{-# 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.Algebra.Structures.Biased 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.Equality
import qualified MAlonzo.Code.Agda.Builtin.Sigma
import qualified MAlonzo.Code.Agda.Primitive
import qualified MAlonzo.Code.Algebra.Consequences.Setoid
import qualified MAlonzo.Code.Algebra.Structures
import qualified MAlonzo.Code.Relation.Binary.Bundles
import qualified MAlonzo.Code.Relation.Binary.Structures

-- Algebra.Structures.Biased._._DistributesOver_
d__DistributesOver__18 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d__DistributesOver__18 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d__DistributesOver__18 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._._DistributesOverʳ_
d__DistributesOver'691'__20 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d__DistributesOver'691'__20 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d__DistributesOver'691'__20 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._._DistributesOverˡ_
d__DistributesOver'737'__22 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d__DistributesOver'737'__22 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d__DistributesOver'737'__22 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._.Commutative
d_Commutative_42 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) -> (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d_Commutative_42 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d_Commutative_42 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._.LeftIdentity
d_LeftIdentity_84 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  AgdaAny -> (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d_LeftIdentity_84 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d_LeftIdentity_84 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._.LeftZero
d_LeftZero_92 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  AgdaAny -> (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d_LeftZero_92 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d_LeftZero_92 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._.RightIdentity
d_RightIdentity_114 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  AgdaAny -> (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d_RightIdentity_114 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d_RightIdentity_114 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._.RightZero
d_RightZero_122 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  AgdaAny -> (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d_RightZero_122 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d_RightZero_122 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._.Zero
d_Zero_142 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  AgdaAny -> (AgdaAny -> AgdaAny -> AgdaAny) -> ()
d_Zero_142 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
d_Zero_142 = T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> AgdaAny
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> T_Level_18
forall a. a
erased
-- Algebra.Structures.Biased._.IsAbelianGroup
d_IsAbelianGroup_146 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsAbelianGroup_146 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
-- Algebra.Structures.Biased._.IsCommutativeMonoid
d_IsCommutativeMonoid_158 :: p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeMonoid_158 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 = ()
-- Algebra.Structures.Biased._.IsCommutativeSemiring
d_IsCommutativeSemiring_164 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeSemiring_164 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 = ()
-- Algebra.Structures.Biased._.IsMonoid
d_IsMonoid_196 :: p -> p -> p -> p -> p -> p -> T_Level_18
d_IsMonoid_196 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 = ()
-- Algebra.Structures.Biased._.IsNearSemiring
d_IsNearSemiring_200 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsNearSemiring_200 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
-- Algebra.Structures.Biased._.IsRing
d_IsRing_212 :: p -> p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsRing_212 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 p
a8 = ()
-- Algebra.Structures.Biased._.IsSemigroup
d_IsSemigroup_218 :: p -> p -> p -> p -> p -> T_Level_18
d_IsSemigroup_218 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero
d_IsSemiringWithoutAnnihilatingZero_224 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsSemiringWithoutAnnihilatingZero_224 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7
  = ()
-- Algebra.Structures.Biased._.IsSemiringWithoutOne
d_IsSemiringWithoutOne_226 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsSemiringWithoutOne_226 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
-- Algebra.Structures.Biased._.IsAbelianGroup._//_
d__'47''47'__234 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny
d__'47''47'__234 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
d__'47''47'__234 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 AgdaAny -> AgdaAny
v6 ~T_IsAbelianGroup_1132
v7
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__234 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny
v6
du__'47''47'__234 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__234 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__234 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny
v1
  = ((AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du__'47''47'__1098 ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0)
      ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v1)
-- Algebra.Structures.Biased._.IsAbelianGroup.comm
d_comm_238 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny
d_comm_238 :: T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_238 T_IsAbelianGroup_1132
v0
  = (T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_1146 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0)
-- Algebra.Structures.Biased._.IsAbelianGroup.identityʳ
d_identity'691'_242 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny
d_identity'691'_242 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
d_identity'691'_242 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny
du_identity'691'_242 T_IsAbelianGroup_1132
v7
du_identity'691'_242 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny
du_identity'691'_242 :: T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny
du_identity'691'_242 T_IsAbelianGroup_1132
v0
  = let v1 :: T_IsGroup_1036
v1
          = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsMonoid_686 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_728
         ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036
v1)))
-- Algebra.Structures.Biased._.IsAbelianGroup.identityˡ
d_identity'737'_244 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny
d_identity'737'_244 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
d_identity'737'_244 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny
du_identity'737'_244 T_IsAbelianGroup_1132
v7
du_identity'737'_244 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny
du_identity'737'_244 :: T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny
du_identity'737'_244 T_IsAbelianGroup_1132
v0
  = let v1 :: T_IsGroup_1036
v1
          = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsMonoid_686 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_726
         ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036
v1)))
-- Algebra.Structures.Biased._.IsAbelianGroup.inverseʳ
d_inverse'691'_248 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny
d_inverse'691'_248 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
d_inverse'691'_248 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny
du_inverse'691'_248 T_IsAbelianGroup_1132
v7
du_inverse'691'_248 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny
du_inverse'691'_248 :: T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny
du_inverse'691'_248 T_IsAbelianGroup_1132
v0
  = (T_IsGroup_1036 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsGroup_1036 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'691'_1108
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.inverseˡ
d_inverse'737'_250 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny
d_inverse'737'_250 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
d_inverse'737'_250 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny
du_inverse'737'_250 T_IsAbelianGroup_1132
v7
du_inverse'737'_250 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny
du_inverse'737'_250 :: T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny
du_inverse'737'_250 T_IsAbelianGroup_1132
v0
  = (T_IsGroup_1036 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsGroup_1036 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'737'_1106
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeMagma
d_isCommutativeMagma_252 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_212
d_isCommutativeMagma_252 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsCommutativeMagma_212
d_isCommutativeMagma_252 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> T_IsCommutativeMagma_212
du_isCommutativeMagma_252 T_IsAbelianGroup_1132
v7
du_isCommutativeMagma_252 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_212
du_isCommutativeMagma_252 :: T_IsAbelianGroup_1132 -> T_IsCommutativeMagma_212
du_isCommutativeMagma_252 T_IsAbelianGroup_1132
v0
  = let v1 :: t
v1
          = (T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> t
forall a b. a -> b
coe
              T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
              (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> T_IsCommutativeMagma_212
forall a b. a -> b
coe
      ((T_IsCommutativeSemigroup_548 -> T_IsCommutativeMagma_212)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsCommutativeSemigroup_548 -> T_IsCommutativeMagma_212
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_586
         ((T_IsCommutativeMonoid_736 -> T_IsCommutativeSemigroup_548)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsCommutativeMonoid_736 -> T_IsCommutativeSemigroup_548
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_786
            (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
forall a. a
v1)))
-- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeMonoid
d_isCommutativeMonoid_254 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_isCommutativeMonoid_254 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsCommutativeMonoid_736
d_isCommutativeMonoid_254 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsCommutativeMonoid_736
du_isCommutativeMonoid_254
du_isCommutativeMonoid_254 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
du_isCommutativeMonoid_254 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsCommutativeMonoid_736
du_isCommutativeMonoid_254 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 AgdaAny -> AgdaAny
v2 T_IsAbelianGroup_1132
v3
  = (T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736)
-> T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe
      T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204 T_IsAbelianGroup_1132
v3
-- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeSemigroup
d_isCommutativeSemigroup_256 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
d_isCommutativeSemigroup_256 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsCommutativeSemigroup_548
d_isCommutativeSemigroup_256 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> T_IsCommutativeSemigroup_548
du_isCommutativeSemigroup_256 T_IsAbelianGroup_1132
v7
du_isCommutativeSemigroup_256 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
du_isCommutativeSemigroup_256 :: T_IsAbelianGroup_1132 -> T_IsCommutativeSemigroup_548
du_isCommutativeSemigroup_256 T_IsAbelianGroup_1132
v0
  = (T_IsCommutativeMonoid_736 -> T_IsCommutativeSemigroup_548)
-> AgdaAny -> T_IsCommutativeSemigroup_548
forall a b. a -> b
coe
      T_IsCommutativeMonoid_736 -> T_IsCommutativeSemigroup_548
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_786
      ((T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
         (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.isGroup
d_isGroup_260 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1036
d_isGroup_260 :: T_IsAbelianGroup_1132 -> T_IsGroup_1036
d_isGroup_260 T_IsAbelianGroup_1132
v0
  = (T_IsAbelianGroup_1132 -> T_IsGroup_1036)
-> AgdaAny -> T_IsGroup_1036
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0)
-- Algebra.Structures.Biased._.IsAbelianGroup.isInvertibleMagma
d_isInvertibleMagma_262 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_924
d_isInvertibleMagma_262 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsInvertibleMagma_924
d_isInvertibleMagma_262 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> T_IsInvertibleMagma_924
du_isInvertibleMagma_262 T_IsAbelianGroup_1132
v7
du_isInvertibleMagma_262 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_924
du_isInvertibleMagma_262 :: T_IsAbelianGroup_1132 -> T_IsInvertibleMagma_924
du_isInvertibleMagma_262 T_IsAbelianGroup_1132
v0
  = (T_IsGroup_1036 -> T_IsInvertibleMagma_924)
-> AgdaAny -> T_IsInvertibleMagma_924
forall a b. a -> b
coe
      T_IsGroup_1036 -> T_IsInvertibleMagma_924
MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1122
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_264 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_976
d_isInvertibleUnitalMagma_264 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsInvertibleUnitalMagma_976
d_isInvertibleUnitalMagma_264 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> T_IsInvertibleUnitalMagma_976
du_isInvertibleUnitalMagma_264 T_IsAbelianGroup_1132
v7
du_isInvertibleUnitalMagma_264 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_976
du_isInvertibleUnitalMagma_264 :: T_IsAbelianGroup_1132 -> T_IsInvertibleUnitalMagma_976
du_isInvertibleUnitalMagma_264 T_IsAbelianGroup_1132
v0
  = (T_IsGroup_1036 -> T_IsInvertibleUnitalMagma_976)
-> AgdaAny -> T_IsInvertibleUnitalMagma_976
forall a b. a -> b
coe
      T_IsGroup_1036 -> T_IsInvertibleUnitalMagma_976
MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1124
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.isPartialEquivalence
d_isPartialEquivalence_270 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_270 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_270 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_270 T_IsAbelianGroup_1132
v7
du_isPartialEquivalence_270 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_270 :: T_IsAbelianGroup_1132 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_270 T_IsAbelianGroup_1132
v0
  = let v1 :: T_IsGroup_1036
v1
          = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_686
v2
             = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_472
v3
                = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_176
v4 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               ((T_IsEquivalence_26 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsEquivalence_26 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
                  ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v4))))))
-- Algebra.Structures.Biased._.IsAbelianGroup.isUnitalMagma
d_isUnitalMagma_274 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
d_isUnitalMagma_274 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_IsUnitalMagma_642
d_isUnitalMagma_274 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132 -> T_IsUnitalMagma_642
du_isUnitalMagma_274 T_IsAbelianGroup_1132
v7
du_isUnitalMagma_274 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
du_isUnitalMagma_274 :: T_IsAbelianGroup_1132 -> T_IsUnitalMagma_642
du_isUnitalMagma_274 T_IsAbelianGroup_1132
v0
  = let v1 :: T_IsGroup_1036
v1
          = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> T_IsUnitalMagma_642
forall a b. a -> b
coe
      ((T_IsMonoid_686 -> T_IsUnitalMagma_642) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> T_IsUnitalMagma_642
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
         ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036
v1)))
-- Algebra.Structures.Biased._.IsAbelianGroup.reflexive
d_reflexive_278 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_278 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_278 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_278 T_IsAbelianGroup_1132
v7
du_reflexive_278 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_278 :: T_IsAbelianGroup_1132
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_278 T_IsAbelianGroup_1132
v0
  = let v1 :: T_IsGroup_1036
v1
          = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_686
v2
             = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_472
v3
                = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_176
v4 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v3) in
             (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
               (\ AgdaAny
v5 AgdaAny
v6 AgdaAny
v7 ->
                  (T_IsEquivalence_26 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                    T_IsEquivalence_26 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                    ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v4))
                    AgdaAny
v5))))
-- Algebra.Structures.Biased._.IsAbelianGroup.setoid
d_setoid_280 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
d_setoid_280 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> T_Setoid_44
d_setoid_280 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7 = T_IsAbelianGroup_1132 -> T_Setoid_44
du_setoid_280 T_IsAbelianGroup_1132
v7
du_setoid_280 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
du_setoid_280 :: T_IsAbelianGroup_1132 -> T_Setoid_44
du_setoid_280 T_IsAbelianGroup_1132
v0
  = let v1 :: T_IsGroup_1036
v1
          = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> T_Setoid_44
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_686
v2
             = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_472
v3
                = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
               ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v3)))))
-- Algebra.Structures.Biased._.IsAbelianGroup.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_286 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_unique'691''45''8315''185'_286 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_unique'691''45''8315''185'_286 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'691''45''8315''185'_286 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
du_unique'691''45''8315''185'_286 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_unique'691''45''8315''185'_286 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'691''45''8315''185'_286 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 AgdaAny -> AgdaAny
