'42'n_2910 Integer
v0 Integer
v1 T__'8804'__18
v2 T__'8804'__18
v3
  = let v4 :: Integer
v4 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
    Any -> T__'8804'__18
forall a b. a -> b
coe
      ((Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
         Any -> Any -> Any
forall a b. a -> b -> b
seq (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
         (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v3 of
            MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v7
              -> (Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
                   Any -> Any -> Any
forall a b. a -> b -> b
seq (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v7)
                   ((T__IsRelatedTo__70 -> Any) -> Any -> Any
forall a b. a -> b
coe
                      T__IsRelatedTo__70 -> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin'45'strict__176
                      (((Any -> Any -> Any -> Any -> Any -> Any)
 -> T_Σ_14
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
forall a b. a -> b
coe
                         (Any -> Any -> Any -> Any -> Any -> Any)
-> T_Σ_14
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''60'_202
                         (\ Any
v8 Any
v9 Any
v10 Any
v11 Any
v12 -> (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans_1862 Any
v9 Any
v11 Any
v12)
                         (T_Σ_14 -> Any
forall a b. a -> b
coe
                            T_Σ_14
MAlonzo.Code.Relation.Binary.PropositionalEquality.Core.du_resp'8322'_144)
                         (\ Any
v8 Any
v9 Any
v10 Any
v11 Any
v12 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'737'_1874 Any
v11 Any
v12)
                         (Integer -> Any
forall a b. a -> b
coe Integer
v0) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v4))
                         ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                         ((T_IsPreorder_70
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
forall a b. a -> b
coe
                            T_IsPreorder_70
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''8804'_228
                            (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                            (\ Any
v8 Any
v9 Any
v10 Any
v11 Any
v12 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 Any
v11 Any
v12)
                            ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v4))
                            ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe Integer
v1) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v4) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
                            ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                            ((T_IsPreorder_70 -> Any -> T__IsRelatedTo__70) -> Any -> Any -> Any
forall a b. a -> b
coe
                               T_IsPreorder_70 -> Any -> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du__'8718'_346
                               (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
                            ((Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                               Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'691''45''8804'_2422 (Integer -> Any
forall a b. a -> b
coe Integer
v1)
                               ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
mulInt (Integer -> Any
forall a b. a -> b
coe Integer
v4) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                               ((Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                                  Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'8804'm'42'n_2886 (Integer -> Any
forall a b. a -> b
coe Integer
v4) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
                                  (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
du_0'60'1'43'n_1944))))
                         ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                            T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                            ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                               T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                               ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'n'43'm_2328 (Integer -> Any
forall a b. a -> b
coe Integer
v4))))))
            T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError))
-- Data.Nat.Properties.*-cancelʳ-<
d_'42''45'cancel'691''45''60'_2920 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'cancel'691''45''60'_2920 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'cancel'691''45''60'_2920 Integer
v0 Integer
v1 Integer
v2 ~T__'8804'__18
v3
  = Integer -> Integer -> Integer -> T__'8804'__18
du_'42''45'cancel'691''45''60'_2920 Integer
v0 Integer
v1 Integer
v2
du_'42''45'cancel'691''45''60'_2920 ::
  Integer ->
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'42''45'cancel'691''45''60'_2920 :: Integer -> Integer -> Integer -> T__'8804'__18
du_'42''45'cancel'691''45''60'_2920 Integer
v0 Integer
v1 Integer
v2
  = let v3 :: b
v3
          = let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
            Any -> b
forall a b. a -> b
coe
              (let v4 :: Integer
v4 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
               Any -> Any
forall a b. a -> b
coe
                 ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                    T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                    ((Integer -> Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                       Integer -> Integer -> Integer -> T__'8804'__18
du_'42''45'cancel'691''45''60'_2920 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v3)
                       (Integer -> Any
forall a b. a -> b
coe Integer
v4)))) in
    Any -> T__'8804'__18
forall a b. a -> b
coe
      ((Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
         Any -> Any -> Any
forall a b. a -> b -> b
seq (Integer -> Any
forall a b. a -> b
coe Integer
v0)
         (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
            Integer
0 -> case Integer -> Any
forall a b. a -> b
coe Integer
v2 of
                   Any
_ | (Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe Integer -> Integer -> Bool
geqInt (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) ->
                       T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
du_0'60'1'43'n_1944
                   Any
_ -> Any -> Any
forall a b. a -> b
coe Any
forall a. a
v3
            Integer
_ -> let v4 :: Integer
v4 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                 Any -> Any
forall a b. a -> b
coe
                   (case Integer -> Any
forall a b. a -> b
coe Integer
v2 of
                      Any
_ | (Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe Integer -> Integer -> Bool
geqInt (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) ->
                          let v5 :: Integer
v5 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                          Any -> Any
forall a b. a -> b
coe
                            ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                               T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                               ((Integer -> Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                                  Integer -> Integer -> Integer -> T__'8804'__18
du_'42''45'cancel'691''45''60'_2920 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v4) (Integer -> Any
forall a b. a -> b
coe Integer
v5)))
                      Any
_ -> Any -> Any
forall a b. a -> b
coe Any
forall a. a
v3)))
-- Data.Nat.Properties.*-cancelˡ-<
d_'42''45'cancel'737''45''60'_2936 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'42''45'cancel'737''45''60'_2936 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'cancel'737''45''60'_2936 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3
  = (Integer -> Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> Integer -> Integer -> T__'8804'__18
du_'42''45'cancel'691''45''60'_2920 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v2)
-- Data.Nat.Properties.*-cancel-<
d_'42''45'cancel'45''60'_2952 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'cancel'45''60'_2952 :: T_Σ_14
d_'42''45'cancel'45''60'_2952
  = (Any -> Any -> T_Σ_14)
-> Any -> (Any -> Any -> Any -> Any -> Any) -> T_Σ_14
forall a b. a -> b
coe
      Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
      ((Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Any
forall a b. a -> b
coe Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'42''45'cancel'737''45''60'_2936)
      (\ Any
v0 Any
v1 Any
v2 Any
v3 -> (Integer -> Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer -> T__'8804'__18
du_'42''45'cancel'691''45''60'_2920 Any
v0 Any
v1 Any
v2)
-- Data.Nat.Properties.^-identityʳ
d_'94''45'identity'691'_2954 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'94''45'identity'691'_2954 :: Integer -> T__'8801'__12
d_'94''45'identity'691'_2954 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.^-zeroˡ
d_'94''45'zero'737'_2958 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'94''45'zero'737'_2958 :: Integer -> T__'8801'__12
d_'94''45'zero'737'_2958 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.^-distribˡ-+-*
d_'94''45'distrib'737''45''43''45''42'_2968 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'94''45'distrib'737''45''43''45''42'_2968 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'94''45'distrib'737''45''43''45''42'_2968 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.^-semigroup-morphism
d_'94''45'semigroup'45'morphism_2986 ::
  Integer -> MAlonzo.Code.Algebra.Morphism.T_IsSemigroupMorphism_148
d_'94''45'semigroup'45'morphism_2986 :: Integer -> T_IsSemigroupMorphism_148
d_'94''45'semigroup'45'morphism_2986 ~Integer
v0
  = T_IsSemigroupMorphism_148
du_'94''45'semigroup'45'morphism_2986
du_'94''45'semigroup'45'morphism_2986 ::
  MAlonzo.Code.Algebra.Morphism.T_IsSemigroupMorphism_148
du_'94''45'semigroup'45'morphism_2986 :: T_IsSemigroupMorphism_148
du_'94''45'semigroup'45'morphism_2986
  = ((Any -> Any -> Any -> Any)
 -> (Any -> Any -> Any) -> T_IsSemigroupMorphism_148)
-> Any -> Any -> T_IsSemigroupMorphism_148
forall a b. a -> b
coe
      (Any -> Any -> Any -> Any)
-> (Any -> Any -> Any) -> T_IsSemigroupMorphism_148
MAlonzo.Code.Algebra.Morphism.C_IsSemigroupMorphism'46'constructor_1081
      Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.^-monoid-morphism
d_'94''45'monoid'45'morphism_2994 ::
  Integer -> MAlonzo.Code.Algebra.Morphism.T_IsMonoidMorphism_298
d_'94''45'monoid'45'morphism_2994 :: Integer -> T_IsMonoidMorphism_298
d_'94''45'monoid'45'morphism_2994 ~Integer
v0
  = T_IsMonoidMorphism_298
du_'94''45'monoid'45'morphism_2994
du_'94''45'monoid'45'morphism_2994 ::
  MAlonzo.Code.Algebra.Morphism.T_IsMonoidMorphism_298
du_'94''45'monoid'45'morphism_2994 :: T_IsMonoidMorphism_298
du_'94''45'monoid'45'morphism_2994
  = (T_IsSemigroupMorphism_148 -> Any -> T_IsMonoidMorphism_298)
-> Any -> Any -> T_IsMonoidMorphism_298
forall a b. a -> b
coe
      T_IsSemigroupMorphism_148 -> Any -> T_IsMonoidMorphism_298
MAlonzo.Code.Algebra.Morphism.C_IsMonoidMorphism'46'constructor_2137
      (T_IsSemigroupMorphism_148 -> Any
forall a b. a -> b
coe T_IsSemigroupMorphism_148
du_'94''45'semigroup'45'morphism_2986) Any
forall a. a
erased
-- Data.Nat.Properties.^-*-assoc
d_'94''45''42''45'assoc_3002 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'94''45''42''45'assoc_3002 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'94''45''42''45'assoc_3002 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m^n≡0⇒m≡0
d_m'94'n'8801'0'8658'm'8801'0_3024 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'94'n'8801'0'8658'm'8801'0_3024 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_m'94'n'8801'0'8658'm'8801'0_3024 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m^n≡1⇒n≡0∨m≡1
d_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_3036 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_3036 :: Integer -> Integer -> T__'8801'__12 -> T__'8846'__30
d_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_3036 ~Integer
v0 Integer
v1 ~T__'8801'__12
v2
  = Integer -> T__'8846'__30
du_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_3036 Integer
v1
du_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_3036 ::
  Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
du_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_3036 :: Integer -> T__'8846'__30
du_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_3036 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Any -> T__'8846'__30) -> Any -> T__'8846'__30
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
forall a. a
erased
      Integer
_ -> (Any -> T__'8846'__30) -> Any -> T__'8846'__30
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
forall a. a
erased
-- Data.Nat.Properties.m≤n⇒m⊔n≡n
d_m'8804'n'8658'm'8852'n'8801'n_3050 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8804'n'8658'm'8852'n'8801'n_3050 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8804'n'8658'm'8852'n'8801'n_3050 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m≥n⇒m⊔n≡m
d_m'8805'n'8658'm'8852'n'8801'm_3060 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8805'n'8658'm'8852'n'8801'm_3060 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8805'n'8658'm'8852'n'8801'm_3060 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m≤n⇒m⊓n≡m
d_m'8804'n'8658'm'8851'n'8801'm_3074 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8804'n'8658'm'8851'n'8801'm_3074 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8804'n'8658'm'8851'n'8801'm_3074 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m≥n⇒m⊓n≡n
d_m'8805'n'8658'm'8851'n'8801'n_3084 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8805'n'8658'm'8851'n'8801'n_3084 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8805'n'8658'm'8851'n'8801'n_3084 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-operator
d_'8851''45'operator_3094 ::
  MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.T_MinOperator_84
d_'8851''45'operator_3094 :: T_MinOperator_84
d_'8851''45'operator_3094
  = ((Any -> Any -> Any)
 -> (Any -> Any -> Any -> Any)
 -> (Any -> Any -> Any -> Any)
 -> T_MinOperator_84)
-> Any -> Any -> Any -> T_MinOperator_84
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> (Any -> Any -> Any -> Any)
-> (Any -> Any -> Any -> Any)
-> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.C_MinOperator'46'constructor_983
      ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d__'8851'__116) Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.⊔-operator
d_'8852''45'operator_3096 ::
  MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.T_MaxOperator_114
d_'8852''45'operator_3096 :: T_MaxOperator_114
d_'8852''45'operator_3096
  = ((Any -> Any -> Any)
 -> (Any -> Any -> Any -> Any)
 -> (Any -> Any -> Any -> Any)
 -> T_MaxOperator_114)
-> Any -> Any -> Any -> T_MaxOperator_114
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> (Any -> Any -> Any -> Any)
-> (Any -> Any -> Any -> Any)
-> T_MaxOperator_114
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.C_MaxOperator'46'constructor_1521
      ((Integer -> Integer -> Integer) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d__'8852'__106) Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.antimono-≤-distrib-⊓
d_antimono'45''8804''45'distrib'45''8851'_3100 ::
  (Integer -> Integer) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18) ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_antimono'45''8804''45'distrib'45''8851'_3100 :: (Integer -> Integer)
-> (Integer -> Integer -> T__'8801'__12 -> T__'8801'__12)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
d_antimono'45''8804''45'distrib'45''8851'_3100 = (Integer -> Integer)
-> (Integer -> Integer -> T__'8801'__12 -> T__'8801'__12)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.antimono-≤-distrib-⊔
d_antimono'45''8804''45'distrib'45''8852'_3102 ::
  (Integer -> Integer) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18) ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_antimono'45''8804''45'distrib'45''8852'_3102 :: (Integer -> Integer)
-> (Integer -> Integer -> T__'8801'__12 -> T__'8801'__12)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
d_antimono'45''8804''45'distrib'45''8852'_3102 = (Integer -> Integer)
-> (Integer -> Integer -> T__'8801'__12 -> T__'8801'__12)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.mono-≤-distrib-⊓
d_mono'45''8804''45'distrib'45''8851'_3104 ::
  (Integer -> Integer) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18) ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_mono'45''8804''45'distrib'45''8851'_3104 :: (Integer -> Integer)
-> (Integer -> Integer -> T__'8801'__12 -> T__'8801'__12)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
d_mono'45''8804''45'distrib'45''8851'_3104 = (Integer -> Integer)
-> (Integer -> Integer -> T__'8801'__12 -> T__'8801'__12)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.mono-≤-distrib-⊔
d_mono'45''8804''45'distrib'45''8852'_3106 ::
  (Integer -> Integer) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18) ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_mono'45''8804''45'distrib'45''8852'_3106 :: (Integer -> Integer)
-> (Integer -> Integer -> T__'8801'__12 -> T__'8801'__12)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
d_mono'45''8804''45'distrib'45''8852'_3106 = (Integer -> Integer)
-> (Integer -> Integer -> T__'8801'__12 -> T__'8801'__12)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤x
d_x'8851'y'8804'x_3108 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8804'x_3108 :: Integer -> Integer -> T__'8804'__18
d_x'8851'y'8804'x_3108
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'x_1626
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.x≤y⇒x⊓z≤y
d_x'8804'y'8658'x'8851'z'8804'y_3110 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8804'y'8658'x'8851'z'8804'y_3110 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_x'8804'y'8658'x'8851'z'8804'y_3110
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'x'8851'z'8804'y_1982
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.x≤y⇒z⊓x≤y
d_x'8804'y'8658'z'8851'x'8804'y_3112 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8804'y'8658'z'8851'x'8804'y_3112 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_x'8804'y'8658'z'8851'x'8804'y_3112
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'z'8851'x'8804'y_1994
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.x≤y⇒x⊓z≤y
d_x'8804'y'8658'x'8851'z'8804'y_3114 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8804'y'8658'x'8851'z'8804'y_3114 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_x'8804'y'8658'x'8851'z'8804'y_3114
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'x'8851'z'8804'y_1982
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.x≤y⇒z⊓x≤y
d_x'8804'y'8658'z'8851'x'8804'y_3116 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8804'y'8658'z'8851'x'8804'y_3116 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_x'8804'y'8658'z'8851'x'8804'y_3116
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'z'8851'x'8804'y_1994
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.x≤y⊓z⇒x≤y
d_x'8804'y'8851'z'8658'x'8804'y_3118 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8804'y'8851'z'8658'x'8804'y_3118 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_x'8804'y'8851'z'8658'x'8804'y_3118
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'y_2006
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.x≤y⊓z⇒x≤z
d_x'8804'y'8851'z'8658'x'8804'z_3120 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8804'y'8851'z'8658'x'8804'z_3120 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_x'8804'y'8851'z'8658'x'8804'z_3120
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'z_2020
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤y
d_x'8851'y'8804'y_3122 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8804'y_3122 :: Integer -> Integer -> T__'8804'__18
d_x'8851'y'8804'y_3122
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_1652
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≈x⇒x≤y
d_x'8851'y'8776'x'8658'x'8804'y_3124 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8776'x'8658'x'8804'y_3124 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_x'8851'y'8776'x'8658'x'8804'y_3124
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'x'8658'x'8804'y_1890
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≈y⇒y≤x
d_x'8851'y'8776'y'8658'y'8804'x_3126 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8776'y'8658'y'8804'x_3126 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_x'8851'y'8776'y'8658'y'8804'x_3126
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'y'8658'y'8804'x_1922
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤x
d_x'8851'y'8804'x_3128 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8804'x_3128 :: Integer -> Integer -> T__'8804'__18
d_x'8851'y'8804'x_3128
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'x_1626
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤x⊔y
d_x'8851'y'8804'x'8852'y_3130 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8804'x'8852'y_3130 :: Integer -> Integer -> T__'8804'__18
d_x'8851'y'8804'x'8852'y_3130
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> Any -> Any -> Any)
-> Any -> Any -> Any -> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_x'8851'y'8804'x'8852'y_2072
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤y
d_x'8851'y'8804'y_3132 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8804'y_3132 :: Integer -> Integer -> T__'8804'__18
d_x'8851'y'8804'y_3132
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_1652
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≈x⇒x≤y
d_x'8851'y'8776'x'8658'x'8804'y_3134 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8776'x'8658'x'8804'y_3134 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_x'8851'y'8776'x'8658'x'8804'y_3134
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'x'8658'x'8804'y_1890
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.x⊓y≈y⇒y≤x
d_x'8851'y'8776'y'8658'y'8804'x_3136 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8851'y'8776'y'8658'y'8804'x_3136 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_x'8851'y'8776'y'8658'y'8804'x_3136
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'y'8658'y'8804'x_1922
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.x≤y⊓z⇒x≤y
d_x'8804'y'8851'z'8658'x'8804'y_3138 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8804'y'8851'z'8658'x'8804'y_3138 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_x'8804'y'8851'z'8658'x'8804'y_3138
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'y_2006
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.x≤y⊓z⇒x≤z
d_x'8804'y'8851'z'8658'x'8804'z_3140 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_x'8804'y'8851'z'8658'x'8804'z_3140 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_x'8804'y'8851'z'8658'x'8804'z_3140
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'z_2020
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-absorbs-⊔
d_'8851''45'absorbs'45''8852'_3142 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'absorbs'45''8852'_3142 :: Integer -> Integer -> T__'8801'__12
d_'8851''45'absorbs'45''8852'_3142 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-assoc
d_'8851''45'assoc_3144 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'assoc_3144 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8851''45'assoc_3144 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-band
d_'8851''45'band_3146 :: MAlonzo.Code.Algebra.Bundles.T_Band_266
d_'8851''45'band_3146 :: T_Band_266
d_'8851''45'band_3146
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_Band_266
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_Band_266)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_Band_266
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'band_1872
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-comm
d_'8851''45'comm_3148 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'comm_3148 :: Integer -> Integer -> T__'8801'__12
d_'8851''45'comm_3148 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-commutativeSemigroup
d_'8851''45'commutativeSemigroup_3150 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemigroup_332
d_'8851''45'commutativeSemigroup_3150 :: T_CommutativeSemigroup_332
d_'8851''45'commutativeSemigroup_3150
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_CommutativeSemigroup_332
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> T_CommutativeSemigroup_332)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> T_CommutativeSemigroup_332
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'commutativeSemigroup_1874
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-distrib-⊔
d_'8851''45'distrib'45''8852'_3158 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'8851''45'distrib'45''8852'_3158 :: T_Σ_14
d_'8851''45'distrib'45''8852'_3158
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14)
-> Any -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45'distrib'45''8852'_1876
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊓-distribʳ-⊔
