{-# 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.Base
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_170 :: p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeMonoid_170 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 = ()
-- Algebra.Structures.Biased._.IsCommutativeSemiring
d_IsCommutativeSemiring_182 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeSemiring_182 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 = ()
-- Algebra.Structures.Biased._.IsMonoid
d_IsMonoid_246 :: p -> p -> p -> p -> p -> p -> T_Level_18
d_IsMonoid_246 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 = ()
-- Algebra.Structures.Biased._.IsNearSemiring
d_IsNearSemiring_254 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsNearSemiring_254 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
-- Algebra.Structures.Biased._.IsRing
d_IsRing_278 :: p -> p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsRing_278 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 p
a8 = ()
-- Algebra.Structures.Biased._.IsSemigroup
d_IsSemigroup_290 :: p -> p -> p -> p -> p -> T_Level_18
d_IsSemigroup_290 p
a0 p
a1 p
a2 p
a3 p
a4 = ()
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero
d_IsSemiringWithoutAnnihilatingZero_302 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsSemiringWithoutAnnihilatingZero_302 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7
  = ()
-- Algebra.Structures.Biased._.IsSemiringWithoutOne
d_IsSemiringWithoutOne_306 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsSemiringWithoutOne_306 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
-- Algebra.Structures.Biased._.IsAbelianGroup._//_
d__'47''47'__320 ::
  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_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny
d__'47''47'__320 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
d__'47''47'__320 ~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_1172
v7
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__320 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny
v6
du__'47''47'__320 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__320 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__320 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'__1136 ((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_324 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny
d_comm_324 :: T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_324 T_IsAbelianGroup_1172
v0
  = (T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_1186 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0)
-- Algebra.Structures.Biased._.IsAbelianGroup.identityʳ
d_identity'691'_328 ::
  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_1172 ->
  AgdaAny -> AgdaAny
d_identity'691'_328 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
d_identity'691'_328 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny
du_identity'691'_328 T_IsAbelianGroup_1172
v7
du_identity'691'_328 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny
du_identity'691'_328 :: T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny
du_identity'691'_328 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsMonoid_712 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_754
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074
v1)))
-- Algebra.Structures.Biased._.IsAbelianGroup.identityˡ
d_identity'737'_330 ::
  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_1172 ->
  AgdaAny -> AgdaAny
d_identity'737'_330 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
d_identity'737'_330 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny
du_identity'737'_330 T_IsAbelianGroup_1172
v7
du_identity'737'_330 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny
du_identity'737'_330 :: T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny
du_identity'737'_330 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsMonoid_712 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_752
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074
v1)))
-- Algebra.Structures.Biased._.IsAbelianGroup.inverseʳ
d_inverse'691'_334 ::
  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_1172 ->
  AgdaAny -> AgdaAny
d_inverse'691'_334 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
d_inverse'691'_334 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny
du_inverse'691'_334 T_IsAbelianGroup_1172
v7
du_inverse'691'_334 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny
du_inverse'691'_334 :: T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny
du_inverse'691'_334 T_IsAbelianGroup_1172
v0
  = (T_IsGroup_1074 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsGroup_1074 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'691'_1146
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.inverseˡ
d_inverse'737'_336 ::
  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_1172 ->
  AgdaAny -> AgdaAny
d_inverse'737'_336 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
d_inverse'737'_336 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny
du_inverse'737'_336 T_IsAbelianGroup_1172
v7
du_inverse'737'_336 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny
du_inverse'737'_336 :: T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny
du_inverse'737'_336 T_IsAbelianGroup_1172
v0
  = (T_IsGroup_1074 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsGroup_1074 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'737'_1144
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeMagma
d_isCommutativeMagma_338 ::
  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_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_338 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_338 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_338 T_IsAbelianGroup_1172
v7
du_isCommutativeMagma_338 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_338 :: T_IsAbelianGroup_1172 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_338 T_IsAbelianGroup_1172
v0
  = let v1 :: AgdaAny
v1
          = (T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
              T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
              (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
         ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
            (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v1)))
-- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeMonoid
d_isCommutativeMonoid_340 ::
  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_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_340 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_340 ~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_1172
-> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_340
du_isCommutativeMonoid_340 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_isCommutativeMonoid_340 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_340 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 AgdaAny -> AgdaAny
v2 T_IsAbelianGroup_1172
v3
  = (T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764)
-> T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244 T_IsAbelianGroup_1172
v3
-- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeSemigroup
d_isCommutativeSemigroup_342 ::
  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_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_342 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_342 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_342 T_IsAbelianGroup_1172
v7
du_isCommutativeSemigroup_342 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_342 :: T_IsAbelianGroup_1172 -> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_342 T_IsAbelianGroup_1172
v0
  = (T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> AgdaAny -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
      ((T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
         (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.isGroup
d_isGroup_346 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074
d_isGroup_346 :: T_IsAbelianGroup_1172 -> T_IsGroup_1074
d_isGroup_346 T_IsAbelianGroup_1172
v0
  = (T_IsAbelianGroup_1172 -> T_IsGroup_1074)
-> AgdaAny -> T_IsGroup_1074
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0)
-- Algebra.Structures.Biased._.IsAbelianGroup.isInvertibleMagma
d_isInvertibleMagma_348 ::
  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_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_348 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_IsInvertibleMagma_958
d_isInvertibleMagma_348 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_348 T_IsAbelianGroup_1172
v7
du_isInvertibleMagma_348 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
du_isInvertibleMagma_348 :: T_IsAbelianGroup_1172 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_348 T_IsAbelianGroup_1172
v0
  = (T_IsGroup_1074 -> T_IsInvertibleMagma_958)
-> AgdaAny -> T_IsInvertibleMagma_958
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1160
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_350 ::
  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_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_350 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_350 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_350 T_IsAbelianGroup_1172