v2 T_IsAbelianGroup_1132
v3
  = ((AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_IsGroup_1036
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsGroup_1036
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.du_unique'691''45''8315''185'_1120
      ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v1) ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2)
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v3))
-- Algebra.Structures.Biased._.IsAbelianGroup.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_288 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_unique'737''45''8315''185'_288 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_unique'737''45''8315''185'_288 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'737''45''8315''185'_288 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
du_unique'737''45''8315''185'_288 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_unique'737''45''8315''185'_288 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'737''45''8315''185'_288 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 AgdaAny -> AgdaAny
v2 T_IsAbelianGroup_1132
v3
  = ((AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_IsGroup_1036
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsGroup_1036
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.du_unique'737''45''8315''185'_1114
      ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v1) ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2)
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v3))
-- Algebra.Structures.Biased._.IsAbelianGroup.∙-congʳ
d_'8729''45'cong'691'_294 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_294 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_294 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_294 T_IsAbelianGroup_1132
v7
du_'8729''45'cong'691'_294 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_294 :: T_IsAbelianGroup_1132
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_294 T_IsAbelianGroup_1132
v0
  = let v1 :: T_IsGroup_1036
v1
          = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_686
v2
             = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_472
v3
                = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
               ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v3)))))
-- Algebra.Structures.Biased._.IsAbelianGroup.∙-congˡ
d_'8729''45'cong'737'_296 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_296 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1132
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_296 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1132
v7
  = T_IsAbelianGroup_1132
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_296 T_IsAbelianGroup_1132
v7
du_'8729''45'cong'737'_296 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_296 :: T_IsAbelianGroup_1132
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_296 T_IsAbelianGroup_1132
v0
  = let v1 :: T_IsGroup_1036
v1
          = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_686
v2
             = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_472
v3
                = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
               ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v3)))))
-- Algebra.Structures.Biased._.IsCommutativeMonoid.comm
d_comm_528 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736 ->
  AgdaAny -> AgdaAny -> AgdaAny
d_comm_528 :: T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_528 T_IsCommutativeMonoid_736
v0
  = (T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_748 (T_IsCommutativeMonoid_736 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v0)
-- Algebra.Structures.Biased._.IsCommutativeMonoid.isMonoid
d_isMonoid_544 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
d_isMonoid_544 :: T_IsCommutativeMonoid_736 -> T_IsMonoid_686
d_isMonoid_544 T_IsCommutativeMonoid_736
v0
  = (T_IsCommutativeMonoid_736 -> T_IsMonoid_686)
-> AgdaAny -> T_IsMonoid_686
forall a b. a -> b
coe T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (T_IsCommutativeMonoid_736 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v0)
-- Algebra.Structures.Biased._.IsCommutativeSemiring.*-comm
d_'42''45'comm_728 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678 ->
  AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'comm_728 :: T_IsCommutativeSemiring_1678 -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'comm_728 T_IsCommutativeSemiring_1678
v0
  = (T_IsCommutativeSemiring_1678 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring_1678 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'comm_1694 (T_IsCommutativeSemiring_1678 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring_1678
v0)
-- Algebra.Structures.Biased._.IsCommutativeSemiring.isSemiring
d_isSemiring_798 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1570
d_isSemiring_798 :: T_IsCommutativeSemiring_1678 -> T_IsSemiring_1570
d_isSemiring_798 T_IsCommutativeSemiring_1678
v0
  = (T_IsCommutativeSemiring_1678 -> T_IsSemiring_1570)
-> AgdaAny -> T_IsSemiring_1570
forall a b. a -> b
coe T_IsCommutativeSemiring_1678 -> T_IsSemiring_1570
MAlonzo.Code.Algebra.Structures.d_isSemiring_1692 (T_IsCommutativeSemiring_1678 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring_1678
v0)
-- Algebra.Structures.Biased._.IsMonoid.identity
d_identity_1600 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1600 :: T_IsMonoid_686 -> T_Σ_14
d_identity_1600 T_IsMonoid_686
v0
  = (T_IsMonoid_686 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsMonoid_686 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_698 (T_IsMonoid_686 -> AgdaAny
forall a b. a -> b
coe T_IsMonoid_686
v0)
-- Algebra.Structures.Biased._.IsMonoid.identityʳ
d_identity'691'_1602 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny -> AgdaAny
d_identity'691'_1602 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_686
-> AgdaAny
-> AgdaAny
d_identity'691'_1602 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_686 -> AgdaAny -> AgdaAny
du_identity'691'_1602
du_identity'691'_1602 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny -> AgdaAny
du_identity'691'_1602 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_686 -> AgdaAny -> AgdaAny
du_identity'691'_1602 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsMonoid_686
v2
  = (T_IsMonoid_686 -> AgdaAny -> AgdaAny)
-> T_IsMonoid_686 -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_728 T_IsMonoid_686
v2
-- Algebra.Structures.Biased._.IsMonoid.identityˡ
d_identity'737'_1604 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny -> AgdaAny
d_identity'737'_1604 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_686
-> AgdaAny
-> AgdaAny
d_identity'737'_1604 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_686 -> AgdaAny -> AgdaAny
du_identity'737'_1604
du_identity'737'_1604 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny -> AgdaAny
du_identity'737'_1604 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_686 -> AgdaAny -> AgdaAny
du_identity'737'_1604 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsMonoid_686
v2
  = (T_IsMonoid_686 -> AgdaAny -> AgdaAny)
-> T_IsMonoid_686 -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_726 T_IsMonoid_686
v2
-- Algebra.Structures.Biased._.IsMonoid.isPartialEquivalence
d_isPartialEquivalence_1610 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1610 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_686
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1610 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 T_IsMonoid_686
v6
  = T_IsMonoid_686 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1610 T_IsMonoid_686
v6
du_isPartialEquivalence_1610 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1610 :: T_IsMonoid_686 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1610 T_IsMonoid_686
v0
  = let v1 :: T_IsSemigroup_472
v1
          = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_176
v2 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsEquivalence_26 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsEquivalence_26 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
            ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v2))))
-- Algebra.Structures.Biased._.IsMonoid.isSemigroup
d_isSemigroup_1612 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
d_isSemigroup_1612 :: T_IsMonoid_686 -> T_IsSemigroup_472
d_isSemigroup_1612 T_IsMonoid_686
v0
  = (T_IsMonoid_686 -> T_IsSemigroup_472)
-> AgdaAny -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> AgdaAny
forall a b. a -> b
coe T_IsMonoid_686
v0)
-- Algebra.Structures.Biased._.IsMonoid.isUnitalMagma
d_isUnitalMagma_1614 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
d_isUnitalMagma_1614 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_686
-> T_IsUnitalMagma_642
d_isUnitalMagma_1614 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_686 -> T_IsUnitalMagma_642
du_isUnitalMagma_1614
du_isUnitalMagma_1614 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
du_isUnitalMagma_1614 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_686 -> T_IsUnitalMagma_642
du_isUnitalMagma_1614 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsMonoid_686
v2
  = (T_IsMonoid_686 -> T_IsUnitalMagma_642)
-> T_IsMonoid_686 -> T_IsUnitalMagma_642
forall a b. a -> b
coe T_IsMonoid_686 -> T_IsUnitalMagma_642
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730 T_IsMonoid_686
v2
-- Algebra.Structures.Biased._.IsMonoid.reflexive
d_reflexive_1618 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_1618 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_686
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_1618 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 T_IsMonoid_686
v6 = T_IsMonoid_686 -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_1618 T_IsMonoid_686
v6
du_reflexive_1618 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_1618 :: T_IsMonoid_686 -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_1618 T_IsMonoid_686
v0
  = let v1 :: T_IsSemigroup_472
v1
          = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_176
v2 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v1) in
       (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
         (\ AgdaAny
v3 AgdaAny
v4 AgdaAny
v5 ->
            (T_IsEquivalence_26 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
              T_IsEquivalence_26 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
              ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v2))
              AgdaAny
v3))
-- Algebra.Structures.Biased._.IsMonoid.setoid
d_setoid_1620 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
d_setoid_1620 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_686
-> T_Setoid_44
d_setoid_1620 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 T_IsMonoid_686
v6 = T_IsMonoid_686 -> T_Setoid_44
du_setoid_1620 T_IsMonoid_686
v6
du_setoid_1620 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
du_setoid_1620 :: T_IsMonoid_686 -> T_Setoid_44
du_setoid_1620 T_IsMonoid_686
v0
  = let v1 :: T_IsSemigroup_472
v1
          = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v0) in
    AgdaAny -> T_Setoid_44
forall a b. a -> b
coe
      ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v1)))
-- Algebra.Structures.Biased._.IsMonoid.∙-congʳ
d_'8729''45'cong'691'_1628 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_1628 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_686
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_1628 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 T_IsMonoid_686
v6
  = T_IsMonoid_686
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_1628 T_IsMonoid_686
v6
du_'8729''45'cong'691'_1628 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_1628 :: T_IsMonoid_686
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_1628 T_IsMonoid_686
v0
  = let v1 :: T_IsSemigroup_472
v1
          = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v1)))
-- Algebra.Structures.Biased._.IsMonoid.∙-congˡ
d_'8729''45'cong'737'_1630 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_1630 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_686
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_1630 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny
v5 T_IsMonoid_686
v6
  = T_IsMonoid_686
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_1630 T_IsMonoid_686
v6
du_'8729''45'cong'737'_1630 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_1630 :: T_IsMonoid_686
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_1630 T_IsMonoid_686
v0
  = let v1 :: T_IsSemigroup_472
v1
          = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v1)))
-- Algebra.Structures.Biased._.IsNearSemiring.*-assoc
d_'42''45'assoc_1700 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_1700 :: T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_1700 T_IsNearSemiring_1218
v0
  = (T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1240 (T_IsNearSemiring_1218 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218
v0)
-- Algebra.Structures.Biased._.IsNearSemiring.*-cong
d_'42''45'cong_1702 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'cong_1702 :: T_IsNearSemiring_1218
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'42''45'cong_1702 T_IsNearSemiring_1218
v0
  = (T_IsNearSemiring_1218
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1238 (T_IsNearSemiring_1218 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218
v0)
-- Algebra.Structures.Biased._.IsNearSemiring.+-isMonoid
d_'43''45'isMonoid_1728 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
d_'43''45'isMonoid_1728 :: T_IsNearSemiring_1218 -> T_IsMonoid_686
d_'43''45'isMonoid_1728 T_IsNearSemiring_1218
v0
  = (T_IsNearSemiring_1218 -> T_IsMonoid_686)
-> AgdaAny -> T_IsMonoid_686
forall a b. a -> b
coe
      T_IsNearSemiring_1218 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_'43''45'isMonoid_1236 (T_IsNearSemiring_1218 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218
v0)
-- Algebra.Structures.Biased._.IsNearSemiring.distribʳ
d_distrib'691'_1734 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_1734 :: T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_1734 T_IsNearSemiring_1218
v0
  = (T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_distrib'691'_1242 (T_IsNearSemiring_1218 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218
v0)
-- Algebra.Structures.Biased._.IsNearSemiring.zeroˡ
d_zero'737'_1750 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
  AgdaAny -> AgdaAny
d_zero'737'_1750 :: T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny
d_zero'737'_1750 T_IsNearSemiring_1218
v0
  = (T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_zero'737'_1244 (T_IsNearSemiring_1218 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1218
v0)
-- Algebra.Structures.Biased._.IsRing.*-assoc
d_'42''45'assoc_2118 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2118 :: T_IsRing_2650 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2118 T_IsRing_2650
v0
  = (T_IsRing_2650 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing_2650 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_2676 (T_IsRing_2650 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2650
v0)
-- Algebra.Structures.Biased._.IsRing.*-cong
d_'42''45'cong_2120 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'cong_2120 :: T_IsRing_2650
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'42''45'cong_2120 T_IsRing_2650
v0
  = (T_IsRing_2650
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe T_IsRing_2650
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'cong_2674 (T_IsRing_2650 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2650
v0)
-- Algebra.Structures.Biased._.IsRing.*-identity
d_'42''45'identity_2126 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_2126 :: T_IsRing_2650 -> T_Σ_14
d_'42''45'identity_2126 T_IsRing_2650
v0
  = (T_IsRing_2650 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsRing_2650 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_2678 (T_IsRing_2650 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2650
v0)
-- Algebra.Structures.Biased._.IsRing.+-isAbelianGroup
d_'43''45'isAbelianGroup_2154 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_2154 :: T_IsRing_2650 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_2154 T_IsRing_2650
v0
  = (T_IsRing_2650 -> T_IsAbelianGroup_1132)
-> AgdaAny -> T_IsAbelianGroup_1132
forall a b. a -> b
coe
      T_IsRing_2650 -> T_IsAbelianGroup_1132
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2672
      (T_IsRing_2650 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2650
v0)
-- Algebra.Structures.Biased._.IsRing.distrib
d_distrib_2184 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2184 :: T_IsRing_2650 -> T_Σ_14
d_distrib_2184 T_IsRing_2650
v0
  = (T_IsRing_2650 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsRing_2650 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_2680 (T_IsRing_2650 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2650
v0)
-- Algebra.Structures.Biased._.IsSemigroup.assoc
d_assoc_2344 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_2344 :: T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_2344 T_IsSemigroup_472
v0
  = (T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v0)
-- Algebra.Structures.Biased._.IsSemigroup.isMagma
d_isMagma_2348 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_176
d_isMagma_2348 :: T_IsSemigroup_472 -> T_IsMagma_176
d_isMagma_2348 T_IsSemigroup_472
v0
  = (T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> T_IsMagma_176
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-assoc
d_'42''45'assoc_2484 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2484 :: T_IsSemiringWithoutAnnihilatingZero_1468
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2484 T_IsSemiringWithoutAnnihilatingZero_1468
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1468
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1468
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1492 (T_IsSemiringWithoutAnnihilatingZero_1468 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1468
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-cong
d_'42''45'cong_2486 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'cong_2486 :: T_IsSemiringWithoutAnnihilatingZero_1468
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'42''45'cong_2486 T_IsSemiringWithoutAnnihilatingZero_1468
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1468
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1468
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1490 (T_IsSemiringWithoutAnnihilatingZero_1468 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1468
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-identity