d_'8851''45'distrib'691''45''8852'_3160 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'distrib'691''45''8852'_3160 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8851''45'distrib'691''45''8852'_3160 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-distribˡ-⊔
d_'8851''45'distrib'737''45''8852'_3162 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'distrib'737''45''8852'_3162 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8851''45'distrib'737''45''8852'_3162 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-glb
d_'8851''45'glb_3164 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'glb_3164 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'glb_3164
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_2100
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-idem
d_'8851''45'idem_3166 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'idem_3166 :: Integer -> T__'8801'__12
d_'8851''45'idem_3166 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isBand
d_'8851''45'isBand_3174 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_230
d_'8851''45'isBand_3174 :: T_IsBand_230
d_'8851''45'isBand_3174
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsBand_230
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsBand_230)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsBand_230
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isBand_1852
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isCommutativeSemigroup
d_'8851''45'isCommutativeSemigroup_3176 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270
d_'8851''45'isCommutativeSemigroup_3176 :: T_IsCommutativeSemigroup_270
d_'8851''45'isCommutativeSemigroup_3176
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsCommutativeSemigroup_270
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> T_IsCommutativeSemigroup_270)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> T_IsCommutativeSemigroup_270
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isCommutativeSemigroup_1854
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isMagma
d_'8851''45'isMagma_3178 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_'8851''45'isMagma_3178 :: T_IsMagma_86
d_'8851''45'isMagma_3178
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsMagma_86
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsMagma_86)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsMagma_86
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isMagma_1848
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isSelectiveMagma
d_'8851''45'isSelectiveMagma_3182 ::
  MAlonzo.Code.Algebra.Structures.T_IsSelectiveMagma_158
d_'8851''45'isSelectiveMagma_3182 :: T_IsSelectiveMagma_158
d_'8851''45'isSelectiveMagma_3182
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsSelectiveMagma_158
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSelectiveMagma_158)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSelectiveMagma_158
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSelectiveMagma_1858
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isSemigroup
d_'8851''45'isSemigroup_3184 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_'8851''45'isSemigroup_3184 :: T_IsSemigroup_194
d_'8851''45'isSemigroup_3184
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemigroup_194)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemigroup_1850
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isSemilattice
d_'8851''45'isSemilattice_3186 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312
d_'8851''45'isSemilattice_3186 :: T_IsSemilattice_312
d_'8851''45'isSemilattice_3186
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsSemilattice_312
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemilattice_312)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemilattice_312
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemilattice_1856
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-magma
d_'8851''45'magma_3188 :: MAlonzo.Code.Algebra.Bundles.T_Magma_36
d_'8851''45'magma_3188 :: T_Magma_36
d_'8851''45'magma_3188
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_Magma_36
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_Magma_36)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_Magma_36
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'magma_1868
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-mono-≤
d_'8851''45'mono'45''8804'_3190 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'mono'45''8804'_3190 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'mono'45''8804'_3190
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'45''8804'_2028
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-monoʳ-≤
d_'8851''45'mono'691''45''8804'_3194 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'mono'691''45''8804'_3194 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8851''45'mono'691''45''8804'_3194
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'691''45''8804'_2088
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-monoˡ-≤
d_'8851''45'mono'737''45''8804'_3196 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'mono'737''45''8804'_3196 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8851''45'mono'737''45''8804'_3196
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'737''45''8804'_2078
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-sel
d_'8851''45'sel_3200 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'8851''45'sel_3200 :: Integer -> Integer -> T__'8846'__30
d_'8851''45'sel_3200
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8846'__30
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> T__'8846'__30)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> T__'8846'__30
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'sel_1806
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-selectiveMagma
d_'8851''45'selectiveMagma_3202 ::
  MAlonzo.Code.Algebra.Bundles.T_SelectiveMagma_90
d_'8851''45'selectiveMagma_3202 :: T_SelectiveMagma_90
d_'8851''45'selectiveMagma_3202
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_SelectiveMagma_90
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_SelectiveMagma_90)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_SelectiveMagma_90
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'selectiveMagma_1878
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-semigroup
d_'8851''45'semigroup_3204 ::
  MAlonzo.Code.Algebra.Bundles.T_Semigroup_206
d_'8851''45'semigroup_3204 :: T_Semigroup_206
d_'8851''45'semigroup_3204
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_Semigroup_206
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_Semigroup_206)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_Semigroup_206
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'semigroup_1870
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-semilattice
d_'8851''45'semilattice_3206 ::
  MAlonzo.Code.Algebra.Bundles.T_Semilattice_402
d_'8851''45'semilattice_3206 :: T_Semilattice_402
d_'8851''45'semilattice_3206
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_Semilattice_402
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_Semilattice_402)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_Semilattice_402
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'semilattice_1876
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-triangulate
d_'8851''45'triangulate_3208 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'triangulate_3208 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8851''45'triangulate_3208 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-⊔-absorptive
d_'8851''45''8852''45'absorptive_3216 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'8851''45''8852''45'absorptive_3216 :: T_Σ_14
d_'8851''45''8852''45'absorptive_3216
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14)
-> Any -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'absorptive_1956
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊓-⊔-distributiveLattice
d_'8851''45''8852''45'distributiveLattice_3218 ::
  MAlonzo.Code.Algebra.Bundles.T_DistributiveLattice_1228
d_'8851''45''8852''45'distributiveLattice_3218 :: T_DistributiveLattice_1228
d_'8851''45''8852''45'distributiveLattice_3218
  = (T_TotalPreorder_204
 -> T_MinOperator_84
 -> T_MaxOperator_114
 -> T_DistributiveLattice_1228)
-> Any -> Any -> Any -> T_DistributiveLattice_1228
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84
-> T_MaxOperator_114
-> T_DistributiveLattice_1228
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'distributiveLattice_1972
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊓-⊔-isDistributiveLattice
d_'8851''45''8852''45'isDistributiveLattice_3220 ::
  MAlonzo.Code.Algebra.Structures.T_IsDistributiveLattice_814
d_'8851''45''8852''45'isDistributiveLattice_3220 :: T_IsDistributiveLattice_814
d_'8851''45''8852''45'isDistributiveLattice_3220
  = (T_TotalPreorder_204
 -> T_MinOperator_84
 -> T_MaxOperator_114
 -> T_IsDistributiveLattice_814)
-> Any -> Any -> Any -> T_IsDistributiveLattice_814
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84
-> T_MaxOperator_114
-> T_IsDistributiveLattice_814
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'isDistributiveLattice_1962
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊓-⊔-isLattice
d_'8851''45''8852''45'isLattice_3222 ::
  MAlonzo.Code.Algebra.Structures.T_IsLattice_740
d_'8851''45''8852''45'isLattice_3222 :: T_IsLattice_740
d_'8851''45''8852''45'isLattice_3222
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_IsLattice_740)
-> Any -> Any -> Any -> T_IsLattice_740
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_IsLattice_740
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'isLattice_1960
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊓-⊔-lattice
d_'8851''45''8852''45'lattice_3224 ::
  MAlonzo.Code.Algebra.Bundles.T_Lattice_1144
d_'8851''45''8852''45'lattice_3224 :: T_Lattice_1144
d_'8851''45''8852''45'lattice_3224
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_Lattice_1144)
-> Any -> Any -> Any -> T_Lattice_1144
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_Lattice_1144
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'lattice_1968
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊔-absorbs-⊓
d_'8852''45'absorbs'45''8851'_3226 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8852''45'absorbs'45''8851'_3226 :: Integer -> Integer -> T__'8801'__12
d_'8852''45'absorbs'45''8851'_3226 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-assoc
d_'8851''45'assoc_3228 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'assoc_3228 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8851''45'assoc_3228 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-band
d_'8851''45'band_3230 :: MAlonzo.Code.Algebra.Bundles.T_Band_266
d_'8851''45'band_3230 :: T_Band_266
d_'8851''45'band_3230
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_Band_266
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_Band_266)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_Band_266
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'band_1872
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-comm
d_'8851''45'comm_3232 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'comm_3232 :: Integer -> Integer -> T__'8801'__12
d_'8851''45'comm_3232 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-commutativeSemigroup
d_'8851''45'commutativeSemigroup_3234 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemigroup_332
d_'8851''45'commutativeSemigroup_3234 :: T_CommutativeSemigroup_332
d_'8851''45'commutativeSemigroup_3234
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_CommutativeSemigroup_332
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> T_CommutativeSemigroup_332)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> T_CommutativeSemigroup_332
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'commutativeSemigroup_1874
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊔-distrib-⊓
d_'8852''45'distrib'45''8851'_3242 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'8852''45'distrib'45''8851'_3242 :: T_Σ_14
d_'8852''45'distrib'45''8851'_3242
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14)
-> Any -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8852''45'distrib'45''8851'_1908
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊔-distribʳ-⊓
d_'8852''45'distrib'691''45''8851'_3244 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8852''45'distrib'691''45''8851'_3244 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8852''45'distrib'691''45''8851'_3244 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊔-distribˡ-⊓
d_'8852''45'distrib'737''45''8851'_3246 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8852''45'distrib'737''45''8851'_3246 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8852''45'distrib'737''45''8851'_3246 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-idem
d_'8851''45'idem_3248 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'idem_3248 :: Integer -> T__'8801'__12
d_'8851''45'idem_3248 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isBand
d_'8851''45'isBand_3256 ::
  MAlonzo.Code.Algebra.Structures.T_IsBand_230
d_'8851''45'isBand_3256 :: T_IsBand_230
d_'8851''45'isBand_3256
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsBand_230
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsBand_230)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsBand_230
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isBand_1852
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isCommutativeSemigroup
d_'8851''45'isCommutativeSemigroup_3258 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_270
d_'8851''45'isCommutativeSemigroup_3258 :: T_IsCommutativeSemigroup_270
d_'8851''45'isCommutativeSemigroup_3258
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsCommutativeSemigroup_270
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> T_IsCommutativeSemigroup_270)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> T_IsCommutativeSemigroup_270
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isCommutativeSemigroup_1854
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isMagma
d_'8851''45'isMagma_3260 ::
  MAlonzo.Code.Algebra.Structures.T_IsMagma_86
d_'8851''45'isMagma_3260 :: T_IsMagma_86
d_'8851''45'isMagma_3260
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsMagma_86
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsMagma_86)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsMagma_86
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isMagma_1848
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isSelectiveMagma
d_'8851''45'isSelectiveMagma_3264 ::
  MAlonzo.Code.Algebra.Structures.T_IsSelectiveMagma_158
d_'8851''45'isSelectiveMagma_3264 :: T_IsSelectiveMagma_158
d_'8851''45'isSelectiveMagma_3264
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsSelectiveMagma_158
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSelectiveMagma_158)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSelectiveMagma_158
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSelectiveMagma_1858
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isSemigroup
d_'8851''45'isSemigroup_3266 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_194
d_'8851''45'isSemigroup_3266 :: T_IsSemigroup_194
d_'8851''45'isSemigroup_3266
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsSemigroup_194
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemigroup_194)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemigroup_1850
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-isSemilattice
d_'8851''45'isSemilattice_3268 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemilattice_312
d_'8851''45'isSemilattice_3268 :: T_IsSemilattice_312
d_'8851''45'isSemilattice_3268
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_IsSemilattice_312
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemilattice_312)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemilattice_312
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemilattice_1856
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-glb
d_'8851''45'glb_3270 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'glb_3270 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'glb_3270
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_2100
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-magma
d_'8851''45'magma_3272 :: MAlonzo.Code.Algebra.Bundles.T_Magma_36
d_'8851''45'magma_3272 :: T_Magma_36
d_'8851''45'magma_3272
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_Magma_36
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_Magma_36)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_Magma_36
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'magma_1868
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-mono-≤
d_'8851''45'mono'45''8804'_3274 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'mono'45''8804'_3274 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'mono'45''8804'_3274
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'45''8804'_2028
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-monoʳ-≤
d_'8851''45'mono'691''45''8804'_3278 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'mono'691''45''8804'_3278 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8851''45'mono'691''45''8804'_3278
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'691''45''8804'_2088
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-monoˡ-≤
d_'8851''45'mono'737''45''8804'_3280 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'mono'737''45''8804'_3280 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8851''45'mono'737''45''8804'_3280
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'737''45''8804'_2078
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-sel
d_'8851''45'sel_3282 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'8851''45'sel_3282 :: Integer -> Integer -> T__'8846'__30
d_'8851''45'sel_3282
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8846'__30
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> T__'8846'__30)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> T__'8846'__30
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'sel_1806
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-selectiveMagma
d_'8851''45'selectiveMagma_3284 ::
  MAlonzo.Code.Algebra.Bundles.T_SelectiveMagma_90
d_'8851''45'selectiveMagma_3284 :: T_SelectiveMagma_90
d_'8851''45'selectiveMagma_3284
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_SelectiveMagma_90
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_SelectiveMagma_90)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_SelectiveMagma_90
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'selectiveMagma_1878
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-semigroup
d_'8851''45'semigroup_3286 ::
  MAlonzo.Code.Algebra.Bundles.T_Semigroup_206
d_'8851''45'semigroup_3286 :: T_Semigroup_206
d_'8851''45'semigroup_3286
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_Semigroup_206
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_Semigroup_206)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_Semigroup_206
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'semigroup_1870
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-semilattice
d_'8851''45'semilattice_3288 ::
  MAlonzo.Code.Algebra.Bundles.T_Semilattice_402
d_'8851''45'semilattice_3288 :: T_Semilattice_402
d_'8851''45'semilattice_3288
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T_Semilattice_402
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_Semilattice_402)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> T_Semilattice_402
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'semilattice_1876
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-⊔-properties.⊓-triangulate
d_'8851''45'triangulate_3290 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'triangulate_3290 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8851''45'triangulate_3290 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-⊔-properties.⊔-⊓-absorptive
d_'8852''45''8851''45'absorptive_3298 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'8852''45''8851''45'absorptive_3298 :: T_Σ_14
d_'8852''45''8851''45'absorptive_3298
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14)
-> Any -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'absorptive_1954
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊔-⊓-distributiveLattice
d_'8852''45''8851''45'distributiveLattice_3300 ::
  MAlonzo.Code.Algebra.Bundles.T_DistributiveLattice_1228
d_'8852''45''8851''45'distributiveLattice_3300 :: T_DistributiveLattice_1228
d_'8852''45''8851''45'distributiveLattice_3300
  = (T_TotalPreorder_204
 -> T_MinOperator_84
 -> T_MaxOperator_114
 -> T_DistributiveLattice_1228)
-> Any -> Any -> Any -> T_DistributiveLattice_1228
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84
-> T_MaxOperator_114
-> T_DistributiveLattice_1228
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'distributiveLattice_1970
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊔-⊓-isDistributiveLattice
d_'8852''45''8851''45'isDistributiveLattice_3302 ::
  MAlonzo.Code.Algebra.Structures.T_IsDistributiveLattice_814
d_'8852''45''8851''45'isDistributiveLattice_3302 :: T_IsDistributiveLattice_814
d_'8852''45''8851''45'isDistributiveLattice_3302
  = (T_TotalPreorder_204
 -> T_MinOperator_84
 -> T_MaxOperator_114
 -> T_IsDistributiveLattice_814)
-> Any -> Any -> Any -> T_IsDistributiveLattice_814
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84
-> T_MaxOperator_114
-> T_IsDistributiveLattice_814
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'isDistributiveLattice_1964
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊔-⊓-isLattice
d_'8852''45''8851''45'isLattice_3304 ::
  MAlonzo.Code.Algebra.Structures.T_IsLattice_740
d_'8852''45''8851''45'isLattice_3304 :: T_IsLattice_740
d_'8852''45''8851''45'isLattice_3304
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_IsLattice_740)
-> Any -> Any -> Any -> T_IsLattice_740
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_IsLattice_740
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'isLattice_1958
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊓-⊔-properties.⊔-⊓-lattice
d_'8852''45''8851''45'lattice_3306 ::
  MAlonzo.Code.Algebra.Bundles.T_Lattice_1144
d_'8852''45''8851''45'lattice_3306 :: T_Lattice_1144
d_'8852''45''8851''45'lattice_3306
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_Lattice_1144)
-> Any -> Any -> Any -> T_Lattice_1144
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_Lattice_1144
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'lattice_1966
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊔-identityˡ
d_'8852''45'identity'737'_3308 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8852''45'identity'737'_3308 :: Integer -> T__'8801'__12
d_'8852''45'identity'737'_3308 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊔-identityʳ
d_'8852''45'identity'691'_3310 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8852''45'identity'691'_3310 :: Integer -> T__'8801'__12
d_'8852''45'identity'691'_3310 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊔-identity
d_'8852''45'identity_3314 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'8852''45'identity_3314 :: T_Σ_14
d_'8852''45'identity_3314
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.⊔-0-isMonoid
d_'8852''45'0'45'isMonoid_3316 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_358
d_'8852''45'0'45'isMonoid_3316 :: T_IsMonoid_358
d_'8852''45'0'45'isMonoid_3316
  = (T_IsSemigroup_194 -> T_Σ_14 -> T_IsMonoid_358)
-> Any -> Any -> T_IsMonoid_358
forall a b. a -> b
coe
      T_IsSemigroup_194 -> T_Σ_14 -> T_IsMonoid_358
MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_7687
      (let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
       Any -> Any
forall a b. a -> b
coe
         (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
          Any -> Any
forall a b. a -> b
coe
            ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemigroup_194)
-> Any -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemigroup_1850
               ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
                  T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
                  (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
               ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
                  T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
                  (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1)))))
      (T_Σ_14 -> Any
forall a b. a -> b
coe T_Σ_14
d_'8852''45'identity_3314)
-- Data.Nat.Properties.⊔-0-isCommutativeMonoid
d_'8852''45'0'45'isCommutativeMonoid_3318 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_406
d_'8852''45'0'45'isCommutativeMonoid_3318 :: T_IsCommutativeMonoid_406
d_'8852''45'0'45'isCommutativeMonoid_3318
  = (T_IsMonoid_358
 -> (Any -> Any -> Any) -> T_IsCommutativeMonoid_406)
-> Any -> Any -> T_IsCommutativeMonoid_406
forall a b. a -> b
coe
      T_IsMonoid_358 -> (Any -> Any -> Any) -> T_IsCommutativeMonoid_406
MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_9361
      (T_IsMonoid_358 -> Any
forall a b. a -> b
coe T_IsMonoid_358
d_'8852''45'0'45'isMonoid_3316)