v7
du_isInvertibleUnitalMagma_350 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_350 :: T_IsAbelianGroup_1172 -> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_350 T_IsAbelianGroup_1172
v0
  = (T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012)
-> AgdaAny -> T_IsInvertibleUnitalMagma_1012
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012
MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1162
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0))
-- Algebra.Structures.Biased._.IsAbelianGroup.isPartialEquivalence
d_isPartialEquivalence_356 ::
  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_1172 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_356 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_356 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_356 T_IsAbelianGroup_1172
v7
du_isPartialEquivalence_356 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_356 :: T_IsAbelianGroup_1172 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_356 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_178
v4 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                  ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v4))))))
-- Algebra.Structures.Biased._.IsAbelianGroup.isUnitalMagma
d_isUnitalMagma_360 ::
  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_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_360 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_IsUnitalMagma_666
d_isUnitalMagma_360 ~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_1172
v7
  = T_IsAbelianGroup_1172 -> T_IsUnitalMagma_666
du_isUnitalMagma_360 T_IsAbelianGroup_1172
v7
du_isUnitalMagma_360 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_360 :: T_IsAbelianGroup_1172 -> T_IsUnitalMagma_666
du_isUnitalMagma_360 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074
v1)))
-- Algebra.Structures.Biased._.IsAbelianGroup.reflexive
d_reflexive_364 ::
  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_1172 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_364 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_364 ~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_1172
v7
  = T_IsAbelianGroup_1172
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_364 T_IsAbelianGroup_1172
v7
du_reflexive_364 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_364 :: T_IsAbelianGroup_1172
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_364 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_178
v4 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v3) in
             (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
               (\ AgdaAny
v5 AgdaAny
v6 AgdaAny
v7 ->
                  (T_IsEquivalence_28 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                    T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_42
                    ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v4))
                    AgdaAny
v5))))
-- Algebra.Structures.Biased._.IsAbelianGroup.setoid
d_setoid_366 ::
  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_1172 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_366 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> T_Setoid_46
d_setoid_366 ~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_1172
v7 = T_IsAbelianGroup_1172 -> T_Setoid_46
du_setoid_366 T_IsAbelianGroup_1172
v7
du_setoid_366 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_366 :: T_IsAbelianGroup_1172 -> T_Setoid_46
du_setoid_366 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488
v3)))))
-- Algebra.Structures.Biased._.IsAbelianGroup.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_372 ::
  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_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_unique'691''45''8315''185'_372 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_unique'691''45''8315''185'_372 ~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_1172
v7
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'691''45''8315''185'_372 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1172
v7
du_unique'691''45''8315''185'_372 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_unique'691''45''8315''185'_372 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'691''45''8315''185'_372 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 AgdaAny -> AgdaAny
v2 T_IsAbelianGroup_1172
v3
  = ((AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_IsGroup_1074
 -> 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_1074
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.du_unique'691''45''8315''185'_1158
      ((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_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3))
-- Algebra.Structures.Biased._.IsAbelianGroup.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_374 ::
  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_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_unique'737''45''8315''185'_374 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_unique'737''45''8315''185'_374 ~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_1172
v7
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'737''45''8315''185'_374 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 AgdaAny -> AgdaAny
v6 T_IsAbelianGroup_1172
v7
du_unique'737''45''8315''185'_374 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  (AgdaAny -> AgdaAny) ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_unique'737''45''8315''185'_374 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'737''45''8315''185'_374 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 AgdaAny -> AgdaAny
v2 T_IsAbelianGroup_1172
v3
  = ((AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_IsGroup_1074
 -> 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_1074
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.du_unique'737''45''8315''185'_1152
      ((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_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v3))
-- Algebra.Structures.Biased._.IsAbelianGroup.∙-congʳ
d_'8729''45'cong'691'_380 ::
  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_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_380 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_380 ~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_1172
v7
  = T_IsAbelianGroup_1172
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_380 T_IsAbelianGroup_1172
v7
du_'8729''45'cong'691'_380 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_380 :: T_IsAbelianGroup_1172
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_380 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_178
v4 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: AgdaAny
v5
                      = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v4) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (let v6 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v6
                         = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v4) in
                   AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'691'_46
                        ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v6)
                        ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                           ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                              T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                              (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v5)))))))))
-- Algebra.Structures.Biased._.IsAbelianGroup.∙-congˡ
d_'8729''45'cong'737'_382 ::
  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_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_382 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsAbelianGroup_1172
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_382 ~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_1172
v7
  = T_IsAbelianGroup_1172
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_382 T_IsAbelianGroup_1172
v7
du_'8729''45'cong'737'_382 ::
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_382 :: T_IsAbelianGroup_1172
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_382 T_IsAbelianGroup_1172
v0
  = let v1 :: T_IsGroup_1074
v1
          = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMonoid_712
v2
             = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsSemigroup_488
v3
                = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsMagma_178
v4 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: AgdaAny
v5
                      = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v4) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (let v6 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v6
                         = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v4) in
                   AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'737'_42
                        ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v6)
                        ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                           ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                              T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                              (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v5)))))))))