d_'42''45'identity_2492 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_2492 :: T_IsSemiringWithoutAnnihilatingZero_1468 -> T_Σ_14
d_'42''45'identity_2492 T_IsSemiringWithoutAnnihilatingZero_1468
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1468 -> T_Σ_14)
-> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1468 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_1494 (T_IsSemiringWithoutAnnihilatingZero_1468 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1468
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_2522 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2522 :: T_IsSemiringWithoutAnnihilatingZero_1468
-> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2522 T_IsSemiringWithoutAnnihilatingZero_1468
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1468
 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1468
-> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1488
      (T_IsSemiringWithoutAnnihilatingZero_1468 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1468
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.distrib
d_distrib_2534 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2534 :: T_IsSemiringWithoutAnnihilatingZero_1468 -> T_Σ_14
d_distrib_2534 T_IsSemiringWithoutAnnihilatingZero_1468
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1468 -> T_Σ_14)
-> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1468 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1496 (T_IsSemiringWithoutAnnihilatingZero_1468 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1468
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.*-assoc
d_'42''45'assoc_2560 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2560 :: T_IsSemiringWithoutOne_1298
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2560 T_IsSemiringWithoutOne_1298
v0
  = (T_IsSemiringWithoutOne_1298
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1320 (T_IsSemiringWithoutOne_1298 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.*-cong
d_'42''45'cong_2562 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'cong_2562 :: T_IsSemiringWithoutOne_1298
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'42''45'cong_2562 T_IsSemiringWithoutOne_1298
v0
  = (T_IsSemiringWithoutOne_1298
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1318 (T_IsSemiringWithoutOne_1298 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_2576 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2576 :: T_IsSemiringWithoutOne_1298 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2576 T_IsSemiringWithoutOne_1298
v0
  = (T_IsSemiringWithoutOne_1298 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_1298 -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1316
      (T_IsSemiringWithoutOne_1298 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.distrib
d_distrib_2584 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2584 :: T_IsSemiringWithoutOne_1298 -> T_Σ_14
d_distrib_2584 T_IsSemiringWithoutOne_1298
v0
  = (T_IsSemiringWithoutOne_1298 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1322 (T_IsSemiringWithoutOne_1298 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.zero
d_zero_2604 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_2604 :: T_IsSemiringWithoutOne_1298 -> T_Σ_14
d_zero_2604 T_IsSemiringWithoutOne_1298
v0
  = (T_IsSemiringWithoutOne_1298 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1324 (T_IsSemiringWithoutOne_1298 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1298
v0)
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ
d_IsCommutativeMonoid'737'_2662 :: p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeMonoid'737'_2662 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 = ()
data T_IsCommutativeMonoid'737'_2662
  = C_IsCommutativeMonoid'737''46'constructor_34899 MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
                                                    (AgdaAny -> AgdaAny)
                                                    (AgdaAny -> AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ.isSemigroup
d_isSemigroup_2674 ::
  T_IsCommutativeMonoid'737'_2662 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
d_isSemigroup_2674 :: T_IsCommutativeMonoid'737'_2662 -> T_IsSemigroup_472
d_isSemigroup_2674 T_IsCommutativeMonoid'737'_2662
v0
  = case T_IsCommutativeMonoid'737'_2662 -> T_IsCommutativeMonoid'737'_2662
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662
v0 of
      C_IsCommutativeMonoid'737''46'constructor_34899 T_IsSemigroup_472
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v1
      T_IsCommutativeMonoid'737'_2662
_ -> T_IsSemigroup_472
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ.identityˡ
d_identity'737'_2676 ::
  T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny
d_identity'737'_2676 :: T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny
d_identity'737'_2676 T_IsCommutativeMonoid'737'_2662
v0
  = case T_IsCommutativeMonoid'737'_2662 -> T_IsCommutativeMonoid'737'_2662
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662
v0 of
      C_IsCommutativeMonoid'737''46'constructor_34899 T_IsSemigroup_472
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2
      T_IsCommutativeMonoid'737'_2662
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ.comm
d_comm_2678 ::
  T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2678 :: T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2678 T_IsCommutativeMonoid'737'_2662
v0
  = case T_IsCommutativeMonoid'737'_2662 -> T_IsCommutativeMonoid'737'_2662
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662
v0 of
      C_IsCommutativeMonoid'737''46'constructor_34899 T_IsSemigroup_472
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> (AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v3
      T_IsCommutativeMonoid'737'_2662
_ -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ.isCommutativeMonoid
d_isCommutativeMonoid_2680 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeMonoid'737'_2662 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_isCommutativeMonoid_2680 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'737'_2662
-> T_IsCommutativeMonoid_736
d_isCommutativeMonoid_2680 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 T_IsCommutativeMonoid'737'_2662
v6
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'737'_2662
-> T_IsCommutativeMonoid_736
du_isCommutativeMonoid_2680 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 T_IsCommutativeMonoid'737'_2662
v6
du_isCommutativeMonoid_2680 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeMonoid'737'_2662 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
du_isCommutativeMonoid_2680 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'737'_2662
-> T_IsCommutativeMonoid_736
du_isCommutativeMonoid_2680 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsCommutativeMonoid'737'_2662
v2
  = (T_IsMonoid_686
 -> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe
      T_IsMonoid_686
-> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_17695
      ((T_IsSemigroup_472 -> T_Σ_14 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> T_Σ_14 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_15873
         ((T_IsCommutativeMonoid'737'_2662 -> T_IsSemigroup_472)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662 -> T_IsSemigroup_472
d_isSemigroup_2674 (T_IsCommutativeMonoid'737'_2662 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662
v2))
         ((T_Setoid_44
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_Σ_14)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_Setoid_44
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'id'737''8658'id_352
            (let v3 :: T_IsSemigroup_472
v3 = T_IsCommutativeMonoid'737'_2662 -> T_IsSemigroup_472
d_isSemigroup_2674 (T_IsCommutativeMonoid'737'_2662 -> T_IsCommutativeMonoid'737'_2662
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662
v2) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                  ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v3))))
            ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) ((T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2678 (T_IsCommutativeMonoid'737'_2662 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662
v2)) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v1)
            ((T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny
d_identity'737'_2676 (T_IsCommutativeMonoid'737'_2662 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662
v2))))
      ((T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2678 (T_IsCommutativeMonoid'737'_2662 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2662
v2))
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ
d_IsCommutativeMonoid'691'_2716 :: p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeMonoid'691'_2716 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 = ()
data T_IsCommutativeMonoid'691'_2716
  = C_IsCommutativeMonoid'691''46'constructor_36341 MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
                                                    (AgdaAny -> AgdaAny)
                                                    (AgdaAny -> AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ.isSemigroup
d_isSemigroup_2728 ::
  T_IsCommutativeMonoid'691'_2716 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
d_isSemigroup_2728 :: T_IsCommutativeMonoid'691'_2716 -> T_IsSemigroup_472
d_isSemigroup_2728 T_IsCommutativeMonoid'691'_2716
v0
  = case T_IsCommutativeMonoid'691'_2716 -> T_IsCommutativeMonoid'691'_2716
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716
v0 of
      C_IsCommutativeMonoid'691''46'constructor_36341 T_IsSemigroup_472
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v1
      T_IsCommutativeMonoid'691'_2716
_ -> T_IsSemigroup_472
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ.identityʳ
d_identity'691'_2730 ::
  T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny
d_identity'691'_2730 :: T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny
d_identity'691'_2730 T_IsCommutativeMonoid'691'_2716
v0
  = case T_IsCommutativeMonoid'691'_2716 -> T_IsCommutativeMonoid'691'_2716
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716
v0 of
      C_IsCommutativeMonoid'691''46'constructor_36341 T_IsSemigroup_472
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2
      T_IsCommutativeMonoid'691'_2716
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ.comm
d_comm_2732 ::
  T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2732 :: T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2732 T_IsCommutativeMonoid'691'_2716
v0
  = case T_IsCommutativeMonoid'691'_2716 -> T_IsCommutativeMonoid'691'_2716
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716
v0 of
      C_IsCommutativeMonoid'691''46'constructor_36341 T_IsSemigroup_472
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> (AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v3
      T_IsCommutativeMonoid'691'_2716
_ -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ.isCommutativeMonoid
d_isCommutativeMonoid_2734 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeMonoid'691'_2716 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_isCommutativeMonoid_2734 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'691'_2716
-> T_IsCommutativeMonoid_736
d_isCommutativeMonoid_2734 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 T_IsCommutativeMonoid'691'_2716
v6
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'691'_2716
-> T_IsCommutativeMonoid_736
du_isCommutativeMonoid_2734 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 T_IsCommutativeMonoid'691'_2716
v6
du_isCommutativeMonoid_2734 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeMonoid'691'_2716 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
du_isCommutativeMonoid_2734 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'691'_2716
-> T_IsCommutativeMonoid_736
du_isCommutativeMonoid_2734 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsCommutativeMonoid'691'_2716
v2
  = (T_IsMonoid_686
 -> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe
      T_IsMonoid_686
-> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_17695
      ((T_IsSemigroup_472 -> T_Σ_14 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> T_Σ_14 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_15873
         ((T_IsCommutativeMonoid'691'_2716 -> T_IsSemigroup_472)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716 -> T_IsSemigroup_472
d_isSemigroup_2728 (T_IsCommutativeMonoid'691'_2716 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716
v2))
         ((T_Setoid_44
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_Σ_14)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_Setoid_44
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'id'691''8658'id_356
            (let v3 :: T_IsSemigroup_472
v3 = T_IsCommutativeMonoid'691'_2716 -> T_IsSemigroup_472
d_isSemigroup_2728 (T_IsCommutativeMonoid'691'_2716 -> T_IsCommutativeMonoid'691'_2716
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716
v2) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                  ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v3))))
            ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) ((T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2732 (T_IsCommutativeMonoid'691'_2716 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716
v2)) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v1)
            ((T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny
d_identity'691'_2730 (T_IsCommutativeMonoid'691'_2716 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716
v2))))
      ((T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2732 (T_IsCommutativeMonoid'691'_2716 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2716
v2))
-- Algebra.Structures.Biased.IsSemiringWithoutOne*
d_IsSemiringWithoutOne'42'_2772 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsSemiringWithoutOne'42'_2772 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
data T_IsSemiringWithoutOne'42'_2772
  = C_IsSemiringWithoutOne'42''46'constructor_37821 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
                                                    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
                                                    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_2788 ::
  T_IsSemiringWithoutOne'42'_2772 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2788 :: T_IsSemiringWithoutOne'42'_2772 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2788 T_IsSemiringWithoutOne'42'_2772
v0
  = case T_IsSemiringWithoutOne'42'_2772 -> T_IsSemiringWithoutOne'42'_2772
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0 of
      C_IsSemiringWithoutOne'42''46'constructor_37821 T_IsCommutativeMonoid_736
v1 T_IsSemigroup_472
v2 T_Σ_14
v3 T_Σ_14
v4
        -> T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v1
      T_IsSemiringWithoutOne'42'_2772
_ -> T_IsCommutativeMonoid_736
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.*-isSemigroup
d_'42''45'isSemigroup_2790 ::
  T_IsSemiringWithoutOne'42'_2772 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
d_'42''45'isSemigroup_2790 :: T_IsSemiringWithoutOne'42'_2772 -> T_IsSemigroup_472
d_'42''45'isSemigroup_2790 T_IsSemiringWithoutOne'42'_2772
v0
  = case T_IsSemiringWithoutOne'42'_2772 -> T_IsSemiringWithoutOne'42'_2772
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0 of
      C_IsSemiringWithoutOne'42''46'constructor_37821 T_IsCommutativeMonoid_736
v1 T_IsSemigroup_472
v2 T_Σ_14
v3 T_Σ_14
v4
        -> T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v2
      T_IsSemiringWithoutOne'42'_2772
_ -> T_IsSemigroup_472
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.distrib
d_distrib_2792 ::
  T_IsSemiringWithoutOne'42'_2772 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2792 :: T_IsSemiringWithoutOne'42'_2772 -> T_Σ_14
d_distrib_2792 T_IsSemiringWithoutOne'42'_2772
v0
  = case T_IsSemiringWithoutOne'42'_2772 -> T_IsSemiringWithoutOne'42'_2772
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0 of
      C_IsSemiringWithoutOne'42''46'constructor_37821 T_IsCommutativeMonoid_736
v1 T_IsSemigroup_472
v2 T_Σ_14
v3 T_Σ_14
v4
        -> T_Σ_14 -> T_Σ_14
forall a b. a -> b
coe T_Σ_14
v3
      T_IsSemiringWithoutOne'42'_2772
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.zero
d_zero_2794 ::
  T_IsSemiringWithoutOne'42'_2772 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_2794 :: T_IsSemiringWithoutOne'42'_2772 -> T_Σ_14
d_zero_2794 T_IsSemiringWithoutOne'42'_2772
v0
  = case T_IsSemiringWithoutOne'42'_2772 -> T_IsSemiringWithoutOne'42'_2772
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0 of
      C_IsSemiringWithoutOne'42''46'constructor_37821 T_IsCommutativeMonoid_736
v1 T_IsSemigroup_472
v2 T_Σ_14
v3 T_Σ_14
v4
        -> T_Σ_14 -> T_Σ_14
forall a b. a -> b
coe T_Σ_14
v4
      T_IsSemiringWithoutOne'42'_2772
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.isSemiringWithoutOne
d_isSemiringWithoutOne_2796 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsSemiringWithoutOne'42'_2772 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298
d_isSemiringWithoutOne_2796 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsSemiringWithoutOne'42'_2772
-> T_IsSemiringWithoutOne_1298
d_isSemiringWithoutOne_2796 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 T_IsSemiringWithoutOne'42'_2772
v7
  = T_IsSemiringWithoutOne'42'_2772 -> T_IsSemiringWithoutOne_1298
du_isSemiringWithoutOne_2796 T_IsSemiringWithoutOne'42'_2772
v7
du_isSemiringWithoutOne_2796 ::
  T_IsSemiringWithoutOne'42'_2772 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298
du_isSemiringWithoutOne_2796 :: T_IsSemiringWithoutOne'42'_2772 -> T_IsSemiringWithoutOne_1298
du_isSemiringWithoutOne_2796 T_IsSemiringWithoutOne'42'_2772
v0
  = (T_IsCommutativeMonoid_736
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutOne_1298)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutOne_1298
forall a b. a -> b
coe
      T_IsCommutativeMonoid_736
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutOne_1298
MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutOne'46'constructor_37629
      ((T_IsSemiringWithoutOne'42'_2772 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2788 (T_IsSemiringWithoutOne'42'_2772 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0))
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsSemiringWithoutOne'42'_2772 -> T_IsSemigroup_472)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772 -> T_IsSemigroup_472
d_'42''45'isSemigroup_2790 (T_IsSemiringWithoutOne'42'_2772 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0))))
      ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
         ((T_IsSemiringWithoutOne'42'_2772 -> T_IsSemigroup_472)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772 -> T_IsSemigroup_472