      (let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
       Any -> Any
forall a b. a -> b
coe
         (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
          Any -> Any
forall a b. a -> b
coe
            ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'comm_1674
               ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
                  T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
                  (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
               ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
                  T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
                  (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1)))))
-- Data.Nat.Properties.⊔-0-monoid
d_'8852''45'0'45'monoid_3320 ::
  MAlonzo.Code.Algebra.Bundles.T_Monoid_506
d_'8852''45'0'45'monoid_3320 :: T_Monoid_506
d_'8852''45'0'45'monoid_3320
  = ((Any -> Any -> Any) -> Any -> T_IsMonoid_358 -> T_Monoid_506)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsMonoid_358
-> T_Monoid_506
forall a b. a -> b
coe
      (Any -> Any -> Any) -> Any -> T_IsMonoid_358 -> T_Monoid_506
MAlonzo.Code.Algebra.Bundles.C_Monoid'46'constructor_8851
      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d__'8852'__106 (Integer
0 :: Integer)
      T_IsMonoid_358
d_'8852''45'0'45'isMonoid_3316
-- Data.Nat.Properties.⊔-0-commutativeMonoid
d_'8852''45'0'45'commutativeMonoid_3322 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeMonoid_582
d_'8852''45'0'45'commutativeMonoid_3322 :: T_CommutativeMonoid_582
d_'8852''45'0'45'commutativeMonoid_3322
  = ((Any -> Any -> Any)
 -> Any -> T_IsCommutativeMonoid_406 -> T_CommutativeMonoid_582)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeMonoid_406
-> T_CommutativeMonoid_582
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> Any -> T_IsCommutativeMonoid_406 -> T_CommutativeMonoid_582
MAlonzo.Code.Algebra.Bundles.C_CommutativeMonoid'46'constructor_10343
      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d__'8852'__106 (Integer
0 :: Integer)
      T_IsCommutativeMonoid_406
d_'8852''45'0'45'isCommutativeMonoid_3318
-- Data.Nat.Properties.mono-≤-distrib-⊔
d_mono'45''8804''45'distrib'45''8852'_3330 ::
  (Integer -> Integer) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18) ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_mono'45''8804''45'distrib'45''8852'_3330 :: (Integer -> Integer)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
d_mono'45''8804''45'distrib'45''8852'_3330 = (Integer -> Integer)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.mono-≤-distrib-⊓
d_mono'45''8804''45'distrib'45''8851'_3340 ::
  (Integer -> Integer) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18) ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_mono'45''8804''45'distrib'45''8851'_3340 :: (Integer -> Integer)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
d_mono'45''8804''45'distrib'45''8851'_3340 = (Integer -> Integer)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.antimono-≤-distrib-⊓
d_antimono'45''8804''45'distrib'45''8851'_3350 ::
  (Integer -> Integer) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18) ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_antimono'45''8804''45'distrib'45''8851'_3350 :: (Integer -> Integer)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
d_antimono'45''8804''45'distrib'45''8851'_3350 = (Integer -> Integer)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.antimono-≤-distrib-⊔
d_antimono'45''8804''45'distrib'45''8852'_3360 ::
  (Integer -> Integer) ->
  (Integer ->
   Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18) ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_antimono'45''8804''45'distrib'45''8852'_3360 :: (Integer -> Integer)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
d_antimono'45''8804''45'distrib'45''8852'_3360 = (Integer -> Integer)
-> (Integer -> Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m<n⇒m<n⊔o
d_m'60'n'8658'm'60'n'8852'o_3370 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8658'm'60'n'8852'o_3370 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8658'm'60'n'8852'o_3370 Integer
v0 Integer
v1
  = let v2 :: T_TotalPreorder_204
v2 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v3 :: T_MaxOperator_114
v3 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'x'8851'z'8804'y_1982
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v2))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v3))
            (Integer -> Any
forall a b. a -> b
coe Integer
v1) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0))))
-- Data.Nat.Properties.m<n⇒m<o⊔n
d_m'60'n'8658'm'60'o'8852'n_3378 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8658'm'60'o'8852'n_3378 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8658'm'60'o'8852'n_3378 Integer
v0 Integer
v1
  = let v2 :: T_TotalPreorder_204
v2 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v3 :: T_MaxOperator_114
v3 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'z'8851'x'8804'y_1994
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v2))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v3))
            (Integer -> Any
forall a b. a -> b
coe Integer
v1) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0))))
-- Data.Nat.Properties.m⊔n<o⇒m<o
d_m'8852'n'60'o'8658'm'60'o_3386 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8852'n'60'o'8658'm'60'o_3386 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'8852'n'60'o'8658'm'60'o_3386 Integer
v0 Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'8852'n'60'o'8658'm'60'o_3386 Integer
v0 Integer
v1 T__'8804'__18
v3
du_m'8852'n'60'o'8658'm'60'o_3386 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'8852'n'60'o'8658'm'60'o_3386 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'8852'n'60'o'8658'm'60'o_3386 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868
      (let v3 :: T_TotalPreorder_204
v3 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
       Any -> Any
forall a b. a -> b
coe
         (let v4 :: T_MaxOperator_114
v4 = T_MaxOperator_114
d_'8852''45'operator_3096 in
          Any -> Any
forall a b. a -> b
coe
            ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'x_1626
               ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
                  T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
                  (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v3))
               ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
                  T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
                  (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v4))
               (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))))
      (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
-- Data.Nat.Properties.m⊔n<o⇒n<o
d_m'8852'n'60'o'8658'n'60'o_3400 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8852'n'60'o'8658'n'60'o_3400 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'8852'n'60'o'8658'n'60'o_3400 Integer
v0 Integer
v1 ~Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'8852'n'60'o'8658'n'60'o_3400 Integer
v0 Integer
v1 T__'8804'__18
v3
du_m'8852'n'60'o'8658'n'60'o_3400 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'8852'n'60'o'8658'n'60'o_3400 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'8852'n'60'o'8658'n'60'o_3400 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868
      (let v3 :: T_TotalPreorder_204
v3 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
       Any -> Any
forall a b. a -> b
coe
         (let v4 :: T_MaxOperator_114
v4 = T_MaxOperator_114
d_'8852''45'operator_3096 in
          Any -> Any
forall a b. a -> b
coe
            ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_1652
               ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
                  T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
                  (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v3))
               ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
                  T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
                  (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v4))
               (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))))
      (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
-- Data.Nat.Properties.⊔-mono-<
d_'8852''45'mono'45''60'_3408 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8852''45'mono'45''60'_3408 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8852''45'mono'45''60'_3408 Integer
v0 Integer
v1 Integer
v2 Integer
v3
  = let v4 :: T_TotalPreorder_204
v4 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      (let v5 :: T_MaxOperator_114
v5 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any)
-> Any -> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'45''8804'_2028
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v4))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v5))
            (Integer -> Any
forall a b. a -> b
coe Integer
v1) ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0)) (Integer -> Any
forall a b. a -> b
coe Integer
v3)
            ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v2))))
-- Data.Nat.Properties.⊔-pres-<m
d_'8852''45'pres'45''60'm_3416 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8852''45'pres'45''60'm_3416 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8852''45'pres'45''60'm_3416 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3 T__'8804'__18
v4
  = (Integer
 -> Integer
 -> Integer
 -> Integer
 -> T__'8804'__18
 -> T__'8804'__18
 -> T__'8804'__18)
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8852''45'mono'45''60'_3408 Integer
v1 Integer
v0 Integer
v2 Integer
v0 T__'8804'__18
v3 T__'8804'__18
v4
-- Data.Nat.Properties.+-distribˡ-⊔
d_'43''45'distrib'737''45''8852'_3426 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'distrib'737''45''8852'_3426 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'43''45'distrib'737''45''8852'_3426 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-distribʳ-⊔
d_'43''45'distrib'691''45''8852'_3438 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'distrib'691''45''8852'_3438 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'43''45'distrib'691''45''8852'_3438 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-distrib-⊔
d_'43''45'distrib'45''8852'_3440 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'43''45'distrib'45''8852'_3440 :: T_Σ_14
d_'43''45'distrib'45''8852'_3440
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.m⊔n≤m+n
d_m'8852'n'8804'm'43'n_3446 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8852'n'8804'm'43'n_3446 :: Integer -> Integer -> T__'8804'__18
d_m'8852'n'8804'm'43'n_3446 Integer
v0 Integer
v1
  = let v2 :: b
v2
          = let v2 :: T_TotalPreorder_204
v2 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
            Any -> b
forall a b. a -> b
coe
              (let v3 :: T_MaxOperator_114
v3 = T_MaxOperator_114
d_'8852''45'operator_3096 in
               Any -> Any
forall a b. a -> b
coe
                 ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> T__'8846'__30)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                    T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> T__'8846'__30
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'sel_1806
                    ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
                       T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
                       (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v2))
                    ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
                       T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
                       (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v3))
                    (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))) in
    Any -> T__'8804'__18
forall a b. a -> b
coe
      (case Any -> T__'8846'__30
forall a b. a -> b
coe Any
forall a. a
v2 of
         MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
v3
           -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
         MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
v3
           -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'n'43'm_2328 (Integer -> Any
forall a b. a -> b
coe Integer
v1)
         T__'8846'__30
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.*-distribˡ-⊔
d_'42''45'distrib'737''45''8852'_3476 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'737''45''8852'_3476 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'737''45''8852'_3476 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distribʳ-⊔
d_'42''45'distrib'691''45''8852'_3498 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'691''45''8852'_3498 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'691''45''8852'_3498 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distrib-⊔
d_'42''45'distrib'45''8852'_3500 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'distrib'45''8852'_3500 :: T_Σ_14
d_'42''45'distrib'45''8852'_3500
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.⊓-zeroˡ
d_'8851''45'zero'737'_3502 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'zero'737'_3502 :: Integer -> T__'8801'__12
d_'8851''45'zero'737'_3502 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-zeroʳ
d_'8851''45'zero'691'_3504 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'zero'691'_3504 :: Integer -> T__'8801'__12
d_'8851''45'zero'691'_3504 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-zero
d_'8851''45'zero_3508 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'8851''45'zero_3508 :: T_Σ_14
d_'8851''45'zero_3508
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.⊔-⊓-isSemiringWithoutOne
d_'8852''45''8851''45'isSemiringWithoutOne_3510 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
d_'8852''45''8851''45'isSemiringWithoutOne_3510 :: T_IsSemiringWithoutOne_952
d_'8852''45''8851''45'isSemiringWithoutOne_3510
  = (T_IsCommutativeMonoid_406
 -> T_IsSemigroup_194
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutOne_952)
-> Any -> Any -> Any -> Any -> T_IsSemiringWithoutOne_952
forall a b. a -> b
coe
      T_IsCommutativeMonoid_406
-> T_IsSemigroup_194
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutOne_952
MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutOne'46'constructor_27777
      (T_IsCommutativeMonoid_406 -> Any
forall a b. a -> b
coe T_IsCommutativeMonoid_406
d_'8852''45'0'45'isCommutativeMonoid_3318)
      ((T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemigroup_194)
-> Any -> Any -> Any
forall a b. a -> b
coe
         T_TotalPreorder_204 -> T_MinOperator_84 -> T_IsSemigroup_194
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemigroup_1850
         (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
         (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094))
      ((T_TotalPreorder_204
 -> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         T_TotalPreorder_204
-> T_MinOperator_84 -> T_MaxOperator_114 -> T_Σ_14
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45'distrib'45''8852'_1876
         (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
         (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096))
      (T_Σ_14 -> Any
forall a b. a -> b
coe T_Σ_14
d_'8851''45'zero_3508)
-- Data.Nat.Properties.⊔-⊓-isCommutativeSemiringWithoutOne
d_'8852''45''8851''45'isCommutativeSemiringWithoutOne_3512 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044
d_'8852''45''8851''45'isCommutativeSemiringWithoutOne_3512 :: T_IsCommutativeSemiringWithoutOne_1044
d_'8852''45''8851''45'isCommutativeSemiringWithoutOne_3512
  = (T_IsSemiringWithoutOne_952
 -> (Any -> Any -> Any) -> T_IsCommutativeSemiringWithoutOne_1044)
-> Any -> Any -> T_IsCommutativeSemiringWithoutOne_1044
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_952
-> (Any -> Any -> Any) -> T_IsCommutativeSemiringWithoutOne_1044
MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemiringWithoutOne'46'constructor_31059
      (T_IsSemiringWithoutOne_952 -> Any
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
d_'8852''45''8851''45'isSemiringWithoutOne_3510)
      ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
         T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'comm_1674
         (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
         (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094))
-- Data.Nat.Properties.⊔-⊓-commutativeSemiringWithoutOne
d_'8852''45''8851''45'commutativeSemiringWithoutOne_3514 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemiringWithoutOne_1598
d_'8852''45''8851''45'commutativeSemiringWithoutOne_3514 :: T_CommutativeSemiringWithoutOne_1598
d_'8852''45''8851''45'commutativeSemiringWithoutOne_3514
  = ((Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> Any
 -> T_IsCommutativeSemiringWithoutOne_1044
 -> T_CommutativeSemiringWithoutOne_1598)
-> (Integer -> Integer -> Integer)
-> (Integer -> Integer -> Integer)
-> Integer
-> T_IsCommutativeSemiringWithoutOne_1044
-> T_CommutativeSemiringWithoutOne_1598
forall a b. a -> b
coe
      (Any -> Any -> Any)
-> (Any -> Any -> Any)
-> Any
-> T_IsCommutativeSemiringWithoutOne_1044
-> T_CommutativeSemiringWithoutOne_1598
MAlonzo.Code.Algebra.Bundles.C_CommutativeSemiringWithoutOne'46'constructor_27945
      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d__'8852'__106
      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d__'8851'__116 (Integer
0 :: Integer)
      T_IsCommutativeSemiringWithoutOne_1044
d_'8852''45''8851''45'isCommutativeSemiringWithoutOne_3512
-- Data.Nat.Properties.m<n⇒m⊓o<n
d_m'60'n'8658'm'8851'o'60'n_3522 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8658'm'8851'o'60'n_3522 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8658'm'8851'o'60'n_3522 Integer
v0 ~Integer
v1 Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'm'8851'o'60'n_3522 Integer
v0 Integer
v2 T__'8804'__18
v3
du_m'60'n'8658'm'8851'o'60'n_3522 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'60'n'8658'm'8851'o'60'n_3522 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'm'8851'o'60'n_3522 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868
      ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'x_1626
         (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
         (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
-- Data.Nat.Properties.m<n⇒o⊓m<n
d_m'60'n'8658'o'8851'm'60'n_3534 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8658'o'8851'm'60'n_3534 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8658'o'8851'm'60'n_3534 Integer
v0 ~Integer
v1 Integer
v2 T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'o'8851'm'60'n_3534 Integer
v0 Integer
v2 T__'8804'__18
v3
du_m'60'n'8658'o'8851'm'60'n_3534 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'60'n'8658'o'8851'm'60'n_3534 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'o'8851'm'60'n_3534 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868
      ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_1652
         (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
         (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0))
      (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
-- Data.Nat.Properties.m<n⊓o⇒m<n
d_m'60'n'8851'o'8658'm'60'n_3546 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8851'o'8658'm'60'n_3546 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8851'o'8658'm'60'n_3546 Integer
v0
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'y_2006
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094)
      ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.m<n⊓o⇒m<o
d_m'60'n'8851'o'8658'm'60'o_3554 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8851'o'8658'm'60'o_3554 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8851'o'8658'm'60'o_3554 Integer
v0
  = (T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'z_2020
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094)
      ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.⊓-mono-<
d_'8851''45'mono'45''60'_3556 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'mono'45''60'_3556 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'mono'45''60'_3556 Integer
v0 Integer
v1 Integer
v2 Integer
v3
  = (T_TotalPreorder_204
 -> T_MinOperator_84
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any
 -> Any)
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'45''8804'_2028
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094)
      ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0)) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
      ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v2)) (Integer -> Any
forall a b. a -> b
coe Integer
v3)
-- Data.Nat.Properties.⊓-pres-m<
d_'8851''45'pres'45'm'60'_3564 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'pres'45'm'60'_3564 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'pres'45'm'60'_3564 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3 T__'8804'__18
v4
  = (Integer
 -> Integer
 -> Integer
 -> Integer
 -> T__'8804'__18
 -> T__'8804'__18
 -> T__'8804'__18)
-> Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'mono'45''60'_3556 Integer
v0 Integer
v1 Integer
v0 Integer
v2 T__'8804'__18
v3 T__'8804'__18
v4
-- Data.Nat.Properties.+-distribˡ-⊓
d_'43''45'distrib'737''45''8851'_3574 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'distrib'737''45''8851'_3574 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'43''45'distrib'737''45''8851'_3574 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-distribʳ-⊓
d_'43''45'distrib'691''45''8851'_3586 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'distrib'691''45''8851'_3586 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'43''45'distrib'691''45''8851'_3586 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-distrib-⊓
d_'43''45'distrib'45''8851'_3588 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'43''45'distrib'45''8851'_3588 :: T_Σ_14
d_'43''45'distrib'45''8851'_3588
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.m⊓n≤m+n
d_m'8851'n'8804'm'43'n_3594 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8851'n'8804'm'43'n_3594 :: Integer -> Integer -> T__'8804'__18
d_m'8851'n'8804'm'43'n_3594 Integer
v0 Integer
v1
  = let v2 :: b
v2
          = let v2 :: T_TotalPreorder_204
v2 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
            Any -> b
forall a b. a -> b
coe
              (let v3 :: T_MinOperator_84
v3 = T_MinOperator_84
d_'8851''45'operator_3094 in
               Any -> Any
forall a b. a -> b
coe
                 ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> T__'8846'__30)
-> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                    T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> T__'8846'__30
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'sel_1806
                    (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v2) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v3) (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))) in
    Any -> T__'8804'__18
forall a b. a -> b
coe
      (case Any -> T__'8846'__30
forall a b. a -> b
coe Any
forall a. a
v2 of
         MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
v3
           -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
         MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
v3
           -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'n'43'm_2328 (Integer -> Any
forall a b. a -> b
coe Integer
v1)
         T__'8846'__30
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.*-distribˡ-⊓
d_'42''45'distrib'737''45''8851'_3624 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'737''45''8851'_3624 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'737''45''8851'_3624 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distribʳ-⊓
d_'42''45'distrib'691''45''8851'_3646 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'691''45''8851'_3646 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'691''45''8851'_3646 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distrib-⊓
d_'42''45'distrib'45''8851'_3648 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'distrib'45''8851'_3648 :: T_Σ_14
d_'42''45'distrib'45''8851'_3648
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.0∸n≡0
d_0'8760'n'8801'0_3650 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_0'8760'n'8801'0_3650 :: Integer -> T__'8801'__12
d_0'8760'n'8801'0_3650 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.n∸n≡0
d_n'8760'n'8801'0_3654 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_n'8760'n'8801'0_3654 :: Integer -> T__'8801'__12
d_n'8760'n'8801'0_3654 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.pred[m∸n]≡m∸[1+n]
d_pred'91'm'8760'n'93''8801'm'8760''91'1'43'n'93'_3662 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_pred'91'm'8760'n'93''8801'm'8760''91'1'43'n'93'_3662 :: Integer -> Integer -> T__'8801'__12
d_pred'91'm'8760'n'93''8801'm'8760''91'1'43'n'93'_3662 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m∸n≤m
d_m'8760'n'8804'm_3676 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8760'n'8804'm_3676 :: Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
      Integer
0 -> (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