-- Algebra.Structures.Biased._.IsCommutativeMonoid.comm
d_comm_614 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  AgdaAny -> AgdaAny -> AgdaAny
d_comm_614 :: T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_614 T_IsCommutativeMonoid_764
v0
  = (T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_776 (T_IsCommutativeMonoid_764 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0)
-- Algebra.Structures.Biased._.IsCommutativeMonoid.isMonoid
d_isMonoid_630 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_630 :: T_IsCommutativeMonoid_764 -> T_IsMonoid_712
d_isMonoid_630 T_IsCommutativeMonoid_764
v0
  = (T_IsCommutativeMonoid_764 -> T_IsMonoid_712)
-> AgdaAny -> T_IsMonoid_712
forall a b. a -> b
coe T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774 (T_IsCommutativeMonoid_764 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v0)
-- Algebra.Structures.Biased._.IsCommutativeSemiring.*-comm
d_'42''45'comm_814 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'comm_814 :: T_IsCommutativeSemiring_1750 -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'comm_814 T_IsCommutativeSemiring_1750
v0
  = (T_IsCommutativeSemiring_1750 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'comm_1766 (T_IsCommutativeSemiring_1750 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)
-- Algebra.Structures.Biased._.IsCommutativeSemiring.isSemiring
d_isSemiring_884 ::
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiring_1640
d_isSemiring_884 :: T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
d_isSemiring_884 T_IsCommutativeSemiring_1750
v0
  = (T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640)
-> AgdaAny -> T_IsSemiring_1640
forall a b. a -> b
coe T_IsCommutativeSemiring_1750 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.d_isSemiring_1764 (T_IsCommutativeSemiring_1750 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring_1750
v0)
-- Algebra.Structures.Biased._.IsMonoid.identity
d_identity_1696 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_1696 :: T_IsMonoid_712 -> T_Σ_14
d_identity_1696 T_IsMonoid_712
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724 (T_IsMonoid_712 -> AgdaAny
forall a b. a -> b
coe T_IsMonoid_712
v0)
-- Algebra.Structures.Biased._.IsMonoid.identityʳ
d_identity'691'_1698 ::
  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_712 ->
  AgdaAny -> AgdaAny
d_identity'691'_1698 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_712
-> AgdaAny
-> AgdaAny
d_identity'691'_1698 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_712 -> AgdaAny -> AgdaAny
du_identity'691'_1698
du_identity'691'_1698 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  AgdaAny -> AgdaAny
du_identity'691'_1698 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_712 -> AgdaAny -> AgdaAny
du_identity'691'_1698 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsMonoid_712
v2
  = (T_IsMonoid_712 -> AgdaAny -> AgdaAny)
-> T_IsMonoid_712 -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_754 T_IsMonoid_712
v2
-- Algebra.Structures.Biased._.IsMonoid.identityˡ
d_identity'737'_1700 ::
  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_712 ->
  AgdaAny -> AgdaAny
d_identity'737'_1700 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_712
-> AgdaAny
-> AgdaAny
d_identity'737'_1700 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_712 -> AgdaAny -> AgdaAny
du_identity'737'_1700
du_identity'737'_1700 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  AgdaAny -> AgdaAny
du_identity'737'_1700 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_712 -> AgdaAny -> AgdaAny
du_identity'737'_1700 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsMonoid_712
v2
  = (T_IsMonoid_712 -> AgdaAny -> AgdaAny)
-> T_IsMonoid_712 -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_752 T_IsMonoid_712
v2
-- Algebra.Structures.Biased._.IsMonoid.isPartialEquivalence
d_isPartialEquivalence_1706 ::
  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_712 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_1706 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_712
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_1706 ~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_712
v6
  = T_IsMonoid_712 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1706 T_IsMonoid_712
v6
du_isPartialEquivalence_1706 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_1706 :: T_IsMonoid_712 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_1706 T_IsMonoid_712
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
            ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v2))))
-- Algebra.Structures.Biased._.IsMonoid.isSemigroup
d_isSemigroup_1708 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_1708 :: T_IsMonoid_712 -> T_IsSemigroup_488
d_isSemigroup_1708 T_IsMonoid_712
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488)
-> AgdaAny -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> AgdaAny
forall a b. a -> b
coe T_IsMonoid_712
v0)
-- Algebra.Structures.Biased._.IsMonoid.isUnitalMagma
d_isUnitalMagma_1710 ::
  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_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_1710 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_712
-> T_IsUnitalMagma_666
d_isUnitalMagma_1710 ~T_Level_18
v0 ~T_Level_18
v1 ~T_Level_18
v2 ~AgdaAny -> AgdaAny -> T_Level_18
v3 = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_712 -> T_IsUnitalMagma_666
du_isUnitalMagma_1710
du_isUnitalMagma_1710 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_1710 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> T_IsMonoid_712 -> T_IsUnitalMagma_666
du_isUnitalMagma_1710 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsMonoid_712
v2
  = (T_IsMonoid_712 -> T_IsUnitalMagma_666)
-> T_IsMonoid_712 -> T_IsUnitalMagma_666
forall a b. a -> b
coe T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756 T_IsMonoid_712
v2
-- Algebra.Structures.Biased._.IsMonoid.reflexive
d_reflexive_1714 ::
  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_712 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_1714 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_712
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_1714 ~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_712
v6 = T_IsMonoid_712 -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_1714 T_IsMonoid_712
v6
du_reflexive_1714 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_1714 :: T_IsMonoid_712 -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_1714 T_IsMonoid_712
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1) in
       (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
         (\ AgdaAny
v3 AgdaAny
v4 AgdaAny
v5 ->
            (T_IsEquivalence_28 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
              T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_42
              ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v2))
              AgdaAny
v3))
-- Algebra.Structures.Biased._.IsMonoid.setoid
d_setoid_1716 ::
  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_712 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_1716 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_712
-> T_Setoid_46
d_setoid_1716 ~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_712
v6 = T_IsMonoid_712 -> T_Setoid_46
du_setoid_1716 T_IsMonoid_712
v6
du_setoid_1716 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_1716 :: T_IsMonoid_712 -> T_Setoid_46
du_setoid_1716 T_IsMonoid_712
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v0) in
    AgdaAny -> T_Setoid_46
forall a b. a -> b
coe
      ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488
v1)))
-- Algebra.Structures.Biased._.IsMonoid.∙-congʳ
d_'8729''45'cong'691'_1724 ::
  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_712 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_1724 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_712
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_1724 ~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_712
v6
  = T_IsMonoid_712
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_1724 T_IsMonoid_712
v6
du_'8729''45'cong'691'_1724 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_1724 :: T_IsMonoid_712
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_1724 T_IsMonoid_712
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: AgdaAny
v3
                = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v4
                   = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v2) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'691'_46
                  ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v4)
                  ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                     ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                        (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v3)))))))
-- Algebra.Structures.Biased._.IsMonoid.∙-congˡ
d_'8729''45'cong'737'_1726 ::
  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_712 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_1726 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsMonoid_712