d_'42''45'isSemigroup_2790 (T_IsSemiringWithoutOne'42'_2772 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0)))
      ((T_IsSemiringWithoutOne'42'_2772 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772 -> T_Σ_14
d_distrib_2792 (T_IsSemiringWithoutOne'42'_2772 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0)) ((T_IsSemiringWithoutOne'42'_2772 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772 -> T_Σ_14
d_zero_2794 (T_IsSemiringWithoutOne'42'_2772 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2772
v0))
-- Algebra.Structures.Biased.IsNearSemiring*
d_IsNearSemiring'42'_2834 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsNearSemiring'42'_2834 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
data T_IsNearSemiring'42'_2834
  = C_IsNearSemiring'42''46'constructor_39635 MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
                                              MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
                                              (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
                                              (AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsNearSemiring*.+-isMonoid
d_'43''45'isMonoid_2850 ::
  T_IsNearSemiring'42'_2834 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
d_'43''45'isMonoid_2850 :: T_IsNearSemiring'42'_2834 -> T_IsMonoid_686
d_'43''45'isMonoid_2850 T_IsNearSemiring'42'_2834
v0
  = case T_IsNearSemiring'42'_2834 -> T_IsNearSemiring'42'_2834
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0 of
      C_IsNearSemiring'42''46'constructor_39635 T_IsMonoid_686
v1 T_IsSemigroup_472
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1
      T_IsNearSemiring'42'_2834
_ -> T_IsMonoid_686
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsNearSemiring*.*-isSemigroup
d_'42''45'isSemigroup_2852 ::
  T_IsNearSemiring'42'_2834 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
d_'42''45'isSemigroup_2852 :: T_IsNearSemiring'42'_2834 -> T_IsSemigroup_472
d_'42''45'isSemigroup_2852 T_IsNearSemiring'42'_2834
v0
  = case T_IsNearSemiring'42'_2834 -> T_IsNearSemiring'42'_2834
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0 of
      C_IsNearSemiring'42''46'constructor_39635 T_IsMonoid_686
v1 T_IsSemigroup_472
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v2
      T_IsNearSemiring'42'_2834
_ -> T_IsSemigroup_472
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsNearSemiring*.distribʳ
d_distrib'691'_2854 ::
  T_IsNearSemiring'42'_2834 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2854 :: T_IsNearSemiring'42'_2834
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2854 T_IsNearSemiring'42'_2834
v0
  = case T_IsNearSemiring'42'_2834 -> T_IsNearSemiring'42'_2834
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0 of
      C_IsNearSemiring'42''46'constructor_39635 T_IsMonoid_686
v1 T_IsSemigroup_472
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3
      T_IsNearSemiring'42'_2834
_ -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsNearSemiring*.zeroˡ
d_zero'737'_2856 :: T_IsNearSemiring'42'_2834 -> AgdaAny -> AgdaAny
d_zero'737'_2856 :: T_IsNearSemiring'42'_2834 -> AgdaAny -> AgdaAny
d_zero'737'_2856 T_IsNearSemiring'42'_2834
v0
  = case T_IsNearSemiring'42'_2834 -> T_IsNearSemiring'42'_2834
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0 of
      C_IsNearSemiring'42''46'constructor_39635 T_IsMonoid_686
v1 T_IsSemigroup_472
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v4
      T_IsNearSemiring'42'_2834
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsNearSemiring*.isNearSemiring
d_isNearSemiring_2858 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsNearSemiring'42'_2834 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218
d_isNearSemiring_2858 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsNearSemiring'42'_2834
-> T_IsNearSemiring_1218
d_isNearSemiring_2858 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 T_IsNearSemiring'42'_2834
v7
  = T_IsNearSemiring'42'_2834 -> T_IsNearSemiring_1218
du_isNearSemiring_2858 T_IsNearSemiring'42'_2834
v7
du_isNearSemiring_2858 ::
  T_IsNearSemiring'42'_2834 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218
du_isNearSemiring_2858 :: T_IsNearSemiring'42'_2834 -> T_IsNearSemiring_1218
du_isNearSemiring_2858 T_IsNearSemiring'42'_2834
v0
  = (T_IsMonoid_686
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> T_IsNearSemiring_1218)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsNearSemiring_1218
forall a b. a -> b
coe
      T_IsMonoid_686
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> T_IsNearSemiring_1218
MAlonzo.Code.Algebra.Structures.C_IsNearSemiring'46'constructor_35025
      ((T_IsNearSemiring'42'_2834 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834 -> T_IsMonoid_686
d_'43''45'isMonoid_2850 (T_IsNearSemiring'42'_2834 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0))
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsNearSemiring'42'_2834 -> T_IsSemigroup_472)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834 -> T_IsSemigroup_472
d_'42''45'isSemigroup_2852 (T_IsNearSemiring'42'_2834 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0))))
      ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
         ((T_IsNearSemiring'42'_2834 -> T_IsSemigroup_472)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834 -> T_IsSemigroup_472
d_'42''45'isSemigroup_2852 (T_IsNearSemiring'42'_2834 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0)))
      ((T_IsNearSemiring'42'_2834
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2854 (T_IsNearSemiring'42'_2834 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0)) ((T_IsNearSemiring'42'_2834 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834 -> AgdaAny -> AgdaAny
d_zero'737'_2856 (T_IsNearSemiring'42'_2834 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2834
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*
d_IsSemiringWithoutAnnihilatingZero'42'_2898 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsSemiringWithoutAnnihilatingZero'42'_2898 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6
                                             p
a7
  = ()
data T_IsSemiringWithoutAnnihilatingZero'42'_2898
  = C_IsSemiringWithoutAnnihilatingZero'42''46'constructor_41443 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                                 MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
                                                                 MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_2914 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2914 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2914 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = case T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0 of
      C_IsSemiringWithoutAnnihilatingZero'42''46'constructor_41443 T_IsCommutativeMonoid_736
v1 T_IsMonoid_686
v2 T_Σ_14
v3
        -> T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v1
      T_IsSemiringWithoutAnnihilatingZero'42'_2898
_ -> T_IsCommutativeMonoid_736
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.*-isMonoid
d_'42''45'isMonoid_2916 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
d_'42''45'isMonoid_2916 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = case T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0 of
      C_IsSemiringWithoutAnnihilatingZero'42''46'constructor_41443 T_IsCommutativeMonoid_736
v1 T_IsMonoid_686
v2 T_Σ_14
v3
        -> T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v2
      T_IsSemiringWithoutAnnihilatingZero'42'_2898
_ -> T_IsMonoid_686
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.distrib
d_distrib_2918 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2918 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_Σ_14
d_distrib_2918 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = case T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0 of
      C_IsSemiringWithoutAnnihilatingZero'42''46'constructor_41443 T_IsCommutativeMonoid_736
v1 T_IsMonoid_686
v2 T_Σ_14
v3
        -> T_Σ_14 -> T_Σ_14
forall a b. a -> b
coe T_Σ_14
v3
      T_IsSemiringWithoutAnnihilatingZero'42'_2898
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_2920 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468
d_isSemiringWithoutAnnihilatingZero_2920 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero_1468
d_isSemiringWithoutAnnihilatingZero_2920 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5
                                         ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero_1468
du_isSemiringWithoutAnnihilatingZero_2920 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_isSemiringWithoutAnnihilatingZero_2920 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468
du_isSemiringWithoutAnnihilatingZero_2920 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero_1468
du_isSemiringWithoutAnnihilatingZero_2920 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = (T_IsCommutativeMonoid_736
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutAnnihilatingZero_1468)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero_1468
forall a b. a -> b
coe
      T_IsCommutativeMonoid_736
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutAnnihilatingZero_1468
MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutAnnihilatingZero'46'constructor_43811
      ((T_IsSemiringWithoutAnnihilatingZero'42'_2898
 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2914 (T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0))
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0)))))
      ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0))))
      ((T_IsMonoid_686 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_698
         ((T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0)))
      ((T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_Σ_14)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_Σ_14
d_distrib_2918 (T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.identityʳ
d_identity'691'_2932 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
d_identity'691'_2932 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny
-> AgdaAny
d_identity'691'_2932 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
du_identity'691'_2932 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_identity'691'_2932 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
du_identity'691'_2932 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
du_identity'691'_2932 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = (T_IsMonoid_686 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_728
      ((T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.identityˡ
d_identity'737'_2934 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
d_identity'737'_2934 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny
-> AgdaAny
d_identity'737'_2934 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
du_identity'737'_2934 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_identity'737'_2934 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
du_identity'737'_2934 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
du_identity'737'_2934 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = (T_IsMonoid_686 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_726
      ((T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.isPartialEquivalence
d_isPartialEquivalence_2940 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_2940 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_2940 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsPartialEquivalence_16
du_isPartialEquivalence_2940 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_isPartialEquivalence_2940 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_2940 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsPartialEquivalence_16
du_isPartialEquivalence_2940 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_176
v3 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsEquivalence_26 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsEquivalence_26 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
               ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v3)))))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.isUnitalMagma
d_isUnitalMagma_2944 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
d_isUnitalMagma_2944 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsUnitalMagma_642
d_isUnitalMagma_2944 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsUnitalMagma_642
du_isUnitalMagma_2944 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_isUnitalMagma_2944 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
du_isUnitalMagma_2944 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsUnitalMagma_642
du_isUnitalMagma_2944 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = (T_IsMonoid_686 -> T_IsUnitalMagma_642)
-> AgdaAny -> T_IsUnitalMagma_642
forall a b. a -> b
coe
      T_IsMonoid_686 -> T_IsUnitalMagma_642
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
      ((T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.reflexive
d_reflexive_2948 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_2948 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_2948 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_2948 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_reflexive_2948 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_2948 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_2948 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_176
v3 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v2) in
          (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
            (\ AgdaAny
v4 AgdaAny
v5 AgdaAny
v6 ->
               (T_IsEquivalence_26 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                 T_IsEquivalence_26 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                 ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v3))
                 AgdaAny
v4)))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.setoid
d_setoid_2950 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
d_setoid_2950 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_Setoid_44
d_setoid_2950 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_Setoid_44
du_setoid_2950 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_setoid_2950 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
du_setoid_2950 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_Setoid_44
du_setoid_2950 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0) in
    AgdaAny -> T_Setoid_44
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.∙-congʳ
d_'8729''45'cong'691'_2958 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_2958 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_2958 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_2958 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_'8729''45'cong'691'_2958 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_2958 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_2958 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.∙-congˡ
d_'8729''45'cong'737'_2960 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_2960 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_2960 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny
v6 ~AgdaAny
v7 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_2960 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v8
du_'8729''45'cong'737'_2960 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_2960 :: T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_2960 T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> T_IsMonoid_686
d_'42''45'isMonoid_2916 (T_IsSemiringWithoutAnnihilatingZero'42'_2898
-> T_IsSemiringWithoutAnnihilatingZero'42'_2898
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_2898
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ
d_IsCommutativeSemiring'737'_2970 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeSemiring'737'_2970 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 = ()
data T_IsCommutativeSemiring'737'_2970
  = C_IsCommutativeSemiring'737''46'constructor_43731 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                      MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                      (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
                                                      (AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_2988 ::
  T_IsCommutativeSemiring'737'_2970 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2988 :: T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2988 T_IsCommutativeSemiring'737'_2970
v0
  = case T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring'737'_2970
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v0 of
      C_IsCommutativeSemiring'737''46'constructor_43731 T_IsCommutativeMonoid_736
v1 T_IsCommutativeMonoid_736
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4
        -> T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v1
      T_IsCommutativeSemiring'737'_2970
_ -> T_IsCommutativeMonoid_736
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_2990 ::
  T_IsCommutativeSemiring'737'_2970 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_2990 :: T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_2990 T_IsCommutativeSemiring'737'_2970
v0
  = case T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring'737'_2970
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v0 of
      C_IsCommutativeSemiring'737''46'constructor_43731 T_IsCommutativeMonoid_736
v1 T_IsCommutativeMonoid_736
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4
        -> T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v2
      T_IsCommutativeSemiring'737'_2970
_ -> T_IsCommutativeMonoid_736
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.distribʳ
d_distrib'691'_2992 ::
  T_IsCommutativeSemiring'737'_2970 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2992 :: T_IsCommutativeSemiring'737'_2970
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2992 T_IsCommutativeSemiring'737'_2970
v0
  = case T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring'737'_2970
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v0 of
      C_IsCommutativeSemiring'737''46'constructor_43731 T_IsCommutativeMonoid_736
v1 T_IsCommutativeMonoid_736
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4
        -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3
      T_IsCommutativeSemiring'737'_2970
_ -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.zeroˡ
d_zero'737'_2994 ::
  T_IsCommutativeSemiring'737'_2970 -> AgdaAny -> AgdaAny
d_zero'737'_2994 :: T_IsCommutativeSemiring'737'_2970 -> AgdaAny -> AgdaAny
d_zero'737'_2994 T_IsCommutativeSemiring'737'_2970
v0
  = case T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring'737'_2970