             Integer -> T__'8804'__18
d_'8804''45'refl_1634
             ((Integer -> Integer -> Integer) -> Integer -> Integer -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 Integer
v0 (Integer
0 :: Integer))
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
                Integer
0 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       Integer -> T__'8804'__18
d_'8804''45'refl_1634
                       ((Integer -> Integer -> Integer) -> Integer -> Integer -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 (Integer
0 :: Integer) Integer
v1)
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676 (Integer -> Any
forall a b. a -> b
coe Integer
v3) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                          ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_n'8804'1'43'n_1724 (Integer -> Any
forall a b. a -> b
coe Integer
v3))))
-- Data.Nat.Properties.m≮m∸n
d_m'8814'm'8760'n_3690 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'8814'm'8760'n_3690 :: Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
d_m'8814'm'8760'n_3690 = Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.1+m≢m∸n
d_1'43'm'8802'm'8760'n_3702 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_1'43'm'8802'm'8760'n_3702 :: Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
d_1'43'm'8802'm'8760'n_3702 = Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.∸-mono
d_'8760''45'mono_3710 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8760''45'mono_3710 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8760''45'mono_3710 Integer
v0 Integer
v1 Integer
v2 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
  = let v6 :: b
v6
          = Any -> b -> b
forall a b. a -> b -> b
seq
              (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5)
              ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> b
forall a b. a -> b
coe
                 T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642
                 ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2)) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v4)) in
    Any -> T__'8804'__18
forall a b. a -> b
coe
      (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v4 of
         T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
           -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v5 of
                T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
                  -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642
                       ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                       (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
                MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v10
                  -> case Integer -> Any
forall a b. a -> b
coe Integer
v2 of
                       Any
_ | (Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe Integer -> Integer -> Bool
geqInt (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) ->
                           case Integer -> Any
forall a b. a -> b
coe Integer
v3 of
                             Any
_ | (Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe Integer -> Integer -> Bool
geqInt (Integer -> Any
forall a b. a -> b
coe Integer
v3) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) ->
                                 T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
                             Any
_ -> Any -> Any
forall a b. a -> b
coe Any
forall a. a
v6
                       Any
_ -> Any -> Any
forall a b. a -> b
coe Any
forall a. a
v6
                T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError
         MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9
           -> let v10 :: b
v10
                    = Any -> b -> b
forall a b. a -> b -> b
seq
                        (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5)
                        ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> b
forall a b. a -> b
coe
                           T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642
                           ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                           ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9)) in
              Any -> Any
forall a b. a -> b
coe
                (case Integer -> Any
forall a b. a -> b
coe Integer
v0 of
                   Any
_ | (Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe Integer -> Integer -> Bool
geqInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) ->
                       let v11 :: Integer
v11 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                       Any -> Any
forall a b. a -> b
coe
                         (let v12 :: b
v12
                                = Any -> b -> b
forall a b. a -> b -> b
seq
                                    (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5)
                                    ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> b
forall a b. a -> b
coe
                                       T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642
                                       ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                                       ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9)) in
                          Any -> Any
forall a b. a -> b
coe
                            (case Integer -> Any
forall a b. a -> b
coe Integer
v1 of
                               Any
_ | (Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe Integer -> Integer -> Bool
geqInt (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) ->
                                   let v13 :: Integer
v13 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                                   Any -> Any
forall a b. a -> b
coe
                                     (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v5 of
                                        T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
                                          -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                                               T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642
                                               ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                                               ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9)
                                        MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v16
                                          -> case Integer -> Any
forall a b. a -> b
coe Integer
v2 of
                                               Any
_ | (Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe Integer -> Integer -> Bool
geqInt (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) ->
                                                   let v17 :: Integer
v17 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                                                   Any -> Any
forall a b. a -> b
coe
                                                     (case Integer -> Any
forall a b. a -> b
coe Integer
v3 of
                                                        Any
_ | (Integer -> Integer -> Bool) -> Any -> Any -> Bool
forall a b. a -> b
coe
                                                              Integer -> Integer -> Bool
geqInt (Integer -> Any
forall a b. a -> b
coe Integer
v3)
                                                              (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) ->
                                                            let v18 :: Integer
v18
                                                                  = Integer -> Integer -> Integer
subInt
                                                                      (Integer -> Integer
forall a b. a -> b
coe Integer
v3)
                                                                      (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                                                            Any -> Any
forall a b. a -> b
coe
                                                              ((Integer
 -> Integer
 -> Integer
 -> Integer
 -> T__'8804'__18
 -> T__'8804'__18
 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                                                                 Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8760''45'mono_3710 (Integer -> Any
forall a b. a -> b
coe Integer
v11)
                                                                 (Integer -> Any
forall a b. a -> b
coe Integer
v13) (Integer -> Any
forall a b. a -> b
coe Integer
v17) (Integer -> Any
forall a b. a -> b
coe Integer
v18)
                                                                 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v9) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v16))
                                                        Any
_ -> Any -> Any
forall a b. a -> b
coe Any
forall a. a
v12)
                                               Any
_ -> Any -> Any
forall a b. a -> b
coe Any
forall a. a
v12
                                        T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
                               Any
_ -> Any -> Any
forall a b. a -> b
coe Any
forall a. a
v12))
                   Any
_ -> Any -> Any
forall a b. a -> b
coe Any
forall a. a
v10)
         T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.∸-monoˡ-≤
d_'8760''45'mono'737''45''8804'_3728 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8760''45'mono'737''45''8804'_3728 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8760''45'mono'737''45''8804'_3728 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3
  = (Integer
 -> Integer
 -> Integer
 -> Integer
 -> T__'8804'__18
 -> T__'8804'__18
 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8760''45'mono_3710 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v2) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v3)
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v2))
-- Data.Nat.Properties.∸-monoʳ-≤
d_'8760''45'mono'691''45''8804'_3740 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8760''45'mono'691''45''8804'_3740 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8760''45'mono'691''45''8804'_3740 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3
  = (Integer
 -> Integer
 -> Integer
 -> Integer
 -> T__'8804'__18
 -> T__'8804'__18
 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8760''45'mono_3710 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v2)) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v3)
-- Data.Nat.Properties.∸-monoʳ-<
d_'8760''45'mono'691''45''60'_3750 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8760''45'mono'691''45''60'_3750 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8760''45'mono'691''45''60'_3750 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3 T__'8804'__18
v4
  = let v5 :: Integer
v5 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
    Any -> T__'8804'__18
forall a b. a -> b
coe
      (case Integer -> Integer
forall a b. a -> b
coe Integer
v2 of
         Integer
0 -> (Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
                Any -> Any -> Any
forall a b. a -> b -> b
seq (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v3)
                ((Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
                   Any -> Any -> Any
forall a b. a -> b -> b
seq (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v4)
                   ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe
                      T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                      (Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676
                         ((Integer -> Integer -> Integer) -> Any -> Any -> Integer
forall a b. a -> b
coe Integer -> Integer -> Integer
subInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer))) (Integer -> Integer
forall a b. a -> b
coe Integer
v5))))
         Integer
_ -> let v6 :: Integer
v6 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
              Any -> Any
forall a b. a -> b
coe
                (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v3 of
                   MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v9
                     -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v4 of
                          MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v12
                            -> let v13 :: Integer
v13 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                               Any -> Any
forall a b. a -> b
coe
                                 ((Integer
 -> Integer
 -> Integer
 -> T__'8804'__18
 -> T__'8804'__18
 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                                    Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8760''45'mono'691''45''60'_3750 (Integer -> Any
forall a b. a -> b
coe Integer
v13) (Integer -> Any
forall a b. a -> b
coe Integer
v5) (Integer -> Any
forall a b. a -> b
coe Integer
v6)
                                    (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v9) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v12))
                          T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError
                   T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError))
-- Data.Nat.Properties.∸-cancelʳ-≤
d_'8760''45'cancel'691''45''8804'_3772 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8760''45'cancel'691''45''8804'_3772 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8760''45'cancel'691''45''8804'_3772 ~Integer
v0 Integer
v1 ~Integer
v2 T__'8804'__18
v3 ~T__'8804'__18
v4
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'8760''45'cancel'691''45''8804'_3772 Integer
v1 T__'8804'__18
v3
du_'8760''45'cancel'691''45''8804'_3772 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8760''45'cancel'691''45''8804'_3772 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'8760''45'cancel'691''45''8804'_3772 Integer
v0 T__'8804'__18
v1
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
      T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
        -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v4
        -> case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
             Integer
0 -> Any -> T__'8804'__18
forall a b. a -> b
coe
                    Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24
             Integer
_ -> let v5 :: Integer
v5 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                  Any -> T__'8804'__18
forall a b. a -> b
coe
                    ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                       ((Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'8760''45'cancel'691''45''8804'_3772 (Integer -> Any
forall a b. a -> b
coe Integer
v5) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v4)))
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.∸-cancelʳ-<
d_'8760''45'cancel'691''45''60'_3792 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8760''45'cancel'691''45''60'_3792 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8760''45'cancel'691''45''60'_3792 Integer
v0 Integer
v1 ~Integer
v2 ~T__'8804'__18
v3
  = Integer -> Integer -> T__'8804'__18
du_'8760''45'cancel'691''45''60'_3792 Integer
v0 Integer
v1
du_'8760''45'cancel'691''45''60'_3792 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8760''45'cancel'691''45''60'_3792 :: Integer -> Integer -> T__'8804'__18
du_'8760''45'cancel'691''45''60'_3792 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> Any -> T__'8804'__18
forall a b. a -> b
coe
             Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
du_0'60'1'43'n_1944
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
du_'8760''45'cancel'691''45''60'_3792 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3))))
-- Data.Nat.Properties.∸-cancelˡ-≡
d_'8760''45'cancel'737''45''8801'_3818 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8760''45'cancel'737''45''8801'_3818 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8801'__12
-> T__'8801'__12
d_'8760''45'cancel'737''45''8801'_3818 = Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∸-cancelʳ-≡
d_'8760''45'cancel'691''45''8801'_3840 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8760''45'cancel'691''45''8801'_3840 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8801'__12
-> T__'8801'__12
d_'8760''45'cancel'691''45''8801'_3840 = Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m∸n≡0⇒m≤n
d_m'8760'n'8801'0'8658'm'8804'n_3854 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8760'n'8801'0'8658'm'8804'n_3854 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_m'8760'n'8801'0'8658'm'8804'n_3854 Integer
v0 ~Integer
v1 ~T__'8801'__12
v2
  = Integer -> T__'8804'__18
du_m'8760'n'8801'0'8658'm'8804'n_3854 Integer
v0
du_m'8760'n'8801'0'8658'm'8804'n_3854 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'8760'n'8801'0'8658'm'8804'n_3854 :: Integer -> T__'8804'__18
du_m'8760'n'8801'0'8658'm'8804'n_3854 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8760'n'8801'0'8658'm'8804'n_3854 (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.m≤n⇒m∸n≡0
d_m'8804'n'8658'm'8760'n'8801'0_3866 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8804'n'8658'm'8760'n'8801'0_3866 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8804'n'8658'm'8760'n'8801'0_3866 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m<n⇒0<n∸m
d_m'60'n'8658'0'60'n'8760'm_3876 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'60'n'8658'0'60'n'8760'm_3876 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_m'60'n'8658'0'60'n'8760'm_3876 Integer
v0 ~Integer
v1 T__'8804'__18
v2
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'0'60'n'8760'm_3876 Integer
v0 T__'8804'__18
v2
du_m'60'n'8658'0'60'n'8760'm_3876 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'60'n'8658'0'60'n'8760'm_3876 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'0'60'n'8760'm_3876 Integer
v0 T__'8804'__18
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
du_0'60'1'43'n_1944
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
                MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v5
                  -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_m'60'n'8658'0'60'n'8760'm_3876 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v5)
                T__'8804'__18
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.m∸n≢0⇒n<m
d_m'8760'n'8802'0'8658'n'60'm_3890 ::
  Integer ->
  Integer ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8760'n'8802'0'8658'n'60'm_3890 :: Integer
-> Integer -> (T__'8801'__12 -> T_'8869'_4) -> T__'8804'__18
d_m'8760'n'8802'0'8658'n'60'm_3890 Integer
v0 Integer
v1 ~T__'8801'__12 -> T_'8869'_4
v2
  = Integer -> Integer -> T__'8804'__18
du_m'8760'n'8802'0'8658'n'60'm_3890 Integer
v0 Integer
v1
du_m'8760'n'8802'0'8658'n'60'm_3890 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_m'8760'n'8802'0'8658'n'60'm_3890 :: Integer -> Integer -> T__'8804'__18
du_m'8760'n'8802'0'8658'n'60'm_3890 Integer
v0 Integer
v1
  = let v2 :: T_Dec_32
v2 = Integer -> Integer -> T_Dec_32
d__'60''63'__1912 (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe Integer
v0) in
    Any -> T__'8804'__18
forall a b. a -> b
coe
      (case T_Dec_32 -> T_Dec_32
forall a b. a -> b
coe T_Dec_32
v2 of
         MAlonzo.Code.Relation.Nullary.C__because__46 Bool
v3 T_Reflects_14
v4
           -> if Bool -> Bool
forall a b. a -> b
coe Bool
v3
                then case T_Reflects_14 -> T_Reflects_14
forall a b. a -> b
coe T_Reflects_14
v4 of
                       MAlonzo.Code.Relation.Nullary.C_of'696'_22 Any
v5 -> Any -> Any
forall a b. a -> b
coe Any
v5
                       T_Reflects_14
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError
                else (Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
                       Any -> Any -> Any
forall a b. a -> b -> b
seq (T_Reflects_14 -> Any
forall a b. a -> b
coe T_Reflects_14
v4)
                       (Any -> Any
forall a b. a -> b
coe
                          Any
MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_24)
         T_Dec_32
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.m>n⇒m∸n≢0
d_m'62'n'8658'm'8760'n'8802'0_3922 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_m'62'n'8658'm'8760'n'8802'0_3922 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
d_m'62'n'8658'm'8760'n'8802'0_3922 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.+-∸-comm
d_'43''45''8760''45'comm_3934 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45''8760''45'comm_3934 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_'43''45''8760''45'comm_3934 = Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∸-+-assoc
d_'8760''45''43''45'assoc_3952 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8760''45''43''45'assoc_3952 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8760''45''43''45'assoc_3952 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.+-∸-assoc
d_'43''45''8760''45'assoc_3976 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45''8760''45'assoc_3976 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_'43''45''8760''45'assoc_3976 = Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m≤n+m∸n
d_m'8804'n'43'm'8760'n_3996 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8804'n'43'm'8760'n_3996 :: Integer -> Integer -> T__'8804'__18
d_m'8804'n'43'm'8760'n_3996 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          (Integer -> Integer -> T__'8804'__18
d_m'8804'n'43'm'8760'n_3996 (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe Integer
v3))))
-- Data.Nat.Properties.m+n∸n≡m
d_m'43'n'8760'n'8801'm_4010 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'43'n'8760'n'8801'm_4010 :: Integer -> Integer -> T__'8801'__12
d_m'43'n'8760'n'8801'm_4010 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m+n∸m≡n
d_m'43'n'8760'm'8801'n_4022 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'43'n'8760'm'8801'n_4022 :: Integer -> Integer -> T__'8801'__12
d_m'43'n'8760'm'8801'n_4022 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m+[n∸m]≡n
d_m'43''91'n'8760'm'93''8801'n_4034 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'43''91'n'8760'm'93''8801'n_4034 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'43''91'n'8760'm'93''8801'n_4034 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m∸n+n≡m
d_m'8760'n'43'n'8801'm_4048 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8760'n'43'n'8801'm_4048 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8760'n'43'n'8801'm_4048 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m∸[m∸n]≡n
d_m'8760''91'm'8760'n'93''8801'n_4060 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8760''91'm'8760'n'93''8801'n_4060 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8760''91'm'8760'n'93''8801'n_4060 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.[m+n]∸[m+o]≡n∸o
d_'91'm'43'n'93''8760''91'm'43'o'93''8801'n'8760'o_4076 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'91'm'43'n'93''8760''91'm'43'o'93''8801'n'8760'o_4076 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'91'm'43'n'93''8760''91'm'43'o'93''8801'n'8760'o_4076 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distribʳ-∸
d_'42''45'distrib'691''45''8760'_4088 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'691''45''8760'_4088 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'691''45''8760'_4088 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distribˡ-∸
d_'42''45'distrib'737''45''8760'_4108 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'737''45''8760'_4108 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'737''45''8760'_4108 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distrib-∸
d_'42''45'distrib'45''8760'_4110 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'distrib'45''8760'_4110 :: T_Σ_14
d_'42''45'distrib'45''8760'_4110
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.even≢odd
d_even'8802'odd_4116 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_even'8802'odd_4116 :: Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
d_even'8802'odd_4116 = Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.m⊓n+n∸m≡n
d_m'8851'n'43'n'8760'm'8801'n_4132 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8851'n'43'n'8760'm'8801'n_4132 :: Integer -> Integer -> T__'8801'__12
d_m'8851'n'43'n'8760'm'8801'n_4132 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.[m∸n]⊓[n∸m]≡0
d_'91'm'8760'n'93''8851''91'n'8760'm'93''8801'0_4146 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'91'm'8760'n'93''8851''91'n'8760'm'93''8801'0_4146 :: Integer -> Integer -> T__'8801'__12
d_'91'm'8760'n'93''8851''91'n'8760'm'93''8801'0_4146 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∸-distribˡ-⊓-⊔
d_'8760''45'distrib'737''45''8851''45''8852'_4162 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8760''45'distrib'737''45''8851''45''8852'_4162 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8760''45'distrib'737''45''8851''45''8852'_4162 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∸-distribʳ-⊓
d_'8760''45'distrib'691''45''8851'_4170 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8760''45'distrib'691''45''8851'_4170 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8760''45'distrib'691''45''8851'_4170 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∸-distribˡ-⊔-⊓