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_1726 ~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_712
v6
  = T_IsMonoid_712
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_1726 T_IsMonoid_712
v6
du_'8729''45'cong'737'_1726 ::
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_1726 :: T_IsMonoid_712
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_1726 T_IsMonoid_712
v0
  = let v1 :: T_IsSemigroup_488
v1
          = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsMagma_178
v2 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: AgdaAny
v3
                = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v4
                   = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v2) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'737'_42
                  ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v4)
                  ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                     ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                        (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v3)))))))
-- Algebra.Structures.Biased._.IsNearSemiring.*-assoc
d_'42''45'assoc_1796 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_1796 :: T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_1796 T_IsNearSemiring_1260
v0
  = (T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1282 (T_IsNearSemiring_1260 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260
v0)
-- Algebra.Structures.Biased._.IsNearSemiring.*-cong
d_'42''45'cong_1798 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'cong_1798 :: T_IsNearSemiring_1260
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'42''45'cong_1798 T_IsNearSemiring_1260
v0
  = (T_IsNearSemiring_1260
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1280 (T_IsNearSemiring_1260 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260
v0)
-- Algebra.Structures.Biased._.IsNearSemiring.+-isMonoid
d_'43''45'isMonoid_1824 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'43''45'isMonoid_1824 :: T_IsNearSemiring_1260 -> T_IsMonoid_712
d_'43''45'isMonoid_1824 T_IsNearSemiring_1260
v0
  = (T_IsNearSemiring_1260 -> T_IsMonoid_712)
-> AgdaAny -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsNearSemiring_1260 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_'43''45'isMonoid_1278 (T_IsNearSemiring_1260 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260
v0)
-- Algebra.Structures.Biased._.IsNearSemiring.distribʳ
d_distrib'691'_1830 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_1830 :: T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_1830 T_IsNearSemiring_1260
v0
  = (T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_distrib'691'_1284 (T_IsNearSemiring_1260 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260
v0)
-- Algebra.Structures.Biased._.IsNearSemiring.zeroˡ
d_zero'737'_1846 ::
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260 ->
  AgdaAny -> AgdaAny
d_zero'737'_1846 :: T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny
d_zero'737'_1846 T_IsNearSemiring_1260
v0
  = (T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_zero'737'_1286 (T_IsNearSemiring_1260 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring_1260
v0)
-- Algebra.Structures.Biased._.IsRing.*-assoc
d_'42''45'assoc_2214 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2214 :: T_IsRing_2740 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2214 T_IsRing_2740
v0
  = (T_IsRing_2740 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing_2740 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_2766 (T_IsRing_2740 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2740
v0)
-- Algebra.Structures.Biased._.IsRing.*-cong
d_'42''45'cong_2216 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'cong_2216 :: T_IsRing_2740
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'42''45'cong_2216 T_IsRing_2740
v0
  = (T_IsRing_2740
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe T_IsRing_2740
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'cong_2764 (T_IsRing_2740 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2740
v0)
-- Algebra.Structures.Biased._.IsRing.*-identity
d_'42''45'identity_2222 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_2222 :: T_IsRing_2740 -> T_Σ_14
d_'42''45'identity_2222 T_IsRing_2740
v0
  = (T_IsRing_2740 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsRing_2740 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_2768 (T_IsRing_2740 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2740
v0)
-- Algebra.Structures.Biased._.IsRing.+-isAbelianGroup
d_'43''45'isAbelianGroup_2250 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_2250 :: T_IsRing_2740 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_2250 T_IsRing_2740
v0
  = (T_IsRing_2740 -> T_IsAbelianGroup_1172)
-> AgdaAny -> T_IsAbelianGroup_1172
forall a b. a -> b
coe
      T_IsRing_2740 -> T_IsAbelianGroup_1172
MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2762
      (T_IsRing_2740 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2740
v0)
-- Algebra.Structures.Biased._.IsRing.distrib
d_distrib_2280 ::
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2280 :: T_IsRing_2740 -> T_Σ_14
d_distrib_2280 T_IsRing_2740
v0
  = (T_IsRing_2740 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsRing_2740 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_2770 (T_IsRing_2740 -> AgdaAny
forall a b. a -> b
coe T_IsRing_2740
v0)
-- Algebra.Structures.Biased._.IsSemigroup.assoc
d_assoc_2442 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_2442 :: T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_2442 T_IsSemigroup_488
v0
  = (T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498 (T_IsSemigroup_488 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488
v0)
-- Algebra.Structures.Biased._.IsSemigroup.isMagma
d_isMagma_2446 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_2446 :: T_IsSemigroup_488 -> T_IsMagma_178
d_isMagma_2446 T_IsSemigroup_488
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> T_IsMagma_178
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-assoc
d_'42''45'assoc_2582 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2582 :: T_IsSemiringWithoutAnnihilatingZero_1536
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2582 T_IsSemiringWithoutAnnihilatingZero_1536
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1560 (T_IsSemiringWithoutAnnihilatingZero_1536 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-cong
d_'42''45'cong_2584 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'cong_2584 :: T_IsSemiringWithoutAnnihilatingZero_1536
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'42''45'cong_2584 T_IsSemiringWithoutAnnihilatingZero_1536
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1558 (T_IsSemiringWithoutAnnihilatingZero_1536 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-identity
d_'42''45'identity_2590 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_'42''45'identity_2590 :: T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
d_'42''45'identity_2590 T_IsSemiringWithoutAnnihilatingZero_1536
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_'42''45'identity_1562 (T_IsSemiringWithoutAnnihilatingZero_1536 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_2620 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_2620 :: T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_2620 T_IsSemiringWithoutAnnihilatingZero_1536
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsSemiringWithoutAnnihilatingZero_1536
-> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1556
      (T_IsSemiringWithoutAnnihilatingZero_1536 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.distrib
d_distrib_2632 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2632 :: T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
d_distrib_2632 T_IsSemiringWithoutAnnihilatingZero_1536
v0
  = (T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14)
-> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1564 (T_IsSemiringWithoutAnnihilatingZero_1536 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero_1536
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.*-assoc
d_'42''45'assoc_2654 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2654 :: T_IsSemiringWithoutOne_1342
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'assoc_2654 T_IsSemiringWithoutOne_1342
v0
  = (T_IsSemiringWithoutOne_1342
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1364 (T_IsSemiringWithoutOne_1342 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.*-cong
d_'42''45'cong_2656 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'42''45'cong_2656 :: T_IsSemiringWithoutOne_1342
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'42''45'cong_2656 T_IsSemiringWithoutOne_1342
v0
  = (T_IsSemiringWithoutOne_1342
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1362 (T_IsSemiringWithoutOne_1342 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_2684 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_2684 :: T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_2684 T_IsSemiringWithoutOne_1342
v0
  = (T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsSemiringWithoutOne_1342 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1360