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v0 of
      C_IsCommutativeSemiring'737''46'constructor_43731 T_IsCommutativeMonoid_736
v1 T_IsCommutativeMonoid_736
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4
        -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v4
      T_IsCommutativeSemiring'737'_2970
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.isCommutativeSemiring
d_isCommutativeSemiring_2996 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsCommutativeSemiring'737'_2970 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
d_isCommutativeSemiring_2996 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring_1678
d_isCommutativeSemiring_2996 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny
v6 ~AgdaAny
v7 T_IsCommutativeSemiring'737'_2970
v8
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring_1678
du_isCommutativeSemiring_2996 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny
v6 T_IsCommutativeSemiring'737'_2970
v8
du_isCommutativeSemiring_2996 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeSemiring'737'_2970 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
du_isCommutativeSemiring_2996 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring_1678
du_isCommutativeSemiring_2996 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny
v2 T_IsCommutativeSemiring'737'_2970
v3
  = (T_IsSemiring_1570
 -> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeSemiring_1678)
-> AgdaAny -> AgdaAny -> T_IsCommutativeSemiring_1678
forall a b. a -> b
coe
      T_IsSemiring_1570
-> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeSemiring_1678
MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemiring'46'constructor_51895
      ((T_IsSemiringWithoutAnnihilatingZero_1468
 -> T_Σ_14 -> T_IsSemiring_1570)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1468
-> T_Σ_14 -> T_IsSemiring_1570
MAlonzo.Code.Algebra.Structures.C_IsSemiring'46'constructor_48071
         ((T_IsCommutativeMonoid_736
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutAnnihilatingZero_1468)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsCommutativeMonoid_736
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutAnnihilatingZero_1468
MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutAnnihilatingZero'46'constructor_43811
            ((T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2988 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3))
            ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
               ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
                  ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                     ((T_IsCommutativeMonoid_736 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                        ((T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_2990 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3))))))
            ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
               ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                  ((T_IsCommutativeMonoid_736 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                     ((T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_2990 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3)))))
            ((T_IsMonoid_686 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_698
               ((T_IsCommutativeMonoid_736 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                  ((T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_2990 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3))))
            ((T_Setoid_44
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
               T_Setoid_44
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'distr'691''8658'distr_528
               (let v4 :: T_IsCommutativeMonoid_736
v4 = T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2988 (T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring'737'_2970
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (let v5 :: T_IsMonoid_686
v5
                         = T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v4) in
                   AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     (let v6 :: T_IsSemigroup_472
v6
                            = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v5) in
                      AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                           ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v6))))))
               ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1) ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0)
               ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
                  ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
                     ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                        ((T_IsCommutativeMonoid_736 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                           ((T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2988 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3))))))
               ((T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_748
                  ((T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_2990 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3)))
               ((T_IsCommutativeSemiring'737'_2970
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2992 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3))))
         ((T_Setoid_44
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_Σ_14)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_Setoid_44
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'ze'737''8658'ze_372
            (let v4 :: T_IsCommutativeMonoid_736
v4 = T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_2988 (T_IsCommutativeSemiring'737'_2970
-> T_IsCommutativeSemiring'737'_2970
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: T_IsMonoid_686
v5
                      = T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v4) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (let v6 :: T_IsSemigroup_472
v6
                         = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v5) in
                   AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                        ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v6))))))
            ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1)
            ((T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_748
               ((T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_2990 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3)))
            (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v2) ((T_IsCommutativeSemiring'737'_2970 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> AgdaAny -> AgdaAny
d_zero'737'_2994 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3))))
      ((T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_748
         ((T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_2990 (T_IsCommutativeSemiring'737'_2970 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_2970
v3)))
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ
d_IsCommutativeSemiring'691'_3098 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeSemiring'691'_3098 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 = ()
data T_IsCommutativeSemiring'691'_3098
  = C_IsCommutativeSemiring'691''46'constructor_48791 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                      MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                      (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
                                                      (AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_3116 ::
  T_IsCommutativeSemiring'691'_3098 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_3116 :: T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_3116 T_IsCommutativeSemiring'691'_3098
v0
  = case T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring'691'_3098
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v0 of
      C_IsCommutativeSemiring'691''46'constructor_48791 T_IsCommutativeMonoid_736
v1 T_IsCommutativeMonoid_736
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4
        -> T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v1
      T_IsCommutativeSemiring'691'_3098
_ -> T_IsCommutativeMonoid_736
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_3118 ::
  T_IsCommutativeSemiring'691'_3098 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_3118 :: T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_3118 T_IsCommutativeSemiring'691'_3098
v0
  = case T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring'691'_3098
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v0 of
      C_IsCommutativeSemiring'691''46'constructor_48791 T_IsCommutativeMonoid_736
v1 T_IsCommutativeMonoid_736
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4
        -> T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v2
      T_IsCommutativeSemiring'691'_3098
_ -> T_IsCommutativeMonoid_736
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.distribˡ
d_distrib'737'_3120 ::
  T_IsCommutativeSemiring'691'_3098 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'737'_3120 :: T_IsCommutativeSemiring'691'_3098
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'737'_3120 T_IsCommutativeSemiring'691'_3098
v0
  = case T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring'691'_3098
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v0 of
      C_IsCommutativeSemiring'691''46'constructor_48791 T_IsCommutativeMonoid_736
v1 T_IsCommutativeMonoid_736
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4
        -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3
      T_IsCommutativeSemiring'691'_3098
_ -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.zeroʳ
d_zero'691'_3122 ::
  T_IsCommutativeSemiring'691'_3098 -> AgdaAny -> AgdaAny
d_zero'691'_3122 :: T_IsCommutativeSemiring'691'_3098 -> AgdaAny -> AgdaAny
d_zero'691'_3122 T_IsCommutativeSemiring'691'_3098
v0
  = case T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring'691'_3098
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v0 of
      C_IsCommutativeSemiring'691''46'constructor_48791 T_IsCommutativeMonoid_736
v1 T_IsCommutativeMonoid_736
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4
        -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v4
      T_IsCommutativeSemiring'691'_3098
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.isCommutativeSemiring
d_isCommutativeSemiring_3124 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsCommutativeSemiring'691'_3098 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
d_isCommutativeSemiring_3124 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring_1678
d_isCommutativeSemiring_3124 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny
v6 ~AgdaAny
v7 T_IsCommutativeSemiring'691'_3098
v8
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring_1678
du_isCommutativeSemiring_3124 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny
v6 T_IsCommutativeSemiring'691'_3098
v8
du_isCommutativeSemiring_3124 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeSemiring'691'_3098 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
du_isCommutativeSemiring_3124 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring_1678
du_isCommutativeSemiring_3124 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny
v2 T_IsCommutativeSemiring'691'_3098
v3
  = (T_IsSemiring_1570
 -> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeSemiring_1678)
-> AgdaAny -> AgdaAny -> T_IsCommutativeSemiring_1678
forall a b. a -> b
coe
      T_IsSemiring_1570
-> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeSemiring_1678
MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemiring'46'constructor_51895
      ((T_IsSemiringWithoutAnnihilatingZero_1468
 -> T_Σ_14 -> T_IsSemiring_1570)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1468
-> T_Σ_14 -> T_IsSemiring_1570
MAlonzo.Code.Algebra.Structures.C_IsSemiring'46'constructor_48071
         ((T_IsCommutativeMonoid_736
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutAnnihilatingZero_1468)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsCommutativeMonoid_736
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutAnnihilatingZero_1468
MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutAnnihilatingZero'46'constructor_43811
            ((T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_3116 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3))
            ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
               ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
                  ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                     ((T_IsCommutativeMonoid_736 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                        ((T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_3118 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3))))))
            ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
               ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                  ((T_IsCommutativeMonoid_736 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                     ((T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_3118 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3)))))
            ((T_IsMonoid_686 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_698
               ((T_IsCommutativeMonoid_736 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                  ((T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_3118 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3))))
            ((T_Setoid_44
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
               T_Setoid_44
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'distr'737''8658'distr_524
               (let v4 :: T_IsCommutativeMonoid_736
v4 = T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_3116 (T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring'691'_3098
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (let v5 :: T_IsMonoid_686
v5
                         = T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v4) in
                   AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     (let v6 :: T_IsSemigroup_472
v6
                            = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v5) in
                      AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                           ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v6))))))
               ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1) ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0)
               ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
                  ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
                     ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                        ((T_IsCommutativeMonoid_736 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                           ((T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_3116 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3))))))
               ((T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_748
                  ((T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_3118 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3)))
               ((T_IsCommutativeSemiring'691'_3098
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'737'_3120 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3))))
         ((T_Setoid_44
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_Σ_14)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_Setoid_44
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'ze'691''8658'ze_376
            (let v4 :: T_IsCommutativeMonoid_736
v4 = T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'43''45'isCommutativeMonoid_3116 (T_IsCommutativeSemiring'691'_3098
-> T_IsCommutativeSemiring'691'_3098
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: T_IsMonoid_686
v5
                      = T_IsCommutativeMonoid_736 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (T_IsCommutativeMonoid_736 -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe T_IsCommutativeMonoid_736
v4) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (let v6 :: T_IsSemigroup_472
v6
                         = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v5) in
                   AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                        ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v6))))))
            ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1)
            ((T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_748
               ((T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_3118 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3)))
            (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v2) ((T_IsCommutativeSemiring'691'_3098 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> AgdaAny -> AgdaAny
d_zero'691'_3122 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3))))
      ((T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsCommutativeMonoid_736 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_748
         ((T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098 -> T_IsCommutativeMonoid_736
d_'42''45'isCommutativeMonoid_3118 (T_IsCommutativeSemiring'691'_3098 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3098
v3)))
-- Algebra.Structures.Biased.IsRing*
d_IsRing'42'_3228 :: p -> p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsRing'42'_3228 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 p
a8 = ()
data T_IsRing'42'_3228
  = C_IsRing'42''46'constructor_53915 MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
                                      MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
                                      MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
                                      MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
-- Algebra.Structures.Biased.IsRing*.+-isAbelianGroup
d_'43''45'isAbelianGroup_3248 ::
  T_IsRing'42'_3228 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3248 :: T_IsRing'42'_3228 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3248 T_IsRing'42'_3228
v0
  = case T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0 of
      C_IsRing'42''46'constructor_53915 T_IsAbelianGroup_1132
v1 T_IsMonoid_686
v2 T_Σ_14
v3 T_Σ_14
v4 -> T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1
      T_IsRing'42'_3228
_ -> T_IsAbelianGroup_1132
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRing*.*-isMonoid
d_'42''45'isMonoid_3250 ::
  T_IsRing'42'_3228 -> MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