d_'8760''45'distrib'737''45''8852''45''8851'_4184 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8760''45'distrib'737''45''8852''45''8851'_4184 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8760''45'distrib'737''45''8852''45''8851'_4184 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∸-distribʳ-⊔
d_'8760''45'distrib'691''45''8852'_4192 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8760''45'distrib'691''45''8852'_4192 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8760''45'distrib'691''45''8852'_4192 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.pred-mono
d_pred'45'mono_4200 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_pred'45'mono_4200 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_pred'45'mono_4200 Integer
v0 Integer
v1 T__'8804'__18
v2
  = (Integer
 -> Integer
 -> Integer
 -> Integer
 -> T__'8804'__18
 -> T__'8804'__18
 -> T__'8804'__18)
-> Any -> Any -> Any -> Any -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8760''45'mono_3710 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer))
      (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v2)
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)))
-- Data.Nat.Properties.pred[n]≤n
d_pred'91'n'93''8804'n_4206 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_pred'91'n'93''8804'n_4206 :: Integer -> T__'8804'__18
d_pred'91'n'93''8804'n_4206 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_n'8804'1'43'n_1724 (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties.≤pred⇒≤
d_'8804'pred'8658''8804'_4214 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804'pred'8658''8804'_4214 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8804'pred'8658''8804'_4214 ~Integer
v0 Integer
v1 T__'8804'__18
v2
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804'pred'8658''8804'_4214 Integer
v1 T__'8804'__18
v2
du_'8804'pred'8658''8804'_4214 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804'pred'8658''8804'_4214 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804'pred'8658''8804'_4214 Integer
v0 T__'8804'__18
v1 = (Any -> Any -> Any) -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Any -> Any -> Any
forall a b. a -> b -> b
seq (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1)
-- Data.Nat.Properties.≤⇒pred≤
d_'8804''8658'pred'8804'_4230 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''8658'pred'8804'_4230 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8804''8658'pred'8804'_4230 Integer
v0 ~Integer
v1 T__'8804'__18
v2
  = Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804''8658'pred'8804'_4230 Integer
v0 T__'8804'__18
v2
du_'8804''8658'pred'8804'_4230 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''8658'pred'8804'_4230 :: Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804''8658'pred'8804'_4230 Integer
v0 T__'8804'__18
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'8804''45'trans_1642 ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_n'8804'1'43'n_1724 (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v1))
-- Data.Nat.Properties.<⇒≤pred
d_'60''8658''8804'pred_4242 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'60''8658''8804'pred_4242 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'60''8658''8804'pred_4242 ~Integer
v0 ~Integer
v1 T__'8804'__18
v2
  = T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'pred_4242 T__'8804'__18
v2
du_'60''8658''8804'pred_4242 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'60''8658''8804'pred_4242 :: T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'pred_4242 T__'8804'__18
v0
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0 of
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v3 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v3
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.suc[pred[n]]≡n
d_suc'91'pred'91'n'93''93''8801'n_4248 ::
  Integer ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_suc'91'pred'91'n'93''93''8801'n_4248 :: Integer -> (T__'8801'__12 -> T_'8869'_4) -> T__'8801'__12
d_suc'91'pred'91'n'93''93''8801'n_4248 = Integer -> (T__'8801'__12 -> T_'8869'_4) -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m≡n⇒∣m-n∣≡0
d_m'8801'n'8658''8739'm'45'n'8739''8801'0_4260 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8801'n'8658''8739'm'45'n'8739''8801'0_4260 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_m'8801'n'8658''8739'm'45'n'8739''8801'0_4260 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣m-n∣≡0⇒m≡n
d_'8739'm'45'n'8739''8801'0'8658'm'8801'n_4268 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739'm'45'n'8739''8801'0'8658'm'8801'n_4268 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8739'm'45'n'8739''8801'0'8658'm'8801'n_4268 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m≤n⇒∣n-m∣≡n∸m
d_m'8804'n'8658''8739'n'45'm'8739''8801'n'8760'm_4282 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8804'n'8658''8739'n'45'm'8739''8801'n'8760'm_4282 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8804'n'8658''8739'n'45'm'8739''8801'n'8760'm_4282 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣m-n∣≡m∸n⇒n≤m
d_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 Integer
v0 Integer
v1 ~T__'8801'__12
v2
  = Integer -> Integer -> T__'8804'__18
du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 Integer
v0 Integer
v1
du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 :: Integer -> Integer -> T__'8804'__18
du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe
                             Integer -> Integer -> T__'8804'__18
du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 (Integer -> Any
forall a b. a -> b
coe Integer
v2)
                             (Integer -> Any
forall a b. a -> b
coe Integer
v3))))
-- Data.Nat.Properties.∣n-n∣≡0
d_'8739'n'45'n'8739''8801'0_4308 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739'n'45'n'8739''8801'0_4308 :: Integer -> T__'8801'__12
d_'8739'n'45'n'8739''8801'0_4308 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣m-m+n∣≡n
d_'8739'm'45'm'43'n'8739''8801'n_4316 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739'm'45'm'43'n'8739''8801'n_4316 :: Integer -> Integer -> T__'8801'__12
d_'8739'm'45'm'43'n'8739''8801'n_4316 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣m+n-m+o∣≡∣n-o∣
d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'8739'_4330 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'8739'_4330 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'8739'_4330
  = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m∸n≤∣m-n∣
d_m'8760'n'8804''8739'm'45'n'8739'_4346 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8760'n'8804''8739'm'45'n'8739'_4346 :: Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804''8739'm'45'n'8739'_4346 Integer
v0 Integer
v1
  = let v2 :: T__'8846'__30
v2 = Integer -> Integer -> T__'8846'__30
d_'8804''45'total_1648 (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe Integer
v1) in
    Any -> T__'8804'__18
forall a b. a -> b
coe
      (case T__'8846'__30 -> T__'8846'__30
forall a b. a -> b
coe T__'8846'__30
v2 of
         MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
v3
           -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
         MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
v3
           -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                Integer -> T__'8804'__18
d_'8804''45'refl_1634
                ((Integer -> Integer -> Integer) -> Integer -> Integer -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 Integer
v0 Integer
v1)
         T__'8846'__30
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.∣m-n∣≤m⊔n
d_'8739'm'45'n'8739''8804'm'8852'n_4376 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8739'm'45'n'8739''8804'm'8852'n_4376 :: Integer -> Integer -> T__'8804'__18
d_'8739'm'45'n'8739''8804'm'8852'n_4376 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
             Integer -> T__'8804'__18
d_'8804''45'refl_1634
             ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       Integer -> T__'8804'__18
d_'8804''45'refl_1634
                       ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                          Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                          (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer)))
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_'8739'm'45'n'8739''8804'm'8852'n_4376 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v3)))
-- Data.Nat.Properties.∣-∣-identityˡ
d_'8739''45''8739''45'identity'737'_4386 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739''45''8739''45'identity'737'_4386 :: Integer -> T__'8801'__12
d_'8739''45''8739''45'identity'737'_4386 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣-∣-identityʳ
d_'8739''45''8739''45'identity'691'_4390 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739''45''8739''45'identity'691'_4390 :: Integer -> T__'8801'__12
d_'8739''45''8739''45'identity'691'_4390 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣-∣-identity
d_'8739''45''8739''45'identity_4394 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'8739''45''8739''45'identity_4394 :: T_Σ_14
d_'8739''45''8739''45'identity_4394
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.∣-∣-comm
d_'8739''45''8739''45'comm_4396 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739''45''8739''45'comm_4396 :: Integer -> Integer -> T__'8801'__12
d_'8739''45''8739''45'comm_4396 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣m-n∣≡[m∸n]∨[n∸m]
d_'8739'm'45'n'8739''8801''91'm'8760'n'93''8744''91'n'8760'm'93'_4410 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'8739'm'45'n'8739''8801''91'm'8760'n'93''8744''91'n'8760'm'93'_4410 :: Integer -> Integer -> T__'8846'__30
d_'8739'm'45'n'8739''8801''91'm'8760'n'93''8744''91'n'8760'm'93'_4410 Integer
v0
                                                                      Integer
v1
  = let v2 :: T__'8846'__30
v2 = Integer -> Integer -> T__'8846'__30
d_'8804''45'total_1648 (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe Integer
v1) in
    Any -> T__'8846'__30
forall a b. a -> b
coe
      (case T__'8846'__30 -> T__'8846'__30
forall a b. a -> b
coe T__'8846'__30
v2 of
         MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
v3
           -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe
                Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                ((T__IsRelatedTo__70 -> Any) -> Any -> Any
forall a b. a -> b
coe
                   T__IsRelatedTo__70 -> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin'45'equality__190
                   ((T_IsPreorder_70 -> Any -> T__IsRelatedTo__70) -> Any -> Any -> Any
forall a b. a -> b
coe
                      T_IsPreorder_70 -> Any -> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du__'8718'_346
                      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                      ((Integer -> Integer -> Integer) -> Integer -> Integer -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 Integer
v1 Integer
v0)))
         MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
v3
           -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
forall a. a
erased
         T__'8846'__30
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.*-distribˡ-∣-∣-aux
d_'42''45'distrib'737''45''8739''45''8739''45'aux_4438 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'737''45''8739''45''8739''45'aux_4438 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_'42''45'distrib'737''45''8739''45''8739''45'aux_4438 = Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distribˡ-∣-∣
d_'42''45'distrib'737''45''8739''45''8739'_4450 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'737''45''8739''45''8739'_4450 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'737''45''8739''45''8739'_4450 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distribʳ-∣-∣
d_'42''45'distrib'691''45''8739''45''8739'_4480 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'691''45''8739''45''8739'_4480 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'691''45''8739''45''8739'_4480 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distrib-∣-∣
d_'42''45'distrib'45''8739''45''8739'_4482 ::
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'distrib'45''8739''45''8739'_4482 :: T_Σ_14
d_'42''45'distrib'45''8739''45''8739'_4482
  = (Any -> Any -> T_Σ_14) -> Any -> Any -> T_Σ_14
forall a b. a -> b
coe Any -> Any -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 Any
forall a. a
erased Any
forall a. a
erased
-- Data.Nat.Properties.m≤n+∣n-m∣
d_m'8804'n'43''8739'n'45'm'8739'_4488 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8804'n'43''8739'n'45'm'8739'_4488 :: Integer -> Integer -> T__'8804'__18
d_m'8804'n'43''8739'n'45'm'8739'_4488 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          (Integer -> Integer -> T__'8804'__18
d_m'8804'n'43''8739'n'45'm'8739'_4488 (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe Integer
v3))))
-- Data.Nat.Properties.m≤n+∣m-n∣
d_m'8804'n'43''8739'm'45'n'8739'_4502 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8804'n'43''8739'm'45'n'8739'_4502 :: Integer -> Integer -> T__'8804'__18
d_m'8804'n'43''8739'm'45'n'8739'_4502 Integer
v0 Integer
v1
  = (Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8804'n'43''8739'n'45'm'8739'_4488 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
-- Data.Nat.Properties.m≤∣m-n∣+n
d_m'8804''8739'm'45'n'8739''43'n_4516 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_m'8804''8739'm'45'n'8739''43'n_4516 :: Integer -> Integer -> T__'8804'__18
d_m'8804''8739'm'45'n'8739''43'n_4516 Integer
v0 Integer
v1
  = (Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8804'n'43''8739'm'45'n'8739'_4502 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
-- Data.Nat.Properties.∣-∣-triangle
d_'8739''45''8739''45'triangle_4524 ::
  Integer ->
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8739''45''8739''45'triangle_4524 :: Integer -> Integer -> Integer -> T__'8804'__18
d_'8739''45''8739''45'triangle_4524 Integer
v0 Integer
v1 Integer
v2
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8804'n'43''8739'n'45'm'8739'_4488 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
      Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
                Integer
0 -> (T_IsPreorder_70
 -> (Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any)
-> Any -> (Any -> Any -> Any -> Any) -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                       T_IsPreorder_70
-> (Any -> Any -> Any -> Any)
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin__160
                       (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                       (\ Any
v4 Any
v5 Any
v6 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 Any
v5 Any
v6)
                       ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                          Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                          (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                       ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                          Integer -> Integer -> Integer
addInt
                          ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                             Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                             (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                          ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                             Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                             (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer))))
                       ((T_IsPreorder_70
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
forall a b. a -> b
coe
                          T_IsPreorder_70
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''8804'_228
                          (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                          (\ Any
v4 Any
v5 Any
v6 Any
v7 Any
v8 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 Any
v7 Any
v8)
                          ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                             Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                             (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                          ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d__'8852'__106 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                          ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                             Integer -> Integer -> Integer
addInt
                             ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                                (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                             ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                                (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer))))
                          ((T_IsPreorder_70
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
forall a b. a -> b
coe
                             T_IsPreorder_70
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''8804'_228
                             (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                             (\ Any
v4 Any
v5 Any
v6 Any
v7 Any
v8 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 Any
v7 Any
v8)
                             ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d__'8852'__106 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                             ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                             ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                Integer -> Integer -> Integer
addInt
                                ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                   Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                                   (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v2))
                                ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                   Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                                   (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer))))
                             ((T_IsPreorder_70 -> Any -> T__IsRelatedTo__70) -> Any -> Any -> Any
forall a b. a -> b
coe
                                T_IsPreorder_70 -> Any -> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du__'8718'_346
                                (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                                ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                   Integer -> Integer -> Integer
addInt
                                   ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                                      (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer)))
                                   (Integer -> Any
forall a b. a -> b
coe Integer
v2)))
                             ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8852'n'8804'm'43'n_3446 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2)))
                          ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_'8739'm'45'n'8739''8804'm'8852'n_4376 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v2)))
                Integer
_ -> let v4 :: Integer
v4 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       (case Integer -> Integer
forall a b. a -> b
coe Integer
v2 of
                          Integer
0 -> (T_IsPreorder_70
 -> (Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any)
-> Any -> (Any -> Any -> Any -> Any) -> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                                 T_IsPreorder_70
-> (Any -> Any -> Any -> Any)
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin__160
                                 (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                                 (\ Any
v5 Any
v6 Any
v7 -> (Integer -> T__'8804'__18 -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'60''8658''8804'_1736 Any
v6 Any
v7)
                                 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                    Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                                    (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer)))
                                 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                    Integer -> Integer -> Integer
addInt
                                    ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                       Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                                       (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                                    ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                       Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142 (Integer -> Any
forall a b. a -> b
coe Integer
v1)
                                       (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer))))
                                 ((T_IsPreorder_70
 -> (Any -> Any -> Any -> Any -> Any -> Any)
 -> Any
 -> Any
 -> Any
 -> T__IsRelatedTo__70
 -> Any
 -> T__IsRelatedTo__70)
-> Any
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> Any
-> Any
-> Any
forall a b. a -> b
coe
                                    T_IsPreorder_70
-> (Any -> Any -> Any -> Any -> Any -> Any)
-> Any
-> Any
-> Any
-> T__IsRelatedTo__70
-> Any
-> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_step'45''8804'_228
                                    (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                                    (\ Any
v5 Any
v6 Any
v7 Any
v8 Any
v9 -> (T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'60''45'trans'691'_1868 Any
v8 Any
v9)
                                    (Integer -> Any
forall a b. a -> b
coe Integer
v0)
                                    ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                       Integer -> Integer -> Integer
addInt
                                       ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                          Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                                          (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                                       (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                                    ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                       Integer -> Integer -> Integer
addInt
                                       ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                          Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                                          (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                                       ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                          Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                                          (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer))))
                                    ((T_IsPreorder_70 -> Any -> T__IsRelatedTo__70) -> Any -> Any -> Any
forall a b. a -> b
coe
                                       T_IsPreorder_70 -> Any -> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du__'8718'_346
                                       (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                                       ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                          Integer -> Integer -> Integer
addInt
                                          ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                             Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                                             (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
                                          ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe
                                             Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
                                             (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe (Integer
0 :: Integer)))))
                                    ((Integer -> Integer -> T__'8804'__18) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8804''8739'm'45'n'8739''43'n_4516 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
                          Integer
_ -> let v5 :: Integer
v5 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v2) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                               Any -> Any
forall a b. a -> b
coe
                                 ((Integer -> Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
                                    Integer -> Integer -> Integer -> T__'8804'__18
d_'8739''45''8739''45'triangle_4524 (Integer -> Any
forall a b. a -> b
coe Integer
v3) (Integer -> Any
forall a b. a -> b
coe Integer
v4)
                                    (Integer -> Any
forall a b. a -> b
coe Integer
v5))))
-- Data.Nat.Properties.∣-∣-isProtoMetric
d_'8739''45''8739''45'isProtoMetric_4552 ::
  MAlonzo.Code.Function.Metric.Structures.T_IsProtoMetric_30