      (T_IsSemiringWithoutOne_1342 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.distrib
d_distrib_2690 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2690 :: T_IsSemiringWithoutOne_1342 -> T_Σ_14
d_distrib_2690 T_IsSemiringWithoutOne_1342
v0
  = (T_IsSemiringWithoutOne_1342 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_distrib_1366 (T_IsSemiringWithoutOne_1342 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v0)
-- Algebra.Structures.Biased._.IsSemiringWithoutOne.zero
d_zero_2712 ::
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_2712 :: T_IsSemiringWithoutOne_1342 -> T_Σ_14
d_zero_2712 T_IsSemiringWithoutOne_1342
v0
  = (T_IsSemiringWithoutOne_1342 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_zero_1368 (T_IsSemiringWithoutOne_1342 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne_1342
v0)
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ
d_IsCommutativeMonoid'737'_2770 :: p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeMonoid'737'_2770 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 = ()
data T_IsCommutativeMonoid'737'_2770
  = C_constructor_2820 MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
                       (AgdaAny -> AgdaAny) (AgdaAny -> AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ.isSemigroup
d_isSemigroup_2782 ::
  T_IsCommutativeMonoid'737'_2770 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_2782 :: T_IsCommutativeMonoid'737'_2770 -> T_IsSemigroup_488
d_isSemigroup_2782 T_IsCommutativeMonoid'737'_2770
v0
  = case T_IsCommutativeMonoid'737'_2770 -> T_IsCommutativeMonoid'737'_2770
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770
v0 of
      C_constructor_2820 T_IsSemigroup_488
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1
      T_IsCommutativeMonoid'737'_2770
_ -> T_IsSemigroup_488
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ.identityˡ
d_identity'737'_2784 ::
  T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny
d_identity'737'_2784 :: T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny
d_identity'737'_2784 T_IsCommutativeMonoid'737'_2770
v0
  = case T_IsCommutativeMonoid'737'_2770 -> T_IsCommutativeMonoid'737'_2770
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770
v0 of
      C_constructor_2820 T_IsSemigroup_488
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2
      T_IsCommutativeMonoid'737'_2770
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ.comm
d_comm_2786 ::
  T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2786 :: T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2786 T_IsCommutativeMonoid'737'_2770
v0
  = case T_IsCommutativeMonoid'737'_2770 -> T_IsCommutativeMonoid'737'_2770
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770
v0 of
      C_constructor_2820 T_IsSemigroup_488
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'_2770
_ -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidˡ.isCommutativeMonoid
d_isCommutativeMonoid_2788 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeMonoid'737'_2770 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_2788 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'737'_2770
-> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_2788 ~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'_2770
v6
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'737'_2770
-> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_2788 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 T_IsCommutativeMonoid'737'_2770
v6
du_isCommutativeMonoid_2788 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeMonoid'737'_2770 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_isCommutativeMonoid_2788 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'737'_2770
-> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_2788 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsCommutativeMonoid'737'_2770
v2
  = (T_IsMonoid_712
 -> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsMonoid_712
-> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.C_constructor_820
      ((T_IsSemigroup_488 -> T_Σ_14 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_Σ_14 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.C_constructor_758
         ((T_IsCommutativeMonoid'737'_2770 -> T_IsSemigroup_488)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770 -> T_IsSemigroup_488
d_isSemigroup_2782 (T_IsCommutativeMonoid'737'_2770 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770
v2))
         ((T_Setoid_46
 -> (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_46
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'id'737''8658'id_372
            ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                  ((T_IsCommutativeMonoid'737'_2770 -> T_IsSemigroup_488)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770 -> T_IsSemigroup_488
d_isSemigroup_2782 (T_IsCommutativeMonoid'737'_2770 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770
v2))))
            ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) ((T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2786 (T_IsCommutativeMonoid'737'_2770 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770
v2)) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v1)
            ((T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny
d_identity'737'_2784 (T_IsCommutativeMonoid'737'_2770 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770
v2))))
      ((T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2786 (T_IsCommutativeMonoid'737'_2770 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'737'_2770
v2))
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ
d_IsCommutativeMonoid'691'_2826 :: p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeMonoid'691'_2826 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 = ()
data T_IsCommutativeMonoid'691'_2826
  = C_constructor_2876 MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
                       (AgdaAny -> AgdaAny) (AgdaAny -> AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ.isSemigroup
d_isSemigroup_2838 ::
  T_IsCommutativeMonoid'691'_2826 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_2838 :: T_IsCommutativeMonoid'691'_2826 -> T_IsSemigroup_488
d_isSemigroup_2838 T_IsCommutativeMonoid'691'_2826
v0
  = case T_IsCommutativeMonoid'691'_2826 -> T_IsCommutativeMonoid'691'_2826
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826
v0 of
      C_constructor_2876 T_IsSemigroup_488
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v1
      T_IsCommutativeMonoid'691'_2826
_ -> T_IsSemigroup_488
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ.identityʳ
d_identity'691'_2840 ::
  T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny
d_identity'691'_2840 :: T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny
d_identity'691'_2840 T_IsCommutativeMonoid'691'_2826
v0
  = case T_IsCommutativeMonoid'691'_2826 -> T_IsCommutativeMonoid'691'_2826
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826
v0 of
      C_constructor_2876 T_IsSemigroup_488
v1 AgdaAny -> AgdaAny
v2 AgdaAny -> AgdaAny -> AgdaAny
v3 -> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny
v2
      T_IsCommutativeMonoid'691'_2826
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ.comm
d_comm_2842 ::
  T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2842 :: T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2842 T_IsCommutativeMonoid'691'_2826
v0
  = case T_IsCommutativeMonoid'691'_2826 -> T_IsCommutativeMonoid'691'_2826
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826
v0 of
      C_constructor_2876 T_IsSemigroup_488
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'_2826
_ -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeMonoidʳ.isCommutativeMonoid
d_isCommutativeMonoid_2844 ::
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  MAlonzo.Code.Agda.Primitive.T_Level_18 ->
  () ->
  (AgdaAny -> AgdaAny -> ()) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeMonoid'691'_2826 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_2844 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'691'_2826
-> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_2844 ~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'_2826
v6
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'691'_2826
-> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_2844 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny
v5 T_IsCommutativeMonoid'691'_2826
v6
du_isCommutativeMonoid_2844 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeMonoid'691'_2826 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_isCommutativeMonoid_2844 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeMonoid'691'_2826
-> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_2844 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny
v1 T_IsCommutativeMonoid'691'_2826
v2
  = (T_IsMonoid_712
 -> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsMonoid_712
-> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.C_constructor_820
      ((T_IsSemigroup_488 -> T_Σ_14 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_Σ_14 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.C_constructor_758
         ((T_IsCommutativeMonoid'691'_2826 -> T_IsSemigroup_488)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826 -> T_IsSemigroup_488