d_'42''45'isMonoid_3250 :: T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 T_IsRing'42'_3228
v0
  = case T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0 of
      C_IsRing'42''46'constructor_53915 T_IsAbelianGroup_1132
v1 T_IsMonoid_686
v2 T_Σ_14
v3 T_Σ_14
v4 -> T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v2
      T_IsRing'42'_3228
_ -> T_IsMonoid_686
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRing*.distrib
d_distrib_3252 ::
  T_IsRing'42'_3228 -> MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_3252 :: T_IsRing'42'_3228 -> T_Σ_14
d_distrib_3252 T_IsRing'42'_3228
v0
  = case T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0 of
      C_IsRing'42''46'constructor_53915 T_IsAbelianGroup_1132
v1 T_IsMonoid_686
v2 T_Σ_14
v3 T_Σ_14
v4 -> T_Σ_14 -> T_Σ_14
forall a b. a -> b
coe T_Σ_14
v3
      T_IsRing'42'_3228
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRing*.zero
d_zero_3254 ::
  T_IsRing'42'_3228 -> MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_3254 :: T_IsRing'42'_3228 -> T_Σ_14
d_zero_3254 T_IsRing'42'_3228
v0
  = case T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0 of
      C_IsRing'42''46'constructor_53915 T_IsAbelianGroup_1132
v1 T_IsMonoid_686
v2 T_Σ_14
v3 T_Σ_14
v4 -> T_Σ_14 -> T_Σ_14
forall a b. a -> b
coe T_Σ_14
v4
      T_IsRing'42'_3228
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRing*.isRing
d_isRing_3256 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRing'42'_3228 -> MAlonzo.Code.Algebra.Structures.T_IsRing_2650
d_isRing_3256 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> T_IsRing_2650
d_isRing_3256 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228 -> T_IsRing_2650
du_isRing_3256 T_IsRing'42'_3228
v9
du_isRing_3256 ::
  T_IsRing'42'_3228 -> MAlonzo.Code.Algebra.Structures.T_IsRing_2650
du_isRing_3256 :: T_IsRing'42'_3228 -> T_IsRing_2650
du_isRing_3256 T_IsRing'42'_3228
v0
  = (T_IsAbelianGroup_1132
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsRing_2650)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsRing_2650
forall a b. a -> b
coe
      T_IsAbelianGroup_1132
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsRing_2650
MAlonzo.Code.Algebra.Structures.C_IsRing'46'constructor_95033
      ((T_IsRing'42'_3228 -> T_IsAbelianGroup_1132) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3248 (T_IsRing'42'_3228 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228
v0))
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsRing'42'_3228 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228
v0)))))
      ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsRing'42'_3228 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228
v0))))
      ((T_IsMonoid_686 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_698
         ((T_IsRing'42'_3228 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228
v0)))
      ((T_IsRing'42'_3228 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228 -> T_Σ_14
d_distrib_3252 (T_IsRing'42'_3228 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228
v0))
-- Algebra.Structures.Biased.IsRing*._._.identityʳ
d_identity'691'_3268 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny -> AgdaAny -> T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
d_identity'691'_3268 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> AgdaAny
-> AgdaAny
d_identity'691'_3268 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
du_identity'691'_3268 T_IsRing'42'_3228
v9
du_identity'691'_3268 :: T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
du_identity'691'_3268 :: T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
du_identity'691'_3268 T_IsRing'42'_3228
v0
  = (T_IsMonoid_686 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_728
      ((T_IsRing'42'_3228 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228
v0))
-- Algebra.Structures.Biased.IsRing*._._.identityˡ
d_identity'737'_3270 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny -> AgdaAny -> T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
d_identity'737'_3270 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> AgdaAny
-> AgdaAny
d_identity'737'_3270 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
du_identity'737'_3270 T_IsRing'42'_3228
v9
du_identity'737'_3270 :: T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
du_identity'737'_3270 :: T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
du_identity'737'_3270 T_IsRing'42'_3228
v0
  = (T_IsMonoid_686 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_726
      ((T_IsRing'42'_3228 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228
v0))
-- Algebra.Structures.Biased.IsRing*._._.isPartialEquivalence
d_isPartialEquivalence_3276 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRing'42'_3228 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_3276 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_3276 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3276 T_IsRing'42'_3228
v9
du_isPartialEquivalence_3276 ::
  T_IsRing'42'_3228 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_3276 :: T_IsRing'42'_3228 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3276 T_IsRing'42'_3228
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_176
v3 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsEquivalence_26 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsEquivalence_26 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
               ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v3)))))
-- Algebra.Structures.Biased.IsRing*._._.isUnitalMagma
d_isUnitalMagma_3280 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRing'42'_3228 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
d_isUnitalMagma_3280 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> T_IsUnitalMagma_642
d_isUnitalMagma_3280 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228 -> T_IsUnitalMagma_642
du_isUnitalMagma_3280 T_IsRing'42'_3228
v9
du_isUnitalMagma_3280 ::
  T_IsRing'42'_3228 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
du_isUnitalMagma_3280 :: T_IsRing'42'_3228 -> T_IsUnitalMagma_642
du_isUnitalMagma_3280 T_IsRing'42'_3228
v0
  = (T_IsMonoid_686 -> T_IsUnitalMagma_642)
-> AgdaAny -> T_IsUnitalMagma_642
forall a b. a -> b
coe
      T_IsMonoid_686 -> T_IsUnitalMagma_642
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
      ((T_IsRing'42'_3228 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3228
v0))
-- Algebra.Structures.Biased.IsRing*._._.reflexive
d_reflexive_3284 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRing'42'_3228 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_3284 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_3284 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228 -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3284 T_IsRing'42'_3228
v9
du_reflexive_3284 ::
  T_IsRing'42'_3228 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_3284 :: T_IsRing'42'_3228 -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3284 T_IsRing'42'_3228
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_176
v3 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v2) in
          (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
            (\ AgdaAny
v4 AgdaAny
v5 AgdaAny
v6 ->
               (T_IsEquivalence_26 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                 T_IsEquivalence_26 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                 ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v3))
                 AgdaAny
v4)))
-- Algebra.Structures.Biased.IsRing*._._.setoid
d_setoid_3286 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRing'42'_3228 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
d_setoid_3286 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> T_Setoid_44
d_setoid_3286 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228 -> T_Setoid_44
du_setoid_3286 T_IsRing'42'_3228
v9
du_setoid_3286 ::
  T_IsRing'42'_3228 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
du_setoid_3286 :: T_IsRing'42'_3228 -> T_Setoid_44
du_setoid_3286 T_IsRing'42'_3228
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0) in
    AgdaAny -> T_Setoid_44
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsRing*._._.∙-congʳ
d_'8729''45'cong'691'_3294 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRing'42'_3228 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_3294 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_3294 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3294 T_IsRing'42'_3228
v9
du_'8729''45'cong'691'_3294 ::
  T_IsRing'42'_3228 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3294 :: T_IsRing'42'_3228
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3294 T_IsRing'42'_3228
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsRing*._._.∙-congˡ
d_'8729''45'cong'737'_3296 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRing'42'_3228 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_3296 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRing'42'_3228
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_3296 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRing'42'_3228
v9
  = T_IsRing'42'_3228
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3296 T_IsRing'42'_3228
v9
du_'8729''45'cong'737'_3296 ::
  T_IsRing'42'_3228 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3296 :: T_IsRing'42'_3228
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3296 T_IsRing'42'_3228
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRing'42'_3228 -> T_IsMonoid_686
d_'42''45'isMonoid_3250 (T_IsRing'42'_3228 -> T_IsRing'42'_3228
forall a b. a -> b
coe T_IsRing'42'_3228
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero
d_IsRingWithoutAnnihilatingZero_3308 :: p -> p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsRingWithoutAnnihilatingZero_3308 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 p
a8
  = ()
data T_IsRingWithoutAnnihilatingZero_3308
  = C_IsRingWithoutAnnihilatingZero'46'constructor_56523 MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
                                                         MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
                                                         MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+-isAbelianGroup
d_'43''45'isAbelianGroup_3326 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 T_IsRingWithoutAnnihilatingZero_3308
v0
  = case T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0 of
      C_IsRingWithoutAnnihilatingZero'46'constructor_56523 T_IsAbelianGroup_1132
v1 T_IsMonoid_686
v2 T_Σ_14
v3
        -> T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1
      T_IsRingWithoutAnnihilatingZero_3308
_ -> T_IsAbelianGroup_1132
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*-isMonoid
d_'42''45'isMonoid_3328 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
d_'42''45'isMonoid_3328 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 T_IsRingWithoutAnnihilatingZero_3308
v0
  = case T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0 of
      C_IsRingWithoutAnnihilatingZero'46'constructor_56523 T_IsAbelianGroup_1132
v1 T_IsMonoid_686
v2 T_Σ_14
v3
        -> T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v2
      T_IsRingWithoutAnnihilatingZero_3308
_ -> T_IsMonoid_686
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.distrib
d_distrib_3330 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_3330 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14
d_distrib_3330 T_IsRingWithoutAnnihilatingZero_3308
v0
  = case T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0 of
      C_IsRingWithoutAnnihilatingZero'46'constructor_56523 T_IsAbelianGroup_1132
v1 T_IsMonoid_686
v2 T_Σ_14
v3
        -> T_Σ_14 -> T_Σ_14
forall a b. a -> b
coe T_Σ_14
v3
      T_IsRingWithoutAnnihilatingZero_3308
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+._//_
d__'47''47'__3334 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny
d__'47''47'__3334 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
d__'47''47'__3334 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 ~T_IsRingWithoutAnnihilatingZero_3308
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__3334 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny
v6
du__'47''47'__3334 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__3334 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__3334 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny
v1
  = ((AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du__'47''47'__1098 ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0)
      ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v1)
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.assoc
d_assoc_3336 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_3336 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_3336 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
      ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
         ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
            ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
               ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.comm
d_comm_3338 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny
d_comm_3338 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny
d_comm_3338 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsAbelianGroup_1132 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_1146
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identity
d_identity_3340 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_3340 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14
d_identity_3340 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMonoid_686 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_686 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_698
      ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
         ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
            ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identityʳ
d_identity'691'_3342 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_identity'691'_3342 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
d_identity'691'_3342 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'691'_3342 T_IsRingWithoutAnnihilatingZero_3308
v9
du_identity'691'_3342 ::
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'691'_3342 :: T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'691'_3342 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1036
v2
             = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMonoid_686 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_728
            ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identityˡ
d_identity'737'_3344 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_identity'737'_3344 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
d_identity'737'_3344 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'737'_3344 T_IsRingWithoutAnnihilatingZero_3308
v9
du_identity'737'_3344 ::
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'737'_3344 :: T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'737'_3344 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1036
v2
             = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMonoid_686 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_726
            ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverse
d_inverse_3346 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_3346 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14
d_inverse_3346 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsGroup_1036 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsGroup_1036 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_1052
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
         ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverseʳ
d_inverse'691'_3348 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_inverse'691'_3348 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
d_inverse'691'_3348 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_inverse'691'_3348 T_IsRingWithoutAnnihilatingZero_3308
v9
du_inverse'691'_3348 ::
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_inverse'691'_3348 :: T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_inverse'691'_3348 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsGroup_1036 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1036 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'691'_1108
         ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverseˡ
d_inverse'737'_3350 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_inverse'737'_3350 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
d_inverse'737'_3350 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_inverse'737'_3350 T_IsRingWithoutAnnihilatingZero_3308
v9
du_inverse'737'_3350 ::
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_inverse'737'_3350 :: T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_inverse'737'_3350 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsGroup_1036 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1036 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'737'_1106
         ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeMagma
d_isCommutativeMagma_3352 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_212
d_isCommutativeMagma_3352 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsCommutativeMagma_212
d_isCommutativeMagma_3352 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsCommutativeMagma_212
du_isCommutativeMagma_3352 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isCommutativeMagma_3352 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_212
du_isCommutativeMagma_3352 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsCommutativeMagma_212
du_isCommutativeMagma_3352 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_IsCommutativeMagma_212
forall a b. a -> b
coe
      (let v2 :: t
v2
             = (T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> t
forall a b. a -> b
coe
                 T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
                 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsCommutativeSemigroup_548 -> T_IsCommutativeMagma_212)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsCommutativeSemigroup_548 -> T_IsCommutativeMagma_212
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_586
            ((T_IsCommutativeMonoid_736 -> T_IsCommutativeSemigroup_548)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsCommutativeMonoid_736 -> T_IsCommutativeSemigroup_548
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_786
               (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
forall a. a
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeMonoid
d_isCommutativeMonoid_3354 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