d_'8739''45''8739''45'isProtoMetric_4552 :: T_IsProtoMetric_30
d_'8739''45''8739''45'isProtoMetric_4552
  = (T_IsPartialOrder_162
 -> T_IsEquivalence_26
 -> (Any -> Any -> Any -> Any -> Any -> Any -> Any)
 -> (Any -> Any -> Any)
 -> T_IsProtoMetric_30)
-> Any -> Any -> Any -> Any -> T_IsProtoMetric_30
forall a b. a -> b
coe
      T_IsPartialOrder_162
-> T_IsEquivalence_26
-> (Any -> Any -> Any -> Any -> Any -> Any -> Any)
-> (Any -> Any -> Any)
-> T_IsProtoMetric_30
MAlonzo.Code.Function.Metric.Structures.C_IsProtoMetric'46'constructor_2109
      (T_IsPartialOrder_162 -> Any
forall a b. a -> b
coe T_IsPartialOrder_162
d_'8804''45'isPartialOrder_1688)
      (T_IsEquivalence_26 -> Any
forall a b. a -> b
coe
         T_IsEquivalence_26
MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_242)
      Any
forall a. a
erased
      ((Any -> Any -> Any) -> Any
forall a b. a -> b
coe (\ Any
v0 Any
v1 -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22))
-- Data.Nat.Properties.∣-∣-isPreMetric
d_'8739''45''8739''45'isPreMetric_4554 ::
  MAlonzo.Code.Function.Metric.Structures.T_IsPreMetric_96
d_'8739''45''8739''45'isPreMetric_4554 :: T_IsPreMetric_96
d_'8739''45''8739''45'isPreMetric_4554
  = (T_IsProtoMetric_30
 -> (Any -> Any -> Any -> Any) -> T_IsPreMetric_96)
-> Any -> Any -> T_IsPreMetric_96
forall a b. a -> b
coe
      T_IsProtoMetric_30
-> (Any -> Any -> Any -> Any) -> T_IsPreMetric_96
MAlonzo.Code.Function.Metric.Structures.C_IsPreMetric'46'constructor_6061
      (T_IsProtoMetric_30 -> Any
forall a b. a -> b
coe T_IsProtoMetric_30
d_'8739''45''8739''45'isProtoMetric_4552) Any
forall a. a
erased
-- Data.Nat.Properties.∣-∣-isQuasiSemiMetric
d_'8739''45''8739''45'isQuasiSemiMetric_4556 ::
  MAlonzo.Code.Function.Metric.Structures.T_IsQuasiSemiMetric_162
d_'8739''45''8739''45'isQuasiSemiMetric_4556 :: T_IsQuasiSemiMetric_162
d_'8739''45''8739''45'isQuasiSemiMetric_4556
  = (T_IsPreMetric_96
 -> (Any -> Any -> Any -> Any) -> T_IsQuasiSemiMetric_162)
-> Any -> Any -> T_IsQuasiSemiMetric_162
forall a b. a -> b
coe
      T_IsPreMetric_96
-> (Any -> Any -> Any -> Any) -> T_IsQuasiSemiMetric_162
MAlonzo.Code.Function.Metric.Structures.C_IsQuasiSemiMetric'46'constructor_9549
      (T_IsPreMetric_96 -> Any
forall a b. a -> b
coe T_IsPreMetric_96
d_'8739''45''8739''45'isPreMetric_4554) Any
forall a. a
erased
-- Data.Nat.Properties.∣-∣-isSemiMetric
d_'8739''45''8739''45'isSemiMetric_4558 ::
  MAlonzo.Code.Function.Metric.Structures.T_IsSemiMetric_232
d_'8739''45''8739''45'isSemiMetric_4558 :: T_IsSemiMetric_232
d_'8739''45''8739''45'isSemiMetric_4558
  = (T_IsQuasiSemiMetric_162
 -> (Any -> Any -> Any) -> T_IsSemiMetric_232)
-> Any -> Any -> T_IsSemiMetric_232
forall a b. a -> b
coe
      T_IsQuasiSemiMetric_162
-> (Any -> Any -> Any) -> T_IsSemiMetric_232
MAlonzo.Code.Function.Metric.Structures.C_IsSemiMetric'46'constructor_13167
      (T_IsQuasiSemiMetric_162 -> Any
forall a b. a -> b
coe T_IsQuasiSemiMetric_162
d_'8739''45''8739''45'isQuasiSemiMetric_4556) Any
forall a. a
erased
-- Data.Nat.Properties.∣-∣-isMetric
d_'8739''45''8739''45'isMetric_4560 ::
  MAlonzo.Code.Function.Metric.Structures.T_IsGeneralMetric_308
d_'8739''45''8739''45'isMetric_4560 :: T_IsGeneralMetric_308
d_'8739''45''8739''45'isMetric_4560
  = (T_IsSemiMetric_232
 -> (Any -> Any -> Any -> Any) -> T_IsGeneralMetric_308)
-> Any -> Any -> T_IsGeneralMetric_308
forall a b. a -> b
coe
      T_IsSemiMetric_232
-> (Any -> Any -> Any -> Any) -> T_IsGeneralMetric_308
MAlonzo.Code.Function.Metric.Structures.C_IsGeneralMetric'46'constructor_17141
      (T_IsSemiMetric_232 -> Any
forall a b. a -> b
coe T_IsSemiMetric_232
d_'8739''45''8739''45'isSemiMetric_4558)
      ((Integer -> Integer -> Integer -> T__'8804'__18) -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer -> T__'8804'__18
d_'8739''45''8739''45'triangle_4524)
-- Data.Nat.Properties.∣-∣-quasiSemiMetric
d_'8739''45''8739''45'quasiSemiMetric_4562 ::
  MAlonzo.Code.Function.Metric.Nat.Bundles.T_QuasiSemiMetric_174
d_'8739''45''8739''45'quasiSemiMetric_4562 :: T_QuasiSemiMetric_174
d_'8739''45''8739''45'quasiSemiMetric_4562
  = ((Any -> Any -> Integer)
 -> T_IsQuasiSemiMetric_162 -> T_QuasiSemiMetric_174)
-> (Integer -> Integer -> Integer)
-> T_IsQuasiSemiMetric_162
-> T_QuasiSemiMetric_174
forall a b. a -> b
coe
      (Any -> Any -> Integer)
-> T_IsQuasiSemiMetric_162 -> T_QuasiSemiMetric_174
MAlonzo.Code.Function.Metric.Nat.Bundles.C_QuasiSemiMetric'46'constructor_3043
      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
      T_IsQuasiSemiMetric_162
d_'8739''45''8739''45'isQuasiSemiMetric_4556
-- Data.Nat.Properties.∣-∣-semiMetric
d_'8739''45''8739''45'semiMetric_4564 ::
  MAlonzo.Code.Function.Metric.Nat.Bundles.T_SemiMetric_266
d_'8739''45''8739''45'semiMetric_4564 :: T_SemiMetric_266
d_'8739''45''8739''45'semiMetric_4564
  = ((Any -> Any -> Integer) -> T_IsSemiMetric_232 -> T_SemiMetric_266)
-> (Integer -> Integer -> Integer)
-> T_IsSemiMetric_232
-> T_SemiMetric_266
forall a b. a -> b
coe
      (Any -> Any -> Integer) -> T_IsSemiMetric_232 -> T_SemiMetric_266
MAlonzo.Code.Function.Metric.Nat.Bundles.C_SemiMetric'46'constructor_4673
      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
      T_IsSemiMetric_232
d_'8739''45''8739''45'isSemiMetric_4558
-- Data.Nat.Properties.∣-∣-preMetric
d_'8739''45''8739''45'preMetric_4566 ::
  MAlonzo.Code.Function.Metric.Nat.Bundles.T_PreMetric_90
d_'8739''45''8739''45'preMetric_4566 :: T_PreMetric_90
d_'8739''45''8739''45'preMetric_4566
  = ((Any -> Any -> Integer) -> T_IsPreMetric_96 -> T_PreMetric_90)
-> (Integer -> Integer -> Integer)
-> T_IsPreMetric_96
-> T_PreMetric_90
forall a b. a -> b
coe
      (Any -> Any -> Integer) -> T_IsPreMetric_96 -> T_PreMetric_90
MAlonzo.Code.Function.Metric.Nat.Bundles.C_PreMetric'46'constructor_1523
      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
      T_IsPreMetric_96
d_'8739''45''8739''45'isPreMetric_4554
-- Data.Nat.Properties.∣-∣-metric
d_'8739''45''8739''45'metric_4568 ::
  MAlonzo.Code.Function.Metric.Nat.Bundles.T_Metric_364
d_'8739''45''8739''45'metric_4568 :: T_Metric_364
d_'8739''45''8739''45'metric_4568
  = ((Any -> Any -> Integer) -> T_IsGeneralMetric_308 -> T_Metric_364)
-> (Integer -> Integer -> Integer)
-> T_IsGeneralMetric_308
-> T_Metric_364
forall a b. a -> b
coe
      (Any -> Any -> Integer) -> T_IsGeneralMetric_308 -> T_Metric_364
MAlonzo.Code.Function.Metric.Nat.Bundles.C_Metric'46'constructor_6373
      Integer -> Integer -> Integer
MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_142
      T_IsGeneralMetric_308
d_'8739''45''8739''45'isMetric_4560
-- Data.Nat.Properties.⌊n/2⌋-mono
d_'8970'n'47'2'8971''45'mono_4570 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8970'n'47'2'8971''45'mono_4570 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8970'n'47'2'8971''45'mono_4570 ~Integer
v0 ~Integer
v1 T__'8804'__18
v2
  = T__'8804'__18 -> T__'8804'__18
du_'8970'n'47'2'8971''45'mono_4570 T__'8804'__18
v2
du_'8970'n'47'2'8971''45'mono_4570 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8970'n'47'2'8971''45'mono_4570 :: T__'8804'__18 -> T__'8804'__18
du_'8970'n'47'2'8971''45'mono_4570 T__'8804'__18
v0
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0 of
      T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
        -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v3
        -> case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v3 of
             T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
               -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
             MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v6
               -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
                    T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                    ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
du_'8970'n'47'2'8971''45'mono_4570 (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v6))
             T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
      T__'8804'__18
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.⌈n/2⌉-mono
d_'8968'n'47'2'8969''45'mono_4574 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8968'n'47'2'8969''45'mono_4574 :: Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8968'n'47'2'8969''45'mono_4574 ~Integer
v0 ~Integer
v1 T__'8804'__18
v2
  = T__'8804'__18 -> T__'8804'__18
du_'8968'n'47'2'8969''45'mono_4574 T__'8804'__18
v2
du_'8968'n'47'2'8969''45'mono_4574 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8968'n'47'2'8969''45'mono_4574 :: T__'8804'__18 -> T__'8804'__18
du_'8968'n'47'2'8969''45'mono_4574 T__'8804'__18
v0
  = (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
      T__'8804'__18 -> T__'8804'__18
du_'8970'n'47'2'8971''45'mono_4570
      ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v0)
-- Data.Nat.Properties.⌊n/2⌋≤⌈n/2⌉
d_'8970'n'47'2'8971''8804''8968'n'47'2'8969'_4580 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8970'n'47'2'8971''8804''8968'n'47'2'8969'_4580 :: Integer -> T__'8804'__18
d_'8970'n'47'2'8971''8804''8968'n'47'2'8969'_4580 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
1 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
2 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                (Integer -> T__'8804'__18
d_'8970'n'47'2'8971''8804''8968'n'47'2'8969'_4580 (Integer -> Integer
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.⌊n/2⌋+⌈n/2⌉≡n
d_'8970'n'47'2'8971''43''8968'n'47'2'8969''8801'n_4586 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8970'n'47'2'8971''43''8968'n'47'2'8969''8801'n_4586 :: Integer -> T__'8801'__12
d_'8970'n'47'2'8971''43''8968'n'47'2'8969''8801'n_4586 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⌊n/2⌋≤n
d_'8970'n'47'2'8971''8804'n_4592 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8970'n'47'2'8971''8804'n_4592 :: Integer -> T__'8804'__18
d_'8970'n'47'2'8971''8804'n_4592 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
1 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
2 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                (Integer -> T__'8804'__18
d_'8970'n'47'2'8971''8804'n_4592 (Integer -> Integer
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.⌊n/2⌋<n
d_'8970'n'47'2'8971''60'n_4598 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8970'n'47'2'8971''60'n_4598 :: Integer -> T__'8804'__18
d_'8970'n'47'2'8971''60'n_4598 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (T__'8804'__18 -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe
             T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
             (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe
                   T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                   (Integer -> T__'8804'__18
d_'8970'n'47'2'8971''8804'n_4592 (Integer -> Integer
forall a b. a -> b
coe Integer
v1))))
-- Data.Nat.Properties.⌈n/2⌉≤n
d_'8968'n'47'2'8969''8804'n_4604 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8968'n'47'2'8969''8804'n_4604 :: Integer -> T__'8804'__18
d_'8968'n'47'2'8969''8804'n_4604 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe
                T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                (Integer -> T__'8804'__18
d_'8970'n'47'2'8971''8804'n_4592 (Integer -> Integer
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.⌈n/2⌉<n
d_'8968'n'47'2'8969''60'n_4610 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8968'n'47'2'8969''60'n_4610 :: Integer -> T__'8804'__18
d_'8968'n'47'2'8969''60'n_4610 Integer
v0
  = (T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe
      T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
      (Integer -> T__'8804'__18
d_'8970'n'47'2'8971''60'n_4598 (Integer -> Integer
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.≤′-trans
d_'8804''8242''45'trans_4614 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
d_'8804''8242''45'trans_4614 :: Integer
-> Integer
-> Integer
-> T__'8804''8242'__154
-> T__'8804''8242'__154
-> T__'8804''8242'__154
d_'8804''8242''45'trans_4614 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804''8242'__154
v3 T__'8804''8242'__154
v4
  = T__'8804''8242'__154
-> T__'8804''8242'__154 -> T__'8804''8242'__154
du_'8804''8242''45'trans_4614 T__'8804''8242'__154
v3 T__'8804''8242'__154
v4
du_'8804''8242''45'trans_4614 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
du_'8804''8242''45'trans_4614 :: T__'8804''8242'__154
-> T__'8804''8242'__154 -> T__'8804''8242'__154
du_'8804''8242''45'trans_4614 T__'8804''8242'__154
v0 T__'8804''8242'__154
v1
  = case T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
v1 of
      T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_158 -> T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
v0
      MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_164 T__'8804''8242'__154
v3
        -> (T__'8804''8242'__154 -> T__'8804''8242'__154)
-> Any -> T__'8804''8242'__154
forall a b. a -> b
coe
             T__'8804''8242'__154 -> T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_164
             ((T__'8804''8242'__154
 -> T__'8804''8242'__154 -> T__'8804''8242'__154)
-> Any -> Any -> Any
forall a b. a -> b
coe T__'8804''8242'__154
-> T__'8804''8242'__154 -> T__'8804''8242'__154
du_'8804''8242''45'trans_4614 (T__'8804''8242'__154 -> Any
forall a b. a -> b
coe T__'8804''8242'__154
v0) (T__'8804''8242'__154 -> Any
forall a b. a -> b
coe T__'8804''8242'__154
v3))
      T__'8804''8242'__154
_ -> T__'8804''8242'__154
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.z≤′n
d_z'8804''8242'n_4624 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
d_z'8804''8242'n_4624 :: Integer -> T__'8804''8242'__154
d_z'8804''8242'n_4624 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_158
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804''8242'__154
forall a b. a -> b
coe
             ((T__'8804''8242'__154 -> T__'8804''8242'__154)
-> T__'8804''8242'__154 -> Any
forall a b. a -> b
coe
                T__'8804''8242'__154 -> T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_164
                (Integer -> T__'8804''8242'__154
d_z'8804''8242'n_4624 (Integer -> Integer
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.s≤′s
d_s'8804''8242's_4632 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
d_s'8804''8242's_4632 :: Integer -> Integer -> T__'8804''8242'__154 -> T__'8804''8242'__154
d_s'8804''8242's_4632 ~Integer
v0 ~Integer
v1 T__'8804''8242'__154
v2 = T__'8804''8242'__154 -> T__'8804''8242'__154
du_s'8804''8242's_4632 T__'8804''8242'__154
v2
du_s'8804''8242's_4632 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
du_s'8804''8242's_4632 :: T__'8804''8242'__154 -> T__'8804''8242'__154
du_s'8804''8242's_4632 T__'8804''8242'__154
v0 = T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
v0
-- Data.Nat.Properties.≤′⇒≤
d_'8804''8242''8658''8804'_4636 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''8242''8658''8804'_4636 :: Integer -> Integer -> T__'8804''8242'__154 -> T__'8804'__18
d_'8804''8242''8658''8804'_4636 Integer
v0 ~Integer
v1 T__'8804''8242'__154
v2
  = Integer -> T__'8804''8242'__154 -> T__'8804'__18
du_'8804''8242''8658''8804'_4636 Integer
v0 T__'8804''8242'__154
v2
du_'8804''8242''8658''8804'_4636 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''8242''8658''8804'_4636 :: Integer -> T__'8804''8242'__154 -> T__'8804'__18
du_'8804''8242''8658''8804'_4636 Integer
v0 T__'8804''8242'__154
v1
  = case T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
v1 of
      T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_158
        -> (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
      MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_164 T__'8804''8242'__154
v3
        -> (Integer -> T__'8804''8242'__154 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804''8242'__154 -> T__'8804'__18
du_'8804''8242''8658''8804'_4636 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (T__'8804''8242'__154 -> Any
forall a b. a -> b
coe T__'8804''8242'__154
v3)
      T__'8804''8242'__154
_ -> T__'8804'__18
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.≤⇒≤′
d_'8804''8658''8804''8242'_4640 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
d_'8804''8658''8804''8242'_4640 :: Integer -> Integer -> T__'8804'__18 -> T__'8804''8242'__154
d_'8804''8658''8804''8242'_4640 ~Integer
v0 Integer
v1 T__'8804'__18
v2
  = Integer -> T__'8804'__18 -> T__'8804''8242'__154
du_'8804''8658''8804''8242'_4640 Integer
v1 T__'8804'__18
v2
du_'8804''8658''8804''8242'_4640 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
du_'8804''8658''8804''8242'_4640 :: Integer -> T__'8804'__18 -> T__'8804''8242'__154
du_'8804''8658''8804''8242'_4640 Integer
v0 T__'8804'__18
v1
  = case T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v1 of
      T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
        -> (Integer -> T__'8804''8242'__154) -> Any -> T__'8804''8242'__154
forall a b. a -> b
coe Integer -> T__'8804''8242'__154
d_z'8804''8242'n_4624 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
      MAlonzo.Code.Data.Nat.Base.C_s'8804's_30 T__'8804'__18
v4
        -> let v5 :: Integer
v5 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804''8242'__154
forall a b. a -> b
coe ((Integer -> T__'8804'__18 -> T__'8804''8242'__154)
-> Any -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804''8242'__154
du_'8804''8658''8804''8242'_4640 (Integer -> Any
forall a b. a -> b
coe Integer
v5) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
v4))
      T__'8804'__18
_ -> T__'8804''8242'__154
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.≤′-step-injective
d_'8804''8242''45'step'45'injective_4652 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8804''8242''45'step'45'injective_4652 :: Integer
-> Integer
-> T__'8804''8242'__154
-> T__'8804''8242'__154
-> T__'8801'__12
-> T__'8801'__12
d_'8804''8242''45'step'45'injective_4652 = Integer
-> Integer
-> T__'8804''8242'__154
-> T__'8804''8242'__154
-> T__'8801'__12
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties._≤′?_
d__'8804''8242''63'__4654 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8804''8242''63'__4654 :: Integer -> Integer -> T_Dec_32
d__'8804''8242''63'__4654 Integer
v0 Integer
v1
  = ((Any -> Any) -> T_Dec_32 -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe
      (Any -> Any) -> T_Dec_32 -> T_Dec_32
MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
      ((Integer -> T__'8804'__18 -> T__'8804''8242'__154) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804''8242'__154
du_'8804''8658''8804''8242'_4640 (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      ((Integer -> Integer -> T_Dec_32) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8804''63'__1676 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties._<′?_
d__'60''8242''63'__4660 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'60''8242''63'__4660 :: Integer -> Integer -> T_Dec_32
d__'60''8242''63'__4660 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe
      Integer -> Integer -> T_Dec_32
d__'8804''8242''63'__4654
      ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0)) (Integer -> Any
forall a b. a -> b
coe Integer
v1)
-- Data.Nat.Properties._≥′?_
d__'8805''8242''63'__4666 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8805''8242''63'__4666 :: Integer -> Integer -> T_Dec_32
d__'8805''8242''63'__4666 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8804''8242''63'__4654 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties._>′?_
d__'62''8242''63'__4668 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'62''8242''63'__4668 :: Integer -> Integer -> T_Dec_32
d__'62''8242''63'__4668 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'60''8242''63'__4660 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.m≤′m+n
d_m'8804''8242'm'43'n_4674 ::
  Integer ->
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
d_m'8804''8242'm'43'n_4674 :: Integer -> Integer -> T__'8804''8242'__154
d_m'8804''8242'm'43'n_4674 Integer
v0 Integer
v1
  = (Integer -> T__'8804'__18 -> T__'8804''8242'__154)
-> Any -> Any -> T__'8804''8242'__154
forall a b. a -> b
coe
      Integer -> T__'8804'__18 -> T__'8804''8242'__154
du_'8804''8658''8804''8242'_4640 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 (Integer -> Any
forall a b. a -> b
coe Integer
v0))
-- Data.Nat.Properties.n≤′m+n
d_n'8804''8242'm'43'n_4684 ::
  Integer ->
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
d_n'8804''8242'm'43'n_4684 :: Integer -> Integer -> T__'8804''8242'__154
d_n'8804''8242'm'43'n_4684 Integer
v0 ~Integer
v1 = Integer -> T__'8804''8242'__154
du_n'8804''8242'm'43'n_4684 Integer
v0
du_n'8804''8242'm'43'n_4684 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
du_n'8804''8242'm'43'n_4684 :: Integer -> T__'8804''8242'__154
du_n'8804''8242'm'43'n_4684 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_158
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804''8242'__154
forall a b. a -> b
coe
             ((T__'8804''8242'__154 -> T__'8804''8242'__154) -> Any -> Any
forall a b. a -> b
coe
                T__'8804''8242'__154 -> T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_164
                ((Integer -> T__'8804''8242'__154) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804''8242'__154
du_n'8804''8242'm'43'n_4684 (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.⌈n/2⌉≤′n
d_'8968'n'47'2'8969''8804''8242'n_4694 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
d_'8968'n'47'2'8969''8804''8242'n_4694 :: Integer -> T__'8804''8242'__154
d_'8968'n'47'2'8969''8804''8242'n_4694 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_158
      Integer
1 -> T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_158
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
2 :: Integer)) in
           Any -> T__'8804''8242'__154
forall a b. a -> b
coe
             ((T__'8804''8242'__154 -> T__'8804''8242'__154)
-> T__'8804''8242'__154 -> Any
forall a b. a -> b
coe
                T__'8804''8242'__154 -> T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_164
                (Integer -> T__'8804''8242'__154
d_'8968'n'47'2'8969''8804''8242'n_4694 (Integer -> Integer
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.⌊n/2⌋≤′n
d_'8970'n'47'2'8971''8804''8242'n_4700 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__154
d_'8970'n'47'2'8971''8804''8242'n_4700 :: Integer -> T__'8804''8242'__154
d_'8970'n'47'2'8971''8804''8242'n_4700 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804''8242'__154 -> T__'8804''8242'__154
forall a b. a -> b
coe T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_158
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804''8242'__154
forall a b. a -> b
coe
             ((T__'8804''8242'__154 -> T__'8804''8242'__154)
-> T__'8804''8242'__154 -> Any
forall a b. a -> b
coe
                T__'8804''8242'__154 -> T__'8804''8242'__154
MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_164
                (Integer -> T__'8804''8242'__154
d_'8968'n'47'2'8969''8804''8242'n_4694 (Integer -> Integer
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.m<ᵇn⇒1+m+[n-1+m]≡n
d_m'60''7495'n'8658'1'43'm'43''91'n'45'1'43'm'93''8801'n_4708 ::
  Integer ->
  Integer ->
  AgdaAny -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'60''7495'n'8658'1'43'm'43''91'n'45'1'43'm'93''8801'n_4708 :: Integer -> Integer -> Any -> T__'8801'__12
d_m'60''7495'n'8658'1'43'm'43''91'n'45'1'43'm'93''8801'n_4708
  = Integer -> Integer -> Any -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m<ᵇ1+m+n
d_m'60''7495'1'43'm'43'n_4720 :: Integer -> Integer -> AgdaAny
d_m'60''7495'1'43'm'43'n_4720 :: Integer -> Integer -> Any
d_m'60''7495'1'43'm'43'n_4720 Integer
v0 ~Integer
v1
  = Integer -> Any
du_m'60''7495'1'43'm'43'n_4720 Integer
v0
du_m'60''7495'1'43'm'43'n_4720 :: Integer -> AgdaAny
du_m'60''7495'1'43'm'43'n_4720 :: Integer -> Any
du_m'60''7495'1'43'm'43'n_4720 Integer
v0
  = (T__'8804'__18 -> Any) -> Any -> Any
forall a b. a -> b
coe
      T__'8804'__18 -> Any
du_'60''8658''60''7495'_1582
      ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
         Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0)))
-- Data.Nat.Properties.<ᵇ⇒<″
d_'60''7495''8658''60''8243'_4728 ::
  Integer ->
  Integer ->
  AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188
d_'60''7495''8658''60''8243'_4728 :: Integer -> Integer -> Any -> T__'8804''8243'__188
d_'60''7495''8658''60''8243'_4728 Integer
v0 Integer
v1 ~Any
v2
  = Integer -> Integer -> T__'8804''8243'__188
du_'60''7495''8658''60''8243'_4728 Integer
v0 Integer
v1
du_'60''7495''8658''60''8243'_4728 ::
  Integer ->
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188
du_'60''7495''8658''60''8243'_4728 :: Integer -> Integer -> T__'8804''8243'__188
du_'60''7495''8658''60''8243'_4728 Integer
v0 Integer
v1
  = (Integer -> T__'8804''8243'__188) -> Any -> T__'8804''8243'__188
forall a b. a -> b
coe
      Integer -> T__'8804''8243'__188
MAlonzo.Code.Data.Nat.Base.C_less'45'than'45'or'45'equal_202
      ((Integer -> Integer -> Integer) -> Integer -> Integer -> Any
forall a b. a -> b
coe
         Integer -> Integer -> Integer
MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 Integer
v1
         (Integer -> Integer -> Integer
addInt (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Integer
forall a b. a -> b
coe Integer
v0)))
-- Data.Nat.Properties.<″⇒<ᵇ