d_isSemigroup_2838 (T_IsCommutativeMonoid'691'_2826 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826
v2))
         ((T_Setoid_46
 -> (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_46
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'id'691''8658'id_376
            ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                  ((T_IsCommutativeMonoid'691'_2826 -> T_IsSemigroup_488)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826 -> T_IsSemigroup_488
d_isSemigroup_2838 (T_IsCommutativeMonoid'691'_2826 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826
v2))))
            ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v0) ((T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2842 (T_IsCommutativeMonoid'691'_2826 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826
v2)) (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v1)
            ((T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny
d_identity'691'_2840 (T_IsCommutativeMonoid'691'_2826 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826
v2))))
      ((T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826 -> AgdaAny -> AgdaAny -> AgdaAny
d_comm_2842 (T_IsCommutativeMonoid'691'_2826 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeMonoid'691'_2826
v2))
-- Algebra.Structures.Biased.IsSemiringWithoutOne*
d_IsSemiringWithoutOne'42'_2884 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsSemiringWithoutOne'42'_2884 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
data T_IsSemiringWithoutOne'42'_2884
  = C_constructor_2940 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
                       MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
                       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_2900 ::
  T_IsSemiringWithoutOne'42'_2884 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_2900 :: T_IsSemiringWithoutOne'42'_2884 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_2900 T_IsSemiringWithoutOne'42'_2884
v0
  = case T_IsSemiringWithoutOne'42'_2884 -> T_IsSemiringWithoutOne'42'_2884
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0 of
      C_constructor_2940 T_IsCommutativeMonoid_764
v1 T_IsSemigroup_488
v2 T_Σ_14
v3 T_Σ_14
v4 -> T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v1
      T_IsSemiringWithoutOne'42'_2884
_ -> T_IsCommutativeMonoid_764
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.*-isSemigroup
d_'42''45'isSemigroup_2902 ::
  T_IsSemiringWithoutOne'42'_2884 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_'42''45'isSemigroup_2902 :: T_IsSemiringWithoutOne'42'_2884 -> T_IsSemigroup_488
d_'42''45'isSemigroup_2902 T_IsSemiringWithoutOne'42'_2884
v0
  = case T_IsSemiringWithoutOne'42'_2884 -> T_IsSemiringWithoutOne'42'_2884
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0 of
      C_constructor_2940 T_IsCommutativeMonoid_764
v1 T_IsSemigroup_488
v2 T_Σ_14
v3 T_Σ_14
v4 -> T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2
      T_IsSemiringWithoutOne'42'_2884
_ -> T_IsSemigroup_488
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.distrib
d_distrib_2904 ::
  T_IsSemiringWithoutOne'42'_2884 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_2904 :: T_IsSemiringWithoutOne'42'_2884 -> T_Σ_14
d_distrib_2904 T_IsSemiringWithoutOne'42'_2884
v0
  = case T_IsSemiringWithoutOne'42'_2884 -> T_IsSemiringWithoutOne'42'_2884
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0 of
      C_constructor_2940 T_IsCommutativeMonoid_764
v1 T_IsSemigroup_488
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'_2884
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.zero
d_zero_2906 ::
  T_IsSemiringWithoutOne'42'_2884 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_2906 :: T_IsSemiringWithoutOne'42'_2884 -> T_Σ_14
d_zero_2906 T_IsSemiringWithoutOne'42'_2884
v0
  = case T_IsSemiringWithoutOne'42'_2884 -> T_IsSemiringWithoutOne'42'_2884
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0 of
      C_constructor_2940 T_IsCommutativeMonoid_764
v1 T_IsSemigroup_488
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'_2884
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutOne*.isSemiringWithoutOne
d_isSemiringWithoutOne_2908 ::
  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'_2884 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_2908 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsSemiringWithoutOne'42'_2884
-> T_IsSemiringWithoutOne_1342
d_isSemiringWithoutOne_2908 ~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'_2884
v7
  = T_IsSemiringWithoutOne'42'_2884 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_2908 T_IsSemiringWithoutOne'42'_2884
v7
du_isSemiringWithoutOne_2908 ::
  T_IsSemiringWithoutOne'42'_2884 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_2908 :: T_IsSemiringWithoutOne'42'_2884 -> T_IsSemiringWithoutOne_1342
du_isSemiringWithoutOne_2908 T_IsSemiringWithoutOne'42'_2884
v0
  = (T_IsCommutativeMonoid_764
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutOne_1342)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutOne_1342
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutOne_1342
MAlonzo.Code.Algebra.Structures.C_constructor_1430
      ((T_IsSemiringWithoutOne'42'_2884 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_2900 (T_IsSemiringWithoutOne'42'_2884 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0))
      ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsSemiringWithoutOne'42'_2884 -> T_IsSemigroup_488)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884 -> T_IsSemigroup_488
d_'42''45'isSemigroup_2902 (T_IsSemiringWithoutOne'42'_2884 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0))))
      ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
         ((T_IsSemiringWithoutOne'42'_2884 -> T_IsSemigroup_488)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884 -> T_IsSemigroup_488
d_'42''45'isSemigroup_2902 (T_IsSemiringWithoutOne'42'_2884 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0)))
      ((T_IsSemiringWithoutOne'42'_2884 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884 -> T_Σ_14
d_distrib_2904 (T_IsSemiringWithoutOne'42'_2884 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0)) ((T_IsSemiringWithoutOne'42'_2884 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884 -> T_Σ_14
d_zero_2906 (T_IsSemiringWithoutOne'42'_2884 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutOne'42'_2884
v0))
-- Algebra.Structures.Biased.IsNearSemiring*
d_IsNearSemiring'42'_2948 :: p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsNearSemiring'42'_2948 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 = ()
data T_IsNearSemiring'42'_2948
  = C_constructor_3004 MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
                       MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
                       (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) (AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsNearSemiring*.+-isMonoid
d_'43''45'isMonoid_2964 ::
  T_IsNearSemiring'42'_2948 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'43''45'isMonoid_2964 :: T_IsNearSemiring'42'_2948 -> T_IsMonoid_712
d_'43''45'isMonoid_2964 T_IsNearSemiring'42'_2948
v0
  = case T_IsNearSemiring'42'_2948 -> T_IsNearSemiring'42'_2948
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0 of
      C_constructor_3004 T_IsMonoid_712
v1 T_IsSemigroup_488
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1
      T_IsNearSemiring'42'_2948
_ -> T_IsMonoid_712
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsNearSemiring*.*-isSemigroup
d_'42''45'isSemigroup_2966 ::
  T_IsNearSemiring'42'_2948 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_'42''45'isSemigroup_2966 :: T_IsNearSemiring'42'_2948 -> T_IsSemigroup_488
d_'42''45'isSemigroup_2966 T_IsNearSemiring'42'_2948
v0
  = case T_IsNearSemiring'42'_2948 -> T_IsNearSemiring'42'_2948
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0 of
      C_constructor_3004 T_IsMonoid_712
v1 T_IsSemigroup_488
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2
      T_IsNearSemiring'42'_2948
_ -> T_IsSemigroup_488
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsNearSemiring*.distribʳ
d_distrib'691'_2968 ::
  T_IsNearSemiring'42'_2948 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2968 :: T_IsNearSemiring'42'_2948
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2968 T_IsNearSemiring'42'_2948
v0
  = case T_IsNearSemiring'42'_2948 -> T_IsNearSemiring'42'_2948
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0 of
      C_constructor_3004 T_IsMonoid_712
v1 T_IsSemigroup_488
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'_2948
_ -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsNearSemiring*.zeroˡ
d_zero'737'_2970 :: T_IsNearSemiring'42'_2948 -> AgdaAny -> AgdaAny
d_zero'737'_2970 :: T_IsNearSemiring'42'_2948 -> AgdaAny -> AgdaAny
d_zero'737'_2970 T_IsNearSemiring'42'_2948
v0
  = case T_IsNearSemiring'42'_2948 -> T_IsNearSemiring'42'_2948
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0 of
      C_constructor_3004 T_IsMonoid_712
v1 T_IsSemigroup_488
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'_2948
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsNearSemiring*.isNearSemiring