d_isCommutativeMonoid_3354 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsCommutativeMonoid_736
d_isCommutativeMonoid_3354 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsCommutativeMonoid_736
du_isCommutativeMonoid_3354 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isCommutativeMonoid_3354 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
du_isCommutativeMonoid_3354 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsCommutativeMonoid_736
du_isCommutativeMonoid_3354 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> T_IsCommutativeMonoid_736
forall a b. a -> b
coe
      T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeSemigroup
d_isCommutativeSemigroup_3356 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
d_isCommutativeSemigroup_3356 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsCommutativeSemigroup_548
d_isCommutativeSemigroup_3356 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8
                              T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308
-> T_IsCommutativeSemigroup_548
du_isCommutativeSemigroup_3356 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isCommutativeSemigroup_3356 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
du_isCommutativeSemigroup_3356 :: T_IsRingWithoutAnnihilatingZero_3308
-> T_IsCommutativeSemigroup_548
du_isCommutativeSemigroup_3356 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_IsCommutativeSemigroup_548
forall a b. a -> b
coe
      ((T_IsCommutativeMonoid_736 -> T_IsCommutativeSemigroup_548)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsCommutativeMonoid_736 -> T_IsCommutativeSemigroup_548
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_786
         ((T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsAbelianGroup_1132 -> T_IsCommutativeMonoid_736
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
            (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isEquivalence
d_isEquivalence_3358 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_3358 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsEquivalence_26
d_isEquivalence_3358 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMagma_176 -> T_IsEquivalence_26)
-> AgdaAny -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
      ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
               ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                  ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isGroup
d_isGroup_3360 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1036
d_isGroup_3360 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsGroup_1036
d_isGroup_3360 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsAbelianGroup_1132 -> T_IsGroup_1036)
-> AgdaAny -> T_IsGroup_1036
forall a b. a -> b
coe
      T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isInvertibleMagma
d_isInvertibleMagma_3362 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_924
d_isInvertibleMagma_3362 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsInvertibleMagma_924
d_isInvertibleMagma_3362 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsInvertibleMagma_924
du_isInvertibleMagma_3362 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isInvertibleMagma_3362 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_924
du_isInvertibleMagma_3362 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsInvertibleMagma_924
du_isInvertibleMagma_3362 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_IsInvertibleMagma_924
forall a b. a -> b
coe
      ((T_IsGroup_1036 -> T_IsInvertibleMagma_924) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1036 -> T_IsInvertibleMagma_924
MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1122
         ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_3364 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_976
d_isInvertibleUnitalMagma_3364 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsInvertibleUnitalMagma_976
d_isInvertibleUnitalMagma_3364 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8
                               T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308
-> T_IsInvertibleUnitalMagma_976
du_isInvertibleUnitalMagma_3364 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isInvertibleUnitalMagma_3364 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_976
du_isInvertibleUnitalMagma_3364 :: T_IsRingWithoutAnnihilatingZero_3308
-> T_IsInvertibleUnitalMagma_976
du_isInvertibleUnitalMagma_3364 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_IsInvertibleUnitalMagma_976
forall a b. a -> b
coe
      ((T_IsGroup_1036 -> T_IsInvertibleUnitalMagma_976)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1036 -> T_IsInvertibleUnitalMagma_976
MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1124
         ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isMagma
d_isMagma_3366 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_176
d_isMagma_3366 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMagma_176
d_isMagma_3366 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> T_IsMagma_176
forall a b. a -> b
coe
      T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
      ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
         ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
            ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
               ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isMonoid
d_isMonoid_3368 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
d_isMonoid_3368 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_isMonoid_3368 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> T_IsMonoid_686
forall a b. a -> b
coe
      T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
         ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isPartialEquivalence
d_isPartialEquivalence_3370 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_3370 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_3370 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3370 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isPartialEquivalence_3370 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_3370 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3370 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1036
v2
             = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_686
v3
                = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_472
v4
                   = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: T_IsMagma_176
v5 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v4) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  ((T_IsEquivalence_26 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsEquivalence_26 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
                     ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v5)))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isSemigroup
d_isSemigroup_3372 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
d_isSemigroup_3372 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsSemigroup_472
d_isSemigroup_3372 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMonoid_686 -> T_IsSemigroup_472)
-> AgdaAny -> T_IsSemigroup_472
forall a b. a -> b
coe
      T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
      ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
         ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
            ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isUnitalMagma
d_isUnitalMagma_3374 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
d_isUnitalMagma_3374 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsUnitalMagma_642
d_isUnitalMagma_3374 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsUnitalMagma_642
du_isUnitalMagma_3374 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isUnitalMagma_3374 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
du_isUnitalMagma_3374 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsUnitalMagma_642
du_isUnitalMagma_3374 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_IsUnitalMagma_642
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1036
v2
             = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMonoid_686 -> T_IsUnitalMagma_642) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsUnitalMagma_642
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
            ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.refl
d_refl_3376 ::
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_refl_3376 :: T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_refl_3376 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsEquivalence_26 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_26 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_34
      ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                  ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                     ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.reflexive
d_reflexive_3378 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_3378 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_3378 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3378 T_IsRingWithoutAnnihilatingZero_3308
v9
du_reflexive_3378 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_3378 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3378 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1036
v2
             = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_686
v3
                = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_472
v4
                   = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: T_IsMagma_176
v5 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v4) in
                (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
                  (\ AgdaAny
v6 AgdaAny
v7 AgdaAny
v8 ->
                     (T_IsEquivalence_26 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                       T_IsEquivalence_26 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                       ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v5))
                       AgdaAny
v6)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.setoid
d_setoid_3380 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
d_setoid_3380 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_Setoid_44
d_setoid_3380 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_Setoid_44
du_setoid_3380 T_IsRingWithoutAnnihilatingZero_3308
v9
du_setoid_3380 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
du_setoid_3380 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_Setoid_44
du_setoid_3380 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_Setoid_44
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1036
v2
             = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_686
v3
                = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_472
v4
                   = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                  ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v4))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.sym
d_sym_3382 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_sym_3382 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_sym_3382 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsEquivalence_26 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_26 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_sym_36
      ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                  ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                     ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.trans
d_trans_3384 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_trans_3384 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_trans_3384 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsEquivalence_26
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_26
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_trans_38
      ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                  ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                     ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_3386 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_unique'691''45''8315''185'_3386 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_unique'691''45''8315''185'_3386 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 ~AgdaAny
v8
                                  T_IsRingWithoutAnnihilatingZero_3308
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'691''45''8315''185'_3386 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3308
v9
du_unique'691''45''8315''185'_3386 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_unique'691''45''8315''185'_3386 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'691''45''8315''185'_3386 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny
v1 AgdaAny
v2 T_IsRingWithoutAnnihilatingZero_3308
v3
  = let v4 :: T_IsAbelianGroup_1132
v4 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v3) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (((AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_IsGroup_1036
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsGroup_1036
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.du_unique'691''45''8315''185'_1120
         ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v2) ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v1)
         ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v4)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_3388 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_unique'737''45''8315''185'_3388 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_unique'737''45''8315''185'_3388 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 ~AgdaAny
v8
                                  T_IsRingWithoutAnnihilatingZero_3308
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'737''45''8315''185'_3388 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3308
v9
du_unique'737''45''8315''185'_3388 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_unique'737''45''8315''185'_3388 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'737''45''8315''185'_3388 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny
v1 AgdaAny
v2 T_IsRingWithoutAnnihilatingZero_3308
v3
  = let v4 :: T_IsAbelianGroup_1132
v4 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v3) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (((AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_IsGroup_1036
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsGroup_1036
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.du_unique'737''45''8315''185'_1114
         ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v2) ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v1)
         ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v4)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.⁻¹-cong
d_'8315''185''45'cong_3390 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8315''185''45'cong_3390 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8315''185''45'cong_3390 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsGroup_1036 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsGroup_1036 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8315''185''45'cong_1054
      ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
         ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-cong
d_'8729''45'cong_3392 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong_3392 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong_3392 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
      ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
               ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                  ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-congʳ
d_'8729''45'cong'691'_3394 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_3394 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_3394 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3394 T_IsRingWithoutAnnihilatingZero_3308
v9
du_'8729''45'cong'691'_3394 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3394 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3394 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1036
v2
             = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_686
v3
                = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_472
v4
                   = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
                  ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v4))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-congˡ
d_'8729''45'cong'737'_3396 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_3396 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_3396 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3396 T_IsRingWithoutAnnihilatingZero_3308
v9
du_'8729''45'cong'737'_3396 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3396 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3396 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsAbelianGroup_1132
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1036
v2
             = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_686
v3
                = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_472
v4
                   = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
                  ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v4))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.assoc
d_assoc_3400 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_3400 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_3400 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
      ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
         ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identity
d_identity_3402 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_3402 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14
d_identity_3402 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMonoid_686 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_686 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_698
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identityʳ
d_identity'691'_3404 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_identity'691'_3404 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
d_identity'691'_3404 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'691'_3404 T_IsRingWithoutAnnihilatingZero_3308
v9
du_identity'691'_3404 ::
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'691'_3404 :: T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'691'_3404 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMonoid_686 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_728
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identityˡ