d_'60''8243''8658''60''7495'_4740 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 -> AgdaAny
d_'60''8243''8658''60''7495'_4740 :: Integer -> Integer -> T__'8804''8243'__188 -> Any
d_'60''8243''8658''60''7495'_4740 Integer
v0 ~Integer
v1 T__'8804''8243'__188
v2
  = Integer -> T__'8804''8243'__188 -> Any
du_'60''8243''8658''60''7495'_4740 Integer
v0 T__'8804''8243'__188
v2
du_'60''8243''8658''60''7495'_4740 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 -> AgdaAny
du_'60''8243''8658''60''7495'_4740 :: Integer -> T__'8804''8243'__188 -> Any
du_'60''8243''8658''60''7495'_4740 Integer
v0 T__'8804''8243'__188
v1
  = (Any -> Any -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
      Any -> Any -> Any
forall a b. a -> b -> b
seq (T__'8804''8243'__188 -> Any
forall a b. a -> b
coe T__'8804''8243'__188
v1)
      ((T__'8804'__18 -> Any) -> Any -> Any
forall a b. a -> b
coe
         T__'8804'__18 -> Any
du_'60''8658''60''7495'_1582
         ((Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
            Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0))))
-- Data.Nat.Properties.≤″⇒≤
d_'8804''8243''8658''8804'_4744 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''8243''8658''8804'_4744 :: Integer -> Integer -> T__'8804''8243'__188 -> T__'8804'__18
d_'8804''8243''8658''8804'_4744 Integer
v0 ~Integer
v1 T__'8804''8243'__188
v2
  = Integer -> T__'8804''8243'__188 -> T__'8804'__18
du_'8804''8243''8658''8804'_4744 Integer
v0 T__'8804''8243'__188
v2
du_'8804''8243''8658''8804'_4744 ::
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''8243''8658''8804'_4744 :: Integer -> T__'8804''8243'__188 -> T__'8804'__18
du_'8804''8243''8658''8804'_4744 Integer
v0 T__'8804''8243'__188
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Any -> Any -> Any) -> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
             Any -> Any -> Any
forall a b. a -> b -> b
seq (T__'8804''8243'__188 -> Any
forall a b. a -> b
coe T__'8804''8243'__188
v1) (T__'8804'__18 -> Any
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22)
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case T__'8804''8243'__188 -> T__'8804''8243'__188
forall a b. a -> b
coe T__'8804''8243'__188
v1 of
                MAlonzo.Code.Data.Nat.Base.C_less'45'than'45'or'45'equal_202 Integer
v3
                  -> (T__'8804'__18 -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe
                       T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                       ((Integer -> T__'8804''8243'__188 -> T__'8804'__18)
-> Any -> Any -> Any
forall a b. a -> b
coe
                          Integer -> T__'8804''8243'__188 -> T__'8804'__18
du_'8804''8243''8658''8804'_4744 (Integer -> Any
forall a b. a -> b
coe Integer
v2)
                          ((Integer -> T__'8804''8243'__188) -> Integer -> Any
forall a b. a -> b
coe
                             Integer -> T__'8804''8243'__188
MAlonzo.Code.Data.Nat.Base.C_less'45'than'45'or'45'equal_202 Integer
v3))
                T__'8804''8243'__188
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.≤⇒≤″
d_'8804''8658''8804''8243'_4748 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188
d_'8804''8658''8804''8243'_4748 :: Integer -> Integer -> T__'8804'__18 -> T__'8804''8243'__188
d_'8804''8658''8804''8243'_4748 Integer
v0 Integer
v1 ~T__'8804'__18
v2
  = Integer -> Integer -> T__'8804''8243'__188
du_'8804''8658''8804''8243'_4748 Integer
v0 Integer
v1
du_'8804''8658''8804''8243'_4748 ::
  Integer ->
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188
du_'8804''8658''8804''8243'_4748 :: Integer -> Integer -> T__'8804''8243'__188
du_'8804''8658''8804''8243'_4748 Integer
v0 Integer
v1
  = (Integer -> T__'8804''8243'__188) -> Any -> T__'8804''8243'__188
forall a b. a -> b
coe
      Integer -> T__'8804''8243'__188
MAlonzo.Code.Data.Nat.Base.C_less'45'than'45'or'45'equal_202
      ((Integer -> Integer -> Integer) -> Integer -> Integer -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 Integer
v1 Integer
v0)
-- Data.Nat.Properties._<″?_
d__'60''8243''63'__4750 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'60''8243''63'__4750 :: Integer -> Integer -> T_Dec_32
d__'60''8243''63'__4750 Integer
v0 Integer
v1
  = ((Any -> Any) -> T_Dec_32 -> T_Dec_32)
-> (Any -> Any) -> Any -> T_Dec_32
forall a b. a -> b
coe
      (Any -> Any) -> T_Dec_32 -> T_Dec_32
MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
      (\ Any
v2 -> (Integer -> Integer -> T__'8804''8243'__188) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804''8243'__188
du_'60''7495''8658''60''8243'_4728 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      ((Bool -> T_Dec_32) -> Any -> Any
forall a b. a -> b
coe
         Bool -> T_Dec_32
MAlonzo.Code.Data.Bool.Properties.d_T'63'_2210
         ((Integer -> Integer -> Bool) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Bool
ltInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties._≤″?_
d__'8804''8243''63'__4756 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8804''8243''63'__4756 :: Integer -> Integer -> T_Dec_32
d__'8804''8243''63'__4756 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Bool -> T_Reflects_14 -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe
             Bool -> T_Reflects_14 -> T_Dec_32
MAlonzo.Code.Relation.Nullary.C__because__46
             (Bool -> Any
forall a b. a -> b
coe Bool
MAlonzo.Code.Agda.Builtin.Bool.C_true_10)
             ((Any -> T_Reflects_14) -> Any -> Any
forall a b. a -> b
coe
                Any -> T_Reflects_14
MAlonzo.Code.Relation.Nullary.C_of'696'_22
                ((Integer -> T__'8804''8243'__188) -> Integer -> Any
forall a b. a -> b
coe
                   Integer -> T__'8804''8243'__188
MAlonzo.Code.Data.Nat.Base.C_less'45'than'45'or'45'equal_202 Integer
v1))
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T_Dec_32
forall a b. a -> b
coe ((Integer -> Integer -> T_Dec_32) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'60''8243''63'__4750 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties._≥″?_
d__'8805''8243''63'__4764 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8805''8243''63'__4764 :: Integer -> Integer -> T_Dec_32
d__'8805''8243''63'__4764 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8804''8243''63'__4756 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties._>″?_
d__'62''8243''63'__4766 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'62''8243''63'__4766 :: Integer -> Integer -> T_Dec_32
d__'62''8243''63'__4766 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'60''8243''63'__4750 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.≤″-irrelevant
d_'8804''8243''45'irrelevant_4768 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8804''8243''45'irrelevant_4768 :: Integer
-> Integer
-> T__'8804''8243'__188
-> T__'8804''8243'__188
-> T__'8801'__12
d_'8804''8243''45'irrelevant_4768 = Integer
-> Integer
-> T__'8804''8243'__188
-> T__'8804''8243'__188
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.<″-irrelevant
d_'60''8243''45'irrelevant_4786 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'60''8243''45'irrelevant_4786 :: Integer
-> Integer
-> T__'8804''8243'__188
-> T__'8804''8243'__188
-> T__'8801'__12
d_'60''8243''45'irrelevant_4786 = Integer
-> Integer
-> T__'8804''8243'__188
-> T__'8804''8243'__188
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.>″-irrelevant
d_'62''8243''45'irrelevant_4788 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'62''8243''45'irrelevant_4788 :: Integer
-> Integer
-> T__'8804''8243'__188
-> T__'8804''8243'__188
-> T__'8801'__12
d_'62''8243''45'irrelevant_4788 = Integer
-> Integer
-> T__'8804''8243'__188
-> T__'8804''8243'__188
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≥″-irrelevant
d_'8805''8243''45'irrelevant_4790 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8805''8243''45'irrelevant_4790 :: Integer
-> Integer
-> T__'8804''8243'__188
-> T__'8804''8243'__188
-> T__'8801'__12
d_'8805''8243''45'irrelevant_4790 = Integer
-> Integer
-> T__'8804''8243'__188
-> T__'8804''8243'__188
-> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≤‴⇒≤″
d_'8804''8244''8658''8804''8243'_4796 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188
d_'8804''8244''8658''8804''8243'_4796 :: Integer -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188
d_'8804''8244''8658''8804''8243'_4796 Integer
v0 Integer
v1 T__'8804''8244'__222
v2
  = case T__'8804''8244'__222 -> T__'8804''8244'__222
forall a b. a -> b
coe T__'8804''8244'__222
v2 of
      T__'8804''8244'__222
MAlonzo.Code.Data.Nat.Base.C_'8804''8244''45'refl_226
        -> (Integer -> T__'8804''8243'__188)
-> Integer -> T__'8804''8243'__188
forall a b. a -> b
coe
             Integer -> T__'8804''8243'__188
MAlonzo.Code.Data.Nat.Base.C_less'45'than'45'or'45'equal_202
             (Integer
0 :: Integer)
      MAlonzo.Code.Data.Nat.Base.C_'8804''8244''45'step_232 T__'8804''8244'__222
v5
        -> (Integer -> T__'8804''8243'__188)
-> Integer -> T__'8804''8243'__188
forall a b. a -> b
coe
             Integer -> T__'8804''8243'__188
MAlonzo.Code.Data.Nat.Base.C_less'45'than'45'or'45'equal_202
             (Integer -> Integer -> Integer
addInt
                (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer))
                ((T__'8804''8243'__188 -> Integer) -> Any -> Integer
forall a b. a -> b
coe
                   T__'8804''8243'__188 -> Integer
MAlonzo.Code.Data.Nat.Base.d_k_198
                   ((Integer
 -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188
d_ind_4808 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (T__'8804''8244'__222 -> Any
forall a b. a -> b
coe T__'8804''8244'__222
v5))))
      T__'8804''8244'__222
_ -> T__'8804''8243'__188
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties._.ind
d_ind_4808 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188
d_ind_4808 :: Integer -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188
d_ind_4808 Integer
v0 Integer
v1 T__'8804''8244'__222
v2
  = (Integer
 -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188)
-> Any -> Any -> Any -> T__'8804''8243'__188
forall a b. a -> b
coe
      Integer -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188
d_'8804''8244''8658''8804''8243'_4796
      ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0)) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (T__'8804''8244'__222 -> Any
forall a b. a -> b
coe T__'8804''8244'__222
v2)
-- Data.Nat.Properties.m≤‴m+k
d_m'8804''8244'm'43'k_4816 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
d_m'8804''8244'm'43'k_4816 :: Integer
-> Integer -> Integer -> T__'8801'__12 -> T__'8804''8244'__222
d_m'8804''8244'm'43'k_4816 ~Integer
v0 ~Integer
v1 Integer
v2 ~T__'8801'__12
v3
  = Integer -> T__'8804''8244'__222
du_m'8804''8244'm'43'k_4816 Integer
v2
du_m'8804''8244'm'43'k_4816 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
du_m'8804''8244'm'43'k_4816 :: Integer -> T__'8804''8244'__222
du_m'8804''8244'm'43'k_4816 Integer
v0
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> T__'8804''8244'__222 -> T__'8804''8244'__222
forall a b. a -> b
coe T__'8804''8244'__222
MAlonzo.Code.Data.Nat.Base.C_'8804''8244''45'refl_226
      Integer
_ -> let v1 :: Integer
v1 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804''8244'__222
forall a b. a -> b
coe
             ((T__'8804''8244'__222 -> T__'8804''8244'__222) -> Any -> Any
forall a b. a -> b
coe
                T__'8804''8244'__222 -> T__'8804''8244'__222
MAlonzo.Code.Data.Nat.Base.C_'8804''8244''45'step_232
                ((Integer -> T__'8804''8244'__222) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804''8244'__222
du_m'8804''8244'm'43'k_4816 (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties.≤″⇒≤‴
d_'8804''8243''8658''8804''8244'_4832 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
d_'8804''8243''8658''8804''8244'_4832 :: Integer -> Integer -> T__'8804''8243'__188 -> T__'8804''8244'__222
d_'8804''8243''8658''8804''8244'_4832 ~Integer
v0 ~Integer
v1 T__'8804''8243'__188
v2
  = T__'8804''8243'__188 -> T__'8804''8244'__222
du_'8804''8243''8658''8804''8244'_4832 T__'8804''8243'__188
v2
du_'8804''8243''8658''8804''8244'_4832 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804''8243'__188 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
du_'8804''8243''8658''8804''8244'_4832 :: T__'8804''8243'__188 -> T__'8804''8244'__222
du_'8804''8243''8658''8804''8244'_4832 T__'8804''8243'__188
v0
  = case T__'8804''8243'__188 -> T__'8804''8243'__188
forall a b. a -> b
coe T__'8804''8243'__188
v0 of
      MAlonzo.Code.Data.Nat.Base.C_less'45'than'45'or'45'equal_202 Integer
v1
        -> (Integer -> T__'8804''8244'__222) -> Any -> T__'8804''8244'__222
forall a b. a -> b
coe Integer -> T__'8804''8244'__222
du_m'8804''8244'm'43'k_4816 (Integer -> Any
forall a b. a -> b
coe Integer
v1)
      T__'8804''8243'__188
_ -> T__'8804''8244'__222
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Data.Nat.Properties.0≤‴n
d_0'8804''8244'n_4840 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
d_0'8804''8244'n_4840 :: Integer -> T__'8804''8244'__222
d_0'8804''8244'n_4840 Integer
v0 = (Integer -> T__'8804''8244'__222) -> Any -> T__'8804''8244'__222
forall a b. a -> b
coe Integer -> T__'8804''8244'__222
du_m'8804''8244'm'43'k_4816 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.<ᵇ⇒<‴
d_'60''7495''8658''60''8244'_4848 ::
  Integer ->
  Integer ->
  AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
d_'60''7495''8658''60''8244'_4848 :: Integer -> Integer -> Any -> T__'8804''8244'__222
d_'60''7495''8658''60''8244'_4848 Integer
v0 Integer
v1 ~Any
v2
  = Integer -> Integer -> T__'8804''8244'__222
du_'60''7495''8658''60''8244'_4848 Integer
v0 Integer
v1
du_'60''7495''8658''60''8244'_4848 ::
  Integer ->
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
du_'60''7495''8658''60''8244'_4848 :: Integer -> Integer -> T__'8804''8244'__222
du_'60''7495''8658''60''8244'_4848 Integer
v0 Integer
v1
  = (T__'8804''8243'__188 -> T__'8804''8244'__222)
-> Any -> T__'8804''8244'__222
forall a b. a -> b
coe
      T__'8804''8243'__188 -> T__'8804''8244'__222
du_'8804''8243''8658''8804''8244'_4832
      ((Integer -> Integer -> T__'8804''8243'__188) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804''8243'__188
du_'60''7495''8658''60''8243'_4728 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties.<‴⇒<ᵇ
d_'60''8244''8658''60''7495'_4860 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222 -> AgdaAny
d_'60''8244''8658''60''7495'_4860 :: Integer -> Integer -> T__'8804''8244'__222 -> Any
d_'60''8244''8658''60''7495'_4860 Integer
v0 Integer
v1 T__'8804''8244'__222
v2
  = (Integer -> T__'8804''8243'__188 -> Any) -> Any -> Any -> Any
forall a b. a -> b
coe
      Integer -> T__'8804''8243'__188 -> Any
du_'60''8243''8658''60''7495'_4740 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
      ((Integer
 -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         Integer -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188
d_'8804''8244''8658''8804''8243'_4796
         ((Integer -> Integer -> Integer) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
addInt (Integer -> Any
forall a b. a -> b
coe (Integer
1 :: Integer)) (Integer -> Any
forall a b. a -> b
coe Integer
v0)) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (T__'8804''8244'__222 -> Any
forall a b. a -> b
coe T__'8804''8244'__222
v2))
-- Data.Nat.Properties._<‴?_
d__'60''8244''63'__4864 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'60''8244''63'__4864 :: Integer -> Integer -> T_Dec_32
d__'60''8244''63'__4864 Integer
v0 Integer
v1
  = ((Any -> Any) -> T_Dec_32 -> T_Dec_32)
-> (Any -> Any) -> Any -> T_Dec_32
forall a b. a -> b
coe
      (Any -> Any) -> T_Dec_32 -> T_Dec_32
MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
      (\ Any
v2 -> (Integer -> Integer -> T__'8804''8244'__222) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804''8244'__222
du_'60''7495''8658''60''8244'_4848 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
      ((Bool -> T_Dec_32) -> Any -> Any
forall a b. a -> b
coe
         Bool -> T_Dec_32
MAlonzo.Code.Data.Bool.Properties.d_T'63'_2210
         ((Integer -> Integer -> Bool) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> Bool
ltInt (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
-- Data.Nat.Properties._≤‴?_
d__'8804''8244''63'__4870 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8804''8244''63'__4870 :: Integer -> Integer -> T_Dec_32
d__'8804''8244''63'__4870 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
      Integer
0 -> (Bool -> T_Reflects_14 -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe
             Bool -> T_Reflects_14 -> T_Dec_32
MAlonzo.Code.Relation.Nullary.C__because__46
             (Bool -> Any
forall a b. a -> b
coe Bool
MAlonzo.Code.Agda.Builtin.Bool.C_true_10)
             ((Any -> T_Reflects_14) -> Any -> Any
forall a b. a -> b
coe
                Any -> T_Reflects_14
MAlonzo.Code.Relation.Nullary.C_of'696'_22
                ((Integer -> T__'8804''8244'__222) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804''8244'__222
d_0'8804''8244'n_4840 (Integer -> Any
forall a b. a -> b
coe Integer
v1)))
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T_Dec_32
forall a b. a -> b
coe ((Integer -> Integer -> T_Dec_32) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'60''8244''63'__4864 (Integer -> Any
forall a b. a -> b
coe Integer
v2) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties._≥‴?_
d__'8805''8244''63'__4878 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'8805''8244''63'__4878 :: Integer -> Integer -> T_Dec_32
d__'8805''8244''63'__4878 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8804''8244''63'__4870 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties._>‴?_
d__'62''8244''63'__4880 ::
  Integer -> Integer -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d__'62''8244''63'__4880 :: Integer -> Integer -> T_Dec_32
d__'62''8244''63'__4880 Integer
v0 Integer
v1
  = (Integer -> Integer -> T_Dec_32) -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'60''8244''63'__4864 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.≤⇒≤‴
d_'8804''8658''8804''8244'_4882 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
d_'8804''8658''8804''8244'_4882 :: Integer -> Integer -> T__'8804'__18 -> T__'8804''8244'__222
d_'8804''8658''8804''8244'_4882 Integer
v0 Integer
v1 ~T__'8804'__18
v2
  = Integer -> Integer -> T__'8804''8244'__222
du_'8804''8658''8804''8244'_4882 Integer
v0 Integer
v1
du_'8804''8658''8804''8244'_4882 ::
  Integer ->
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222
du_'8804''8658''8804''8244'_4882 :: Integer -> Integer -> T__'8804''8244'__222
du_'8804''8658''8804''8244'_4882 Integer
v0 Integer
v1
  = (T__'8804''8243'__188 -> T__'8804''8244'__222)
-> Any -> T__'8804''8244'__222
forall a b. a -> b
coe
      T__'8804''8243'__188 -> T__'8804''8244'__222
du_'8804''8243''8658''8804''8244'_4832
      ((Integer -> Integer -> T__'8804''8243'__188) -> Any -> Any -> Any
forall a b. a -> b
coe Integer -> Integer -> T__'8804''8243'__188
du_'8804''8658''8804''8243'_4748 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1))
-- Data.Nat.Properties.≤‴⇒≤
d_'8804''8244''8658''8804'_4884 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__222 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''8244''8658''8804'_4884 :: Integer -> Integer -> T__'8804''8244'__222 -> T__'8804'__18
d_'8804''8244''8658''8804'_4884 Integer
v0 Integer
v1 T__'8804''8244'__222
v2
  = (Integer -> T__'8804''8243'__188 -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> T__'8804''8243'__188 -> T__'8804'__18
du_'8804''8243''8658''8804'_4744 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
      ((Integer
 -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188)
-> Any -> Any -> Any -> Any
forall a b. a -> b
coe
         Integer -> Integer -> T__'8804''8244'__222 -> T__'8804''8243'__188
d_'8804''8244''8658''8804''8243'_4796 (Integer -> Any
forall a b. a -> b
coe Integer
v0) (Integer -> Any
forall a b. a -> b
coe Integer
v1) (T__'8804''8244'__222 -> Any
forall a b. a -> b
coe T__'8804''8244'__222
v2))
-- Data.Nat.Properties.eq?
d_eq'63'_4890 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  MAlonzo.Code.Function.Injection.T_Injection_88 ->
  AgdaAny -> AgdaAny -> MAlonzo.Code.Relation.Nullary.T_Dec_32
d_eq'63'_4890 :: () -> () -> T_Injection_88 -> Any -> Any -> T_Dec_32
d_eq'63'_4890 ~()
v0 ~()
v1 T_Injection_88
v2 = T_Injection_88 -> Any -> Any -> T_Dec_32
du_eq'63'_4890 T_Injection_88
v2
du_eq'63'_4890 ::
  MAlonzo.Code.Function.Injection.T_Injection_88 ->
  AgdaAny -> AgdaAny -> MAlonzo.Code.Relation.Nullary.T_Dec_32
du_eq'63'_4890 :: T_Injection_88 -> Any -> Any -> T_Dec_32
du_eq'63'_4890 T_Injection_88
v0
  = (T_Injection_88
 -> (Any -> Any -> T_Dec_32) -> Any -> Any -> T_Dec_32)
-> Any -> Any -> Any -> Any -> T_Dec_32
forall a b. a -> b
coe
      T_Injection_88
-> (Any -> Any -> T_Dec_32) -> Any -> Any -> T_Dec_32
MAlonzo.Code.Relation.Nullary.Decidable.du_via'45'injection_68
      (T_Injection_88 -> Any
forall a b. a -> b
coe T_Injection_88
v0) ((Integer -> Integer -> T_Dec_32) -> Any
forall a b. a -> b
coe Integer -> Integer -> T_Dec_32
d__'8799'__1528)
-- Data.Nat.Properties._*-mono_
d__'42''45'mono__4894 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d__'42''45'mono__4894 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d__'42''45'mono__4894 Integer
v0 Integer
v1 Integer
v2 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
  = (Integer
 -> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'42''45'mono'45''8804'_2824 Integer
v1 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
-- Data.Nat.Properties._+-mono_
d__'43''45'mono__4896 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d__'43''45'mono__4896 :: Integer
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d__'43''45'mono__4896 Integer
v0 Integer
v1 Integer
v2 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
  = (Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18)
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8804'__18
du_'43''45'mono'45''8804'_2398 Integer
v3 T__'8804'__18
v4 T__'8804'__18
v5
-- Data.Nat.Properties.+-right-identity
d_'43''45'right'45'identity_4898 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45'right'45'identity_4898 :: Integer -> T__'8801'__12
d_'43''45'right'45'identity_4898 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-right-zero
d_'42''45'right'45'zero_4900 ::
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'right'45'zero_4900 :: Integer -> T__'8801'__12
d_'42''45'right'45'zero_4900 = Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.distribʳ-*-+
d_distrib'691''45''42''45''43'_4902 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_distrib'691''45''42''45''43'_4902 :: Integer -> Integer -> Integer -> T__'8801'__12
d_distrib'691''45''42''45''43'_4902 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.*-distrib-∸ʳ
d_'42''45'distrib'45''8760''691'_4904 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'42''45'distrib'45''8760''691'_4904 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'42''45'distrib'45''8760''691'_4904 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.cancel-+-left
d_cancel'45''43''45'left_4906 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_cancel'45''43''45'left_4906 :: Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_cancel'45''43''45'left_4906 = Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.cancel-+-left-≤
d_cancel'45''43''45'left'45''8804'_4908 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_cancel'45''43''45'left'45''8804'_4908 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_cancel'45''43''45'left'45''8804'_4908 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'43''45'cancel'737''45''8804'_2242 Integer
v0 T__'8804'__18
v3
-- Data.Nat.Properties.cancel-*-right
d_cancel'45''42''45'right_4910 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_cancel'45''42''45'right_4910 :: Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_cancel'45''42''45'right_4910 = Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.cancel-*-right-≤
d_cancel'45''42''45'right'45''8804'_4912 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_cancel'45''42''45'right'45''8804'_4912 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_cancel'45''42''45'right'45''8804'_4912 Integer
v0 Integer
v1 Integer
v2 T__'8804'__18
v3
  = (Integer -> T__'8804'__18) -> Integer -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18
du_'42''45'cancel'691''45''8804'_2792 Integer
v0
-- Data.Nat.Properties.strictTotalOrder
d_strictTotalOrder_4914 ::
  MAlonzo.Code.Relation.Binary.Bundles.T_StrictTotalOrder_864
d_strictTotalOrder_4914 :: T_StrictTotalOrder_864
d_strictTotalOrder_4914 = T_StrictTotalOrder_864 -> T_StrictTotalOrder_864
forall a b. a -> b
coe T_StrictTotalOrder_864
d_'60''45'strictTotalOrder_1934
-- Data.Nat.Properties.isCommutativeSemiring
d_isCommutativeSemiring_4916 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344
d_isCommutativeSemiring_4916 :: T_IsCommutativeSemiring_1344
d_isCommutativeSemiring_4916
  = T_IsCommutativeSemiring_1344 -> T_IsCommutativeSemiring_1344
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
d_'43''45''42''45'isCommutativeSemiring_2624
-- Data.Nat.Properties.commutativeSemiring