d_isNearSemiring_2972 ::
  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'_2948 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
d_isNearSemiring_2972 :: T_Level_18
-> T_Level_18
-> T_Level_18
-> (AgdaAny -> AgdaAny -> T_Level_18)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsNearSemiring'42'_2948
-> T_IsNearSemiring_1260
d_isNearSemiring_2972 ~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'_2948
v7
  = T_IsNearSemiring'42'_2948 -> T_IsNearSemiring_1260
du_isNearSemiring_2972 T_IsNearSemiring'42'_2948
v7
du_isNearSemiring_2972 ::
  T_IsNearSemiring'42'_2948 ->
  MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1260
du_isNearSemiring_2972 :: T_IsNearSemiring'42'_2948 -> T_IsNearSemiring_1260
du_isNearSemiring_2972 T_IsNearSemiring'42'_2948
v0
  = (T_IsMonoid_712
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> T_IsNearSemiring_1260)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsNearSemiring_1260
forall a b. a -> b
coe
      T_IsMonoid_712
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> T_IsNearSemiring_1260
MAlonzo.Code.Algebra.Structures.C_constructor_1334
      ((T_IsNearSemiring'42'_2948 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948 -> T_IsMonoid_712
d_'43''45'isMonoid_2964 (T_IsNearSemiring'42'_2948 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0))
      ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsNearSemiring'42'_2948 -> T_IsSemigroup_488)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948 -> T_IsSemigroup_488
d_'42''45'isSemigroup_2966 (T_IsNearSemiring'42'_2948 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0))))
      ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
         ((T_IsNearSemiring'42'_2948 -> T_IsSemigroup_488)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948 -> T_IsSemigroup_488
d_'42''45'isSemigroup_2966 (T_IsNearSemiring'42'_2948 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0)))
      ((T_IsNearSemiring'42'_2948
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_2968 (T_IsNearSemiring'42'_2948 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0)) ((T_IsNearSemiring'42'_2948 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948 -> AgdaAny -> AgdaAny
d_zero'737'_2970 (T_IsNearSemiring'42'_2948 -> AgdaAny
forall a b. a -> b
coe T_IsNearSemiring'42'_2948
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*
d_IsSemiringWithoutAnnihilatingZero'42'_3014 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsSemiringWithoutAnnihilatingZero'42'_3014 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6
                                             p
a7
  = ()
data T_IsSemiringWithoutAnnihilatingZero'42'_3014
  = C_constructor_3078 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
                       MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
                       MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_3030 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3030 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3030 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = case T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero'42'_3014
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0 of
      C_constructor_3078 T_IsCommutativeMonoid_764
v1 T_IsMonoid_712
v2 T_Σ_14
v3 -> T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v1
      T_IsSemiringWithoutAnnihilatingZero'42'_3014
_ -> T_IsCommutativeMonoid_764
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.*-isMonoid
d_'42''45'isMonoid_3032 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'42''45'isMonoid_3032 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = case T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero'42'_3014
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0 of
      C_constructor_3078 T_IsCommutativeMonoid_764
v1 T_IsMonoid_712
v2 T_Σ_14
v3 -> T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2
      T_IsSemiringWithoutAnnihilatingZero'42'_3014
_ -> T_IsMonoid_712
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.distrib
d_distrib_3034 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_3034 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_Σ_14
d_distrib_3034 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = case T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero'42'_3014
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0 of
      C_constructor_3078 T_IsCommutativeMonoid_764
v1 T_IsMonoid_712
v2 T_Σ_14
v3 -> T_Σ_14 -> T_Σ_14
forall a b. a -> b
coe T_Σ_14
v3
      T_IsSemiringWithoutAnnihilatingZero'42'_3014
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.isSemiringWithoutAnnihilatingZero
d_isSemiringWithoutAnnihilatingZero_3036 ::
  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'_3014 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_3036 :: 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'_3014
-> T_IsSemiringWithoutAnnihilatingZero_1536
d_isSemiringWithoutAnnihilatingZero_3036 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero_1536
du_isSemiringWithoutAnnihilatingZero_3036 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_isSemiringWithoutAnnihilatingZero_3036 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1536
du_isSemiringWithoutAnnihilatingZero_3036 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero_1536
du_isSemiringWithoutAnnihilatingZero_3036 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = (T_IsCommutativeMonoid_764
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsSemiringWithoutAnnihilatingZero_1536
forall a b. a -> b
coe
      T_IsCommutativeMonoid_764
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.C_constructor_1630
      ((T_IsSemiringWithoutAnnihilatingZero'42'_3014
 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3030 (T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0))
      ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0)))))
      ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0))))
      ((T_IsMonoid_712 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
         ((T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0)))
      ((T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_Σ_14)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_Σ_14
d_distrib_3034 (T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.identityʳ
d_identity'691'_3048 ::
  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'_3014 -> AgdaAny -> AgdaAny
d_identity'691'_3048 :: 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'_3014
-> AgdaAny
-> AgdaAny
d_identity'691'_3048 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny -> AgdaAny
du_identity'691'_3048 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_identity'691'_3048 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny -> AgdaAny
du_identity'691'_3048 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny -> AgdaAny
du_identity'691'_3048 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = (T_IsMonoid_712 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_754
      ((T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.identityˡ
d_identity'737'_3050 ::
  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'_3014 -> AgdaAny -> AgdaAny
d_identity'737'_3050 :: 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'_3014
-> AgdaAny
-> AgdaAny
d_identity'737'_3050 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny -> AgdaAny
du_identity'737'_3050 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_identity'737'_3050 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny -> AgdaAny
du_identity'737'_3050 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny -> AgdaAny
du_identity'737'_3050 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = (T_IsMonoid_712 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_752
      ((T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.isPartialEquivalence
d_isPartialEquivalence_3056 ::
  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'_3014 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_3056 :: 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'_3014
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_3056 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsPartialEquivalence_16
du_isPartialEquivalence_3056 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_isPartialEquivalence_3056 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_3056 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsPartialEquivalence_16
du_isPartialEquivalence_3056 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero'42'_3014
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.isUnitalMagma
d_isUnitalMagma_3060 ::
  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'_3014 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_3060 :: 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'_3014
-> T_IsUnitalMagma_666
d_isUnitalMagma_3060 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsUnitalMagma_666
du_isUnitalMagma_3060 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_isUnitalMagma_3060 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_3060 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsUnitalMagma_666
du_isUnitalMagma_3060 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = (T_IsMonoid_712 -> T_IsUnitalMagma_666)