d_identity'737'_3406 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_identity'737'_3406 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
d_identity'737'_3406 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'737'_3406 T_IsRingWithoutAnnihilatingZero_3308
v9
du_identity'737'_3406 ::
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'737'_3406 :: T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_identity'737'_3406 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMonoid_686 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_726
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isEquivalence
d_isEquivalence_3408 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
d_isEquivalence_3408 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsEquivalence_26
d_isEquivalence_3408 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMagma_176 -> T_IsEquivalence_26)
-> AgdaAny -> T_IsEquivalence_26
forall a b. a -> b
coe
      T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
      ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isMagma
d_isMagma_3410 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_176
d_isMagma_3410 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMagma_176
d_isMagma_3410 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> T_IsMagma_176
forall a b. a -> b
coe
      T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
      ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
         ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isPartialEquivalence
d_isPartialEquivalence_3412 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_3412 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_3412 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3412 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isPartialEquivalence_3412 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_3412 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3412 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_176
v3 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsEquivalence_26 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsEquivalence_26 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
               ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v3)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isSemigroup
d_isSemigroup_3414 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
d_isSemigroup_3414 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsSemigroup_472
d_isSemigroup_3414 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMonoid_686 -> T_IsSemigroup_472)
-> AgdaAny -> T_IsSemigroup_472
forall a b. a -> b
coe
      T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isUnitalMagma
d_isUnitalMagma_3416 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
d_isUnitalMagma_3416 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsUnitalMagma_642
d_isUnitalMagma_3416 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsUnitalMagma_642
du_isUnitalMagma_3416 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isUnitalMagma_3416 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
du_isUnitalMagma_3416 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsUnitalMagma_642
du_isUnitalMagma_3416 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMonoid_686 -> T_IsUnitalMagma_642)
-> AgdaAny -> T_IsUnitalMagma_642
forall a b. a -> b
coe
      T_IsMonoid_686 -> T_IsUnitalMagma_642
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.refl
d_refl_3418 ::
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_refl_3418 :: T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_refl_3418 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsEquivalence_26 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_26 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_34
      ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.reflexive
d_reflexive_3420 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_3420 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_3420 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3420 T_IsRingWithoutAnnihilatingZero_3308
v9
du_reflexive_3420 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_3420 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3420 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_176
v3 = T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> T_IsSemigroup_472
forall a b. a -> b
coe T_IsSemigroup_472
v2) in
          (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
            (\ AgdaAny
v4 AgdaAny
v5 AgdaAny
v6 ->
               (T_IsEquivalence_26 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                 T_IsEquivalence_26 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                 ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (T_IsMagma_176 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_176
v3))
                 AgdaAny
v4)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.setoid
d_setoid_3422 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
d_setoid_3422 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_Setoid_44
d_setoid_3422 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_Setoid_44
du_setoid_3422 T_IsRingWithoutAnnihilatingZero_3308
v9
du_setoid_3422 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
du_setoid_3422 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_Setoid_44
du_setoid_3422 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> T_Setoid_44
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.sym
d_sym_3424 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_sym_3424 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_sym_3424 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsEquivalence_26 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_26 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_sym_36
      ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.trans
d_trans_3426 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_trans_3426 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_trans_3426 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsEquivalence_26
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_26
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_trans_38
      ((T_IsMagma_176 -> T_IsEquivalence_26) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176 -> T_IsEquivalence_26
MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-cong
d_'8729''45'cong_3428 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong_3428 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong_3428 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
      ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-congʳ
d_'8729''45'cong'691'_3430 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_3430 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_3430 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3430 T_IsRingWithoutAnnihilatingZero_3308
v9
du_'8729''45'cong'691'_3430 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3430 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3430 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-congˡ
d_'8729''45'cong'737'_3432 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_3432 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_3432 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3432 T_IsRingWithoutAnnihilatingZero_3308
v9
du_'8729''45'cong'737'_3432 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3432 :: T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3432 T_IsRingWithoutAnnihilatingZero_3308
v0
  = let v1 :: T_IsMonoid_686
v1 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_472
v2
             = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_176
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_176
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
            ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zeroˡ
d_zero'737'_3434 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_zero'737'_3434 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
d_zero'737'_3434 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
du_zero'737'_3434 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3308
v9
du_zero'737'_3434 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_zero'737'_3434 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
du_zero'737'_3434 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny -> AgdaAny
v2 AgdaAny
v3 T_IsRingWithoutAnnihilatingZero_3308
v4
  = (T_Setoid_44
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_Setoid_44
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Setoid.du_assoc'8743'distrib'691''8743'id'691''8743'inv'691''8658'ze'737'_594
      (let v5 :: T_IsAbelianGroup_1132
v5 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v6 :: T_IsGroup_1036
v6
                = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v5) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v7 :: T_IsMonoid_686
v7
                   = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v6) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v8 :: T_IsSemigroup_472
v8
                      = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v7) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                     ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v8)))))))
      ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1) ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v3)
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                  ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                     ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4)))))))
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4)))))
      ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
               ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                  ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4))))))
      ((T_Σ_14 -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_Σ_14 -> AgdaAny
MAlonzo.Code.Agda.Builtin.Sigma.d_snd_30
         ((T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14
d_distrib_3330 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4)))
      (let v5 :: T_IsAbelianGroup_1132
v5 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v6 :: T_IsGroup_1036
v6
                = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v5) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsMonoid_686 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_728
               ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036
v6)))))
      (let v5 :: T_IsAbelianGroup_1132
v5 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsGroup_1036 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsGroup_1036 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'691'_1108
            ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v5))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zeroʳ
d_zero'691'_3436 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
d_zero'691'_3436 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
d_zero'691'_3436 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
du_zero'691'_3436 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3308
v9
du_zero'691'_3436 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
du_zero'691'_3436 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
du_zero'691'_3436 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny -> AgdaAny
v2 AgdaAny
v3 T_IsRingWithoutAnnihilatingZero_3308
v4
  = (T_Setoid_44
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_Setoid_44
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Setoid.du_assoc'8743'distrib'737''8743'id'691''8743'inv'691''8658'ze'691'_606
      (let v5 :: T_IsAbelianGroup_1132
v5 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v6 :: T_IsGroup_1036
v6
                = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v5) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v7 :: T_IsMonoid_686
v7
                   = T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> T_IsGroup_1036
forall a b. a -> b
coe T_IsGroup_1036
v6) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v8 :: T_IsSemigroup_472
v8
                      = T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (T_IsMonoid_686 -> T_IsMonoid_686
forall a b. a -> b
coe T_IsMonoid_686
v7) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  ((T_IsMagma_176 -> T_Setoid_44) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMagma_176 -> T_Setoid_44
MAlonzo.Code.Algebra.Structures.du_setoid_200
                     ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480 (T_IsSemigroup_472 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_472
v8)))))))
      ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1) ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v3)
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                  ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                     ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4)))))))
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4)))))
      ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
               ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                  ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4))))))
      ((T_Σ_14 -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_Σ_14 -> AgdaAny
MAlonzo.Code.Agda.Builtin.Sigma.d_fst_28
         ((T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14
d_distrib_3330 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4)))
      (let v5 :: T_IsAbelianGroup_1132
v5 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v6 :: T_IsGroup_1036
v6
                = T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> T_IsAbelianGroup_1132
forall a b. a -> b
coe T_IsAbelianGroup_1132
v5) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsMonoid_686 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_728
               ((T_IsGroup_1036 -> T_IsMonoid_686) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036 -> T_IsMonoid_686
MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (T_IsGroup_1036 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1036
v6)))))
      (let v5 :: T_IsAbelianGroup_1132
v5 = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRingWithoutAnnihilatingZero_3308
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsGroup_1036 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsGroup_1036 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'691'_1108
            ((T_IsAbelianGroup_1132 -> T_IsGroup_1036) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132 -> T_IsGroup_1036
MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (T_IsAbelianGroup_1132 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1132
v5))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zero
d_zero_3438 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_3438 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_Σ_14
d_zero_3438 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_Σ_14
du_zero_3438 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3308
v9
du_zero_3438 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_zero_3438 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_Σ_14
du_zero_3438 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny -> AgdaAny
v2 AgdaAny
v3 T_IsRingWithoutAnnihilatingZero_3308
v4
  = (AgdaAny -> AgdaAny -> T_Σ_14) -> AgdaAny -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      AgdaAny -> AgdaAny -> T_Σ_14
MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
      (((AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> AgdaAny
 -> T_IsRingWithoutAnnihilatingZero_3308
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
du_zero'737'_3434 ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1) ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v3) (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4))
      (((AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> AgdaAny
 -> T_IsRingWithoutAnnihilatingZero_3308
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> AgdaAny
-> AgdaAny
du_zero'691'_3436 ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1) ((AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v3) (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v4))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.isRing
d_isRing_3440 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_2650
d_isRing_3440 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3308
-> T_IsRing_2650
d_isRing_3440 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 ~AgdaAny -> AgdaAny -> AgdaAny
v4 ~AgdaAny -> AgdaAny -> AgdaAny
v5 ~AgdaAny -> AgdaAny
v6 ~AgdaAny
v7 ~AgdaAny
v8 T_IsRingWithoutAnnihilatingZero_3308
v9
  = T_IsRingWithoutAnnihilatingZero_3308 -> T_IsRing_2650
du_isRing_3440 T_IsRingWithoutAnnihilatingZero_3308
v9
du_isRing_3440 ::
  T_IsRingWithoutAnnihilatingZero_3308 ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_2650
du_isRing_3440 :: T_IsRingWithoutAnnihilatingZero_3308 -> T_IsRing_2650
du_isRing_3440 T_IsRingWithoutAnnihilatingZero_3308
v0
  = (T_IsAbelianGroup_1132
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsRing_2650)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsRing_2650
forall a b. a -> b
coe
      T_IsAbelianGroup_1132
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsRing_2650
MAlonzo.Code.Algebra.Structures.C_IsRing'46'constructor_95033
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsAbelianGroup_1132
d_'43''45'isAbelianGroup_3326 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))
      ((T_IsMagma_176
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_176
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
         ((T_IsSemigroup_472 -> T_IsMagma_176) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_472 -> T_IsMagma_176
MAlonzo.Code.Algebra.Structures.d_isMagma_480
            ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
               ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))))
      ((T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_472 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_482
         ((T_IsMonoid_686 -> T_IsSemigroup_472) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_686 -> T_IsSemigroup_472
MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
            ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))))
      ((T_IsMonoid_686 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_686 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_698
         ((T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_IsMonoid_686
d_'42''45'isMonoid_3328 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0)))
      ((T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308 -> T_Σ_14
d_distrib_3330 (T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3308
v0))