d_commutativeSemiring_4918 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemiring_2094
d_commutativeSemiring_4918 :: T_CommutativeSemiring_2094
d_commutativeSemiring_4918
  = T_CommutativeSemiring_2094 -> T_CommutativeSemiring_2094
forall a b. a -> b
coe T_CommutativeSemiring_2094
d_'43''45''42''45'commutativeSemiring_2642
-- Data.Nat.Properties.isDistributiveLattice
d_isDistributiveLattice_4920 ::
  MAlonzo.Code.Algebra.Structures.T_IsDistributiveLattice_814
d_isDistributiveLattice_4920 :: T_IsDistributiveLattice_814
d_isDistributiveLattice_4920
  = (T_TotalPreorder_204
 -> T_MinOperator_84
 -> T_MaxOperator_114
 -> T_IsDistributiveLattice_814)
-> Any -> Any -> Any -> T_IsDistributiveLattice_814
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84
-> T_MaxOperator_114
-> T_IsDistributiveLattice_814
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'isDistributiveLattice_1962
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.distributiveLattice
d_distributiveLattice_4922 ::
  MAlonzo.Code.Algebra.Bundles.T_DistributiveLattice_1228
d_distributiveLattice_4922 :: T_DistributiveLattice_1228
d_distributiveLattice_4922
  = (T_TotalPreorder_204
 -> T_MinOperator_84
 -> T_MaxOperator_114
 -> T_DistributiveLattice_1228)
-> Any -> Any -> Any -> T_DistributiveLattice_1228
forall a b. a -> b
coe
      T_TotalPreorder_204
-> T_MinOperator_84
-> T_MaxOperator_114
-> T_DistributiveLattice_1228
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'distributiveLattice_1972
      (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
d_'8804''45'totalPreorder_1696)
      (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
d_'8851''45'operator_3094) (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
d_'8852''45'operator_3096)
-- Data.Nat.Properties.⊔-⊓-0-isSemiringWithoutOne
d_'8852''45''8851''45'0'45'isSemiringWithoutOne_4924 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_952
d_'8852''45''8851''45'0'45'isSemiringWithoutOne_4924 :: T_IsSemiringWithoutOne_952
d_'8852''45''8851''45'0'45'isSemiringWithoutOne_4924
  = T_IsSemiringWithoutOne_952 -> T_IsSemiringWithoutOne_952
forall a b. a -> b
coe T_IsSemiringWithoutOne_952
d_'8852''45''8851''45'isSemiringWithoutOne_3510
-- Data.Nat.Properties.⊔-⊓-0-isCommutativeSemiringWithoutOne
d_'8852''45''8851''45'0'45'isCommutativeSemiringWithoutOne_4926 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1044
d_'8852''45''8851''45'0'45'isCommutativeSemiringWithoutOne_4926 :: T_IsCommutativeSemiringWithoutOne_1044
d_'8852''45''8851''45'0'45'isCommutativeSemiringWithoutOne_4926
  = T_IsCommutativeSemiringWithoutOne_1044
-> T_IsCommutativeSemiringWithoutOne_1044
forall a b. a -> b
coe T_IsCommutativeSemiringWithoutOne_1044
d_'8852''45''8851''45'isCommutativeSemiringWithoutOne_3512
-- Data.Nat.Properties.⊔-⊓-0-commutativeSemiringWithoutOne
d_'8852''45''8851''45'0'45'commutativeSemiringWithoutOne_4928 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemiringWithoutOne_1598
d_'8852''45''8851''45'0'45'commutativeSemiringWithoutOne_4928 :: T_CommutativeSemiringWithoutOne_1598
d_'8852''45''8851''45'0'45'commutativeSemiringWithoutOne_4928
  = T_CommutativeSemiringWithoutOne_1598
-> T_CommutativeSemiringWithoutOne_1598
forall a b. a -> b
coe T_CommutativeSemiringWithoutOne_1598
d_'8852''45''8851''45'commutativeSemiringWithoutOne_3514
-- Data.Nat.Properties.¬i+1+j≤i
d_'172'i'43'1'43'j'8804'i_4930 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_'172'i'43'1'43'j'8804'i_4930 :: Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
d_'172'i'43'1'43'j'8804'i_4930 = Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.≤-steps
d_'8804''45'steps_4932 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''45'steps_4932 :: Integer -> Integer -> Integer -> T__'8804'__18 -> T__'8804'__18
d_'8804''45'steps_4932 ~Integer
v0 ~Integer
v1 ~Integer
v2 T__'8804'__18
v3 = T__'8804'__18 -> T__'8804'__18
du_'8804''45'steps_4932 T__'8804'__18
v3
du_'8804''45'steps_4932 ::
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_'8804''45'steps_4932 :: T__'8804'__18 -> T__'8804'__18
du_'8804''45'steps_4932 T__'8804'__18
v0 = T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
v0
-- Data.Nat.Properties.i∸k∸j+j∸k≡i+j∸k
d_i'8760'k'8760'j'43'j'8760'k'8801'i'43'j'8760'k_4940 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_i'8760'k'8760'j'43'j'8760'k'8801'i'43'j'8760'k_4940 :: Integer -> Integer -> Integer -> T__'8801'__12
d_i'8760'k'8760'j'43'j'8760'k'8801'i'43'j'8760'k_4940 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.im≡jm+n⇒[i∸j]m≡n
d_im'8801'jm'43'n'8658''91'i'8760'j'93'm'8801'n_4976 ::
  Integer ->
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_im'8801'jm'43'n'8658''91'i'8760'j'93'm'8801'n_4976 :: Integer
-> Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_im'8801'jm'43'n'8658''91'i'8760'j'93'm'8801'n_4976 = Integer
-> Integer -> Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.≤+≢⇒<
d_'8804''43''8802''8658''60'_4992 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8804''43''8802''8658''60'_4992 :: Integer
-> Integer
-> T__'8804'__18
-> (T__'8801'__12 -> T_'8869'_4)
-> T__'8804'__18
d_'8804''43''8802''8658''60'_4992 Integer
v0 Integer
v1 T__'8804'__18
v2 T__'8801'__12 -> T_'8869'_4
v3
  = (Integer -> T__'8804'__18 -> T__'8804'__18)
-> Integer -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18 -> T__'8804'__18
du_'8804''8743''8802''8658''60'_1800 Integer
v1 T__'8804'__18
v2
-- Data.Nat.Properties.≤-irrelevance
d_'8804''45'irrelevance_4994 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8804''45'irrelevance_4994 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
d_'8804''45'irrelevance_4994 = Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.<-irrelevance
d_'60''45'irrelevance_4996 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'60''45'irrelevance_4996 :: Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
d_'60''45'irrelevance_4996 = Integer
-> Integer -> T__'8804'__18 -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.i+1+j≢i
d_i'43'1'43'j'8802'i_4998 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_i'43'1'43'j'8802'i_4998 :: Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
d_i'43'1'43'j'8802'i_4998 = Integer -> Integer -> T__'8801'__12 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.i+j≡0⇒i≡0
d_i'43'j'8801'0'8658'i'8801'0_5000 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_i'43'j'8801'0'8658'i'8801'0_5000 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_i'43'j'8801'0'8658'i'8801'0_5000 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.i+j≡0⇒j≡0
d_i'43'j'8801'0'8658'j'8801'0_5002 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_i'43'j'8801'0'8658'j'8801'0_5002 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_i'43'j'8801'0'8658'j'8801'0_5002 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.i+1+j≰i
d_i'43'1'43'j'8816'i_5004 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Empty.T_'8869'_4
d_i'43'1'43'j'8816'i_5004 :: Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
d_i'43'1'43'j'8816'i_5004 = Integer -> Integer -> T__'8804'__18 -> T_'8869'_4
forall a. a
erased
-- Data.Nat.Properties.i*j≡0⇒i≡0∨j≡0
d_i'42'j'8801'0'8658'i'8801'0'8744'j'8801'0_5006 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_i'42'j'8801'0'8658'i'8801'0'8744'j'8801'0_5006 :: Integer -> Integer -> T__'8801'__12 -> T__'8846'__30
d_i'42'j'8801'0'8658'i'8801'0'8744'j'8801'0_5006 Integer
v0 Integer
v1 T__'8801'__12
v2
  = (Integer -> T__'8846'__30) -> Integer -> T__'8846'__30
forall a b. a -> b
coe Integer -> T__'8846'__30
du_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_2682 Integer
v0
-- Data.Nat.Properties.i*j≡1⇒i≡1
d_i'42'j'8801'1'8658'i'8801'1_5008 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_i'42'j'8801'1'8658'i'8801'1_5008 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_i'42'j'8801'1'8658'i'8801'1_5008 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.i*j≡1⇒j≡1
d_i'42'j'8801'1'8658'j'8801'1_5010 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_i'42'j'8801'1'8658'j'8801'1_5010 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_i'42'j'8801'1'8658'j'8801'1_5010 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.i^j≡0⇒i≡0
d_i'94'j'8801'0'8658'i'8801'0_5012 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_i'94'j'8801'0'8658'i'8801'0_5012 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_i'94'j'8801'0'8658'i'8801'0_5012 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.i^j≡1⇒j≡0∨i≡1
d_i'94'j'8801'1'8658'j'8801'0'8744'i'8801'1_5014 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_i'94'j'8801'1'8658'j'8801'0'8744'i'8801'1_5014 :: Integer -> Integer -> T__'8801'__12 -> T__'8846'__30
d_i'94'j'8801'1'8658'j'8801'0'8744'i'8801'1_5014 Integer
v0 Integer
v1 T__'8801'__12
v2
  = (Integer -> T__'8846'__30) -> Integer -> T__'8846'__30
forall a b. a -> b
coe Integer -> T__'8846'__30
du_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_3036 Integer
v1
-- Data.Nat.Properties.[i+j]∸[i+k]≡j∸k
d_'91'i'43'j'93''8760''91'i'43'k'93''8801'j'8760'k_5016 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'91'i'43'j'93''8760''91'i'43'k'93''8801'j'8760'k_5016 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'91'i'43'j'93''8760''91'i'43'k'93''8801'j'8760'k_5016 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m≢0⇒suc[pred[m]]≡m
d_m'8802'0'8658'suc'91'pred'91'm'93''93''8801'm_5018 ::
  Integer ->
  (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8802'0'8658'suc'91'pred'91'm'93''93''8801'm_5018 :: Integer -> (T__'8801'__12 -> T_'8869'_4) -> T__'8801'__12
d_m'8802'0'8658'suc'91'pred'91'm'93''93''8801'm_5018 = Integer -> (T__'8801'__12 -> T_'8869'_4) -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.n≡m⇒∣n-m∣≡0
d_n'8801'm'8658''8739'n'45'm'8739''8801'0_5020 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_n'8801'm'8658''8739'n'45'm'8739''8801'0_5020 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_n'8801'm'8658''8739'n'45'm'8739''8801'0_5020 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣n-m∣≡0⇒n≡m
d_'8739'n'45'm'8739''8801'0'8658'n'8801'm_5022 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739'n'45'm'8739''8801'0'8658'n'8801'm_5022 :: Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
d_'8739'n'45'm'8739''8801'0'8658'n'8801'm_5022 = Integer -> Integer -> T__'8801'__12 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣n-m∣≡n∸m⇒m≤n
d_'8739'n'45'm'8739''8801'n'8760'm'8658'm'8804'n_5024 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8739'n'45'm'8739''8801'n'8760'm'8658'm'8804'n_5024 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_'8739'n'45'm'8739''8801'n'8760'm'8658'm'8804'n_5024 Integer
v0 Integer
v1 T__'8801'__12
v2
  = (Integer -> Integer -> T__'8804'__18)
-> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_4292 Integer
v0 Integer
v1
-- Data.Nat.Properties.∣n-n+m∣≡m
d_'8739'n'45'n'43'm'8739''8801'm_5026 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739'n'45'n'43'm'8739''8801'm_5026 :: Integer -> Integer -> T__'8801'__12
d_'8739'n'45'n'43'm'8739''8801'm_5026 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣n+m-n+o∣≡∣m-o|
d_'8739'n'43'm'45'n'43'o'8739''8801''8739'm'45'o'124'_5028 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739'n'43'm'45'n'43'o'8739''8801''8739'm'45'o'124'_5028 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8739'n'43'm'45'n'43'o'8739''8801''8739'm'45'o'124'_5028 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.∣m+n-m+o∣≡∣n-o|
d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'124'_5030 ::
  Integer ->
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'124'_5030 :: Integer -> Integer -> Integer -> T__'8801'__12
d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'124'_5030 = Integer -> Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.n∸m≤∣n-m∣
d_n'8760'm'8804''8739'n'45'm'8739'_5032 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8760'm'8804''8739'n'45'm'8739'_5032 :: Integer -> Integer -> T__'8804'__18
d_n'8760'm'8804''8739'n'45'm'8739'_5032
  = (Integer -> Integer -> T__'8804'__18)
-> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804''8739'm'45'n'8739'_4346
-- Data.Nat.Properties.∣n-m∣≤n⊔m
d_'8739'n'45'm'8739''8804'n'8852'm_5034 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8739'n'45'm'8739''8804'n'8852'm_5034 :: Integer -> Integer -> T__'8804'__18
d_'8739'n'45'm'8739''8804'n'8852'm_5034
  = (Integer -> Integer -> T__'8804'__18)
-> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_'8739'm'45'n'8739''8804'm'8852'n_4376
-- Data.Nat.Properties.n≤m+n
d_n'8804'm'43'n_5040 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8804'm'43'n_5040 :: Integer -> Integer -> T__'8804'__18
d_n'8804'm'43'n_5040 ~Integer
v0 Integer
v1 = Integer -> T__'8804'__18
du_n'8804'm'43'n_5040 Integer
v1
du_n'8804'm'43'n_5040 ::
  Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
du_n'8804'm'43'n_5040 :: Integer -> T__'8804'__18
du_n'8804'm'43'n_5040 Integer
v0 = (Integer -> T__'8804'__18) -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> T__'8804'__18
du_m'8804'm'43'n_2316 (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.n≤m+n∸m
d_n'8804'm'43'n'8760'm_5052 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8804'm'43'n'8760'm_5052 :: Integer -> Integer -> T__'8804'__18
d_n'8804'm'43'n'8760'm_5052 Integer
v0 Integer
v1
  = case Integer -> Integer
forall a b. a -> b
coe Integer
v1 of
      Integer
0 -> T__'8804'__18 -> T__'8804'__18
forall a b. a -> b
coe T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_z'8804'n_22
      Integer
_ -> let v2 :: Integer
v2 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v1) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
           Any -> T__'8804'__18
forall a b. a -> b
coe
             (case Integer -> Integer
forall a b. a -> b
coe Integer
v0 of
                Integer
0 -> (Integer -> T__'8804'__18) -> Any -> Any
forall a b. a -> b
coe Integer -> T__'8804'__18
d_'8804''45'refl_1634 (Integer -> Any
forall a b. a -> b
coe Integer
v1)
                Integer
_ -> let v3 :: Integer
v3 = Integer -> Integer -> Integer
subInt (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe (Integer
1 :: Integer)) in
                     Any -> Any
forall a b. a -> b
coe
                       ((T__'8804'__18 -> T__'8804'__18) -> T__'8804'__18 -> Any
forall a b. a -> b
coe
                          T__'8804'__18 -> T__'8804'__18
MAlonzo.Code.Data.Nat.Base.C_s'8804's_30
                          (Integer -> Integer -> T__'8804'__18
d_n'8804'm'43'n'8760'm_5052 (Integer -> Integer
forall a b. a -> b
coe Integer
v3) (Integer -> Integer
forall a b. a -> b
coe Integer
v2))))
-- Data.Nat.Properties.∣n-m∣≡[n∸m]∨[m∸n]
d_'8739'n'45'm'8739''8801''91'n'8760'm'93''8744''91'm'8760'n'93'_5066 ::
  Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
d_'8739'n'45'm'8739''8801''91'n'8760'm'93''8744''91'm'8760'n'93'_5066 :: Integer -> Integer -> T__'8846'__30
d_'8739'n'45'm'8739''8801''91'n'8760'm'93''8744''91'm'8760'n'93'_5066 Integer
v0
                                                                      Integer
v1
  = let v2 :: T__'8846'__30
v2 = Integer -> Integer -> T__'8846'__30
d_'8804''45'total_1648 (Integer -> Integer
forall a b. a -> b
coe Integer
v0) (Integer -> Integer
forall a b. a -> b
coe Integer
v1) in
    Any -> T__'8846'__30
forall a b. a -> b
coe
      (case T__'8846'__30 -> T__'8846'__30
forall a b. a -> b
coe T__'8846'__30
v2 of
         MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
v3
           -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 Any
forall a. a
erased
         MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 Any
v3
           -> (Any -> T__'8846'__30) -> Any -> Any
forall a b. a -> b
coe
                Any -> T__'8846'__30
MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                ((T__IsRelatedTo__70 -> Any) -> Any -> Any
forall a b. a -> b
coe
                   T__IsRelatedTo__70 -> Any
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_begin'45'equality__190
                   ((T_IsPreorder_70 -> Any -> T__IsRelatedTo__70) -> Any -> Any -> Any
forall a b. a -> b
coe
                      T_IsPreorder_70 -> Any -> T__IsRelatedTo__70
MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du__'8718'_346
                      (T_IsPreorder_70 -> Any
forall a b. a -> b
coe T_IsPreorder_70
d_'8804''45'isPreorder_1684)
                      ((Integer -> Integer -> Integer) -> Integer -> Integer -> Any
forall a b. a -> b
coe Integer -> Integer -> Integer
MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 Integer
v0 Integer
v1)))
         T__'8846'__30
_ -> Any
forall a. a
MAlonzo.RTE.mazUnreachableError)
-- Data.Nat.Properties.+-*-suc
d_'43''45''42''45'suc_5088 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'43''45''42''45'suc_5088 :: Integer -> Integer -> T__'8801'__12
d_'43''45''42''45'suc_5088 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.n∸m≤n
d_n'8760'm'8804'n_5094 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8760'm'8804'n_5094 :: Integer -> Integer -> T__'8804'__18
d_n'8760'm'8804'n_5094 Integer
v0 Integer
v1
  = (Integer -> Integer -> T__'8804'__18)
-> Any -> Any -> T__'8804'__18
forall a b. a -> b
coe Integer -> Integer -> T__'8804'__18
d_m'8760'n'8804'm_3676 (Integer -> Any
forall a b. a -> b
coe Integer
v1) (Integer -> Any
forall a b. a -> b
coe Integer
v0)
-- Data.Nat.Properties.∀[m≤n⇒m≢o]⇒o<n
d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'o'60'n_5106 ::
  Integer ->
  Integer ->
  (Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'o'60'n_5106 :: Integer
-> Integer
-> (Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4)
-> T__'8804'__18
d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'o'60'n_5106 Integer
v0 Integer
v1 Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
v2
  = (Integer -> Integer -> T__'8804'__18)
-> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> Integer -> T__'8804'__18
du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_1992 Integer
v0 Integer
v1
-- Data.Nat.Properties.∀[m<n⇒m≢o]⇒o≤n
d_'8704''91'm'60'n'8658'm'8802'o'93''8658'o'8804'n_5114 ::
  Integer ->
  Integer ->
  (Integer ->
   MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
   MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
   MAlonzo.Code.Data.Empty.T_'8869'_4) ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8704''91'm'60'n'8658'm'8802'o'93''8658'o'8804'n_5114 :: Integer
-> Integer
-> (Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4)
-> T__'8804'__18
d_'8704''91'm'60'n'8658'm'8802'o'93''8658'o'8804'n_5114 Integer
v0 Integer
v1 Integer -> T__'8804'__18 -> T__'8801'__12 -> T_'8869'_4
v2
  = (Integer -> Integer -> T__'8804'__18)
-> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe
      Integer -> Integer -> T__'8804'__18
du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_2020 Integer
v0 Integer
v1
-- Data.Nat.Properties.*-+-isSemiring
d_'42''45''43''45'isSemiring_5116 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1238
d_'42''45''43''45'isSemiring_5116 :: T_IsSemiring_1238
d_'42''45''43''45'isSemiring_5116
  = T_IsSemiring_1238 -> T_IsSemiring_1238
forall a b. a -> b
coe T_IsSemiring_1238
d_'43''45''42''45'isSemiring_2622
-- Data.Nat.Properties.*-+-isCommutativeSemiring
d_'42''45''43''45'isCommutativeSemiring_5118 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1344
d_'42''45''43''45'isCommutativeSemiring_5118 :: T_IsCommutativeSemiring_1344
d_'42''45''43''45'isCommutativeSemiring_5118
  = T_IsCommutativeSemiring_1344 -> T_IsCommutativeSemiring_1344
forall a b. a -> b
coe T_IsCommutativeSemiring_1344
d_'43''45''42''45'isCommutativeSemiring_2624
-- Data.Nat.Properties.*-+-semiring
d_'42''45''43''45'semiring_5120 ::
  MAlonzo.Code.Algebra.Bundles.T_Semiring_1932
d_'42''45''43''45'semiring_5120 :: T_Semiring_1932
d_'42''45''43''45'semiring_5120
  = T_Semiring_1932 -> T_Semiring_1932
forall a b. a -> b
coe T_Semiring_1932
d_'43''45''42''45'semiring_2640
-- Data.Nat.Properties.*-+-commutativeSemiring
d_'42''45''43''45'commutativeSemiring_5122 ::
  MAlonzo.Code.Algebra.Bundles.T_CommutativeSemiring_2094
d_'42''45''43''45'commutativeSemiring_5122 :: T_CommutativeSemiring_2094
d_'42''45''43''45'commutativeSemiring_5122
  = T_CommutativeSemiring_2094 -> T_CommutativeSemiring_2094
forall a b. a -> b
coe T_CommutativeSemiring_2094
d_'43''45''42''45'commutativeSemiring_2642
-- Data.Nat.Properties.m≤n⇒n⊔m≡n
d_m'8804'n'8658'n'8852'm'8801'n_5124 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8804'n'8658'n'8852'm'8801'n_5124 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8804'n'8658'n'8852'm'8801'n_5124 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.m≤n⇒n⊓m≡m
d_m'8804'n'8658'n'8851'm'8801'm_5126 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_m'8804'n'8658'n'8851'm'8801'm_5126 :: Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
d_m'8804'n'8658'n'8851'm'8801'm_5126 = Integer -> Integer -> T__'8804'__18 -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.n⊔m≡m⇒n≤m
d_n'8852'm'8801'm'8658'n'8804'm_5128 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8852'm'8801'm'8658'n'8804'm_5128 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_n'8852'm'8801'm'8658'n'8804'm_5128
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'y'8658'y'8804'x_1922
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.n⊔m≡n⇒m≤n
d_n'8852'm'8801'n'8658'm'8804'n_5130 ::
  Integer ->
  Integer ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8852'm'8801'n'8658'm'8804'n_5130 :: Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
d_n'8852'm'8801'n'8658'm'8804'n_5130
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8801'__12 -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'x'8658'x'8804'y_1890
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.n≤m⊔n
d_n'8804'm'8852'n_5132 ::
  Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_n'8804'm'8852'n_5132 :: Integer -> Integer -> T__'8804'__18
d_n'8804'm'8852'n_5132
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any -> Integer -> Integer -> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204 -> T_MinOperator_84 -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_1652
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊔-least
d_'8852''45'least_5134 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8852''45'least_5134 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8852''45'least_5134
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_2100
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-greatest
d_'8851''45'greatest_5136 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'greatest_5136 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'greatest_5136
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_2100
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊔-pres-≤m
d_'8852''45'pres'45''8804'm_5138 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8852''45'pres'45''8804'm_5138 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8852''45'pres'45''8804'm_5138
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MaxOperator_114
v1 = T_MaxOperator_114
d_'8852''45'operator_3096 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_2100
            ((T_TotalPreorder_204 -> T_TotalPreorder_204) -> Any -> Any
forall a b. a -> b
coe
               T_TotalPreorder_204 -> T_TotalPreorder_204
MAlonzo.Code.Relation.Binary.Construct.Converse.du_totalPreorder_718
               (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0))
            ((T_MaxOperator_114 -> T_MinOperator_84) -> Any -> Any
forall a b. a -> b
coe
               T_MaxOperator_114 -> T_MinOperator_84
MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_160
               (T_MaxOperator_114 -> Any
forall a b. a -> b
coe T_MaxOperator_114
v1))))
-- Data.Nat.Properties.⊓-pres-m≤
d_'8851''45'pres'45'm'8804'_5140 ::
  Integer ->
  Integer ->
  Integer ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18 ->
  MAlonzo.Code.Data.Nat.Base.T__'8804'__18
d_'8851''45'pres'45'm'8804'_5140 :: Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
d_'8851''45'pres'45'm'8804'_5140
  = let v0 :: T_TotalPreorder_204
v0 = T_TotalPreorder_204
d_'8804''45'totalPreorder_1696 in
    Any
-> Integer
-> Integer
-> Integer
-> T__'8804'__18
-> T__'8804'__18
-> T__'8804'__18
forall a b. a -> b
coe
      (let v1 :: T_MinOperator_84
v1 = T_MinOperator_84
d_'8851''45'operator_3094 in
       Any -> Any
forall a b. a -> b
coe
         ((T_TotalPreorder_204
 -> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any)
-> Any -> Any -> Any
forall a b. a -> b
coe
            T_TotalPreorder_204
-> T_MinOperator_84 -> Any -> Any -> Any -> Any -> Any -> Any
MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_2100
            (T_TotalPreorder_204 -> Any
forall a b. a -> b
coe T_TotalPreorder_204
v0) (T_MinOperator_84 -> Any
forall a b. a -> b
coe T_MinOperator_84
v1)))
-- Data.Nat.Properties.⊔-abs-⊓
d_'8852''45'abs'45''8851'_5142 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8852''45'abs'45''8851'_5142 :: Integer -> Integer -> T__'8801'__12
d_'8852''45'abs'45''8851'_5142 = Integer -> Integer -> T__'8801'__12
forall a. a
erased
-- Data.Nat.Properties.⊓-abs-⊔
d_'8851''45'abs'45''8852'_5144 ::
  Integer ->
  Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
d_'8851''45'abs'45''8852'_5144 :: Integer -> Integer -> T__'8801'__12
d_'8851''45'abs'45''8852'_5144 = Integer -> Integer -> T__'8801'__12
forall a. a
erased