-> AgdaAny -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
      ((T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> AgdaAny
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.reflexive
d_reflexive_3064 ::
  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'_3014 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_3064 :: 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'_3014
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_3064 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3064 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_reflexive_3064 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_3064 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3064 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero'42'_3014
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
            (\ AgdaAny
v4 AgdaAny
v5 AgdaAny
v6 ->
               (T_IsEquivalence_28 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                 T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_42
                 ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3))
                 AgdaAny
v4)))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.setoid
d_setoid_3066 ::
  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'_3014 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_3066 :: 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'_3014
-> T_Setoid_46
d_setoid_3066 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_Setoid_46
du_setoid_3066 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_setoid_3066 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_3066 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_Setoid_46
du_setoid_3066 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero'42'_3014
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0) in
    AgdaAny -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488
v2))))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.∙-congʳ
d_'8729''45'cong'691'_3074 ::
  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'_3014 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_3074 :: 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'_3014
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_3074 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3074 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_'8729''45'cong'691'_3074 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3074 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3074 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero'42'_3014
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: AgdaAny
v4
                   = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5
                      = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v3) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'691'_46
                     ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5)
                     ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                        ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                           (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v4))))))))
-- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.∙-congˡ
d_'8729''45'cong'737'_3076 ::
  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'_3014 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_3076 :: 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'_3014
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_3076 ~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'_3014
v8
  = T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3076 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v8
du_'8729''45'cong'737'_3076 ::
  T_IsSemiringWithoutAnnihilatingZero'42'_3014 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3076 :: T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3076 T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsSemiringWithoutAnnihilatingZero'42'_3014 -> T_IsMonoid_712
d_'42''45'isMonoid_3032 (T_IsSemiringWithoutAnnihilatingZero'42'_3014
-> T_IsSemiringWithoutAnnihilatingZero'42'_3014
forall a b. a -> b
coe T_IsSemiringWithoutAnnihilatingZero'42'_3014
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: AgdaAny
v4
                   = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5
                      = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v3) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'737'_42
                     ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5)
                     ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                        ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                           (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v4))))))))
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ
d_IsCommutativeSemiring'737'_3088 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeSemiring'737'_3088 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 = ()
data T_IsCommutativeSemiring'737'_3088
  = C_constructor_3208 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
                       MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
                       (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) (AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_3106 ::
  T_IsCommutativeSemiring'737'_3088 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3106 :: T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3106 T_IsCommutativeSemiring'737'_3088
v0
  = case T_IsCommutativeSemiring'737'_3088
-> T_IsCommutativeSemiring'737'_3088
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v0 of
      C_constructor_3208 T_IsCommutativeMonoid_764
v1 T_IsCommutativeMonoid_764
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v1
      T_IsCommutativeSemiring'737'_3088
_ -> T_IsCommutativeMonoid_764
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_3108 ::
  T_IsCommutativeSemiring'737'_3088 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3108 :: T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3108 T_IsCommutativeSemiring'737'_3088
v0
  = case T_IsCommutativeSemiring'737'_3088
-> T_IsCommutativeSemiring'737'_3088
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v0 of
      C_constructor_3208 T_IsCommutativeMonoid_764
v1 T_IsCommutativeMonoid_764
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v2
      T_IsCommutativeSemiring'737'_3088
_ -> T_IsCommutativeMonoid_764
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.distribʳ
d_distrib'691'_3110 ::
  T_IsCommutativeSemiring'737'_3088 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_3110 :: T_IsCommutativeSemiring'737'_3088
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_3110 T_IsCommutativeSemiring'737'_3088
v0
  = case T_IsCommutativeSemiring'737'_3088
-> T_IsCommutativeSemiring'737'_3088
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v0 of
      C_constructor_3208 T_IsCommutativeMonoid_764
v1 T_IsCommutativeMonoid_764
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'_3088
_ -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.zeroˡ
d_zero'737'_3112 ::
  T_IsCommutativeSemiring'737'_3088 -> AgdaAny -> AgdaAny
d_zero'737'_3112 :: T_IsCommutativeSemiring'737'_3088 -> AgdaAny -> AgdaAny
d_zero'737'_3112 T_IsCommutativeSemiring'737'_3088
v0
  = case T_IsCommutativeSemiring'737'_3088
-> T_IsCommutativeSemiring'737'_3088
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v0 of
      C_constructor_3208 T_IsCommutativeMonoid_764
v1 T_IsCommutativeMonoid_764
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'_3088
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringˡ.isCommutativeSemiring
d_isCommutativeSemiring_3114 ::
  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'_3088 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_3114 :: 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'_3088
-> T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_3114 ~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'_3088
v8
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeSemiring'737'_3088
-> T_IsCommutativeSemiring_1750
du_isCommutativeSemiring_3114 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny
v6 T_IsCommutativeSemiring'737'_3088
v8
du_isCommutativeSemiring_3114 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeSemiring'737'_3088 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750
du_isCommutativeSemiring_3114 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeSemiring'737'_3088
-> T_IsCommutativeSemiring_1750
du_isCommutativeSemiring_3114 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny
v2 T_IsCommutativeSemiring'737'_3088
v3
  = (T_IsSemiring_1640
 -> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeSemiring_1750)
-> AgdaAny -> AgdaAny -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe
      T_IsSemiring_1640
-> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.C_constructor_1862
      ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_Σ_14 -> T_IsSemiring_1640)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536
-> T_Σ_14 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.C_constructor_1740
         ((T_IsCommutativeMonoid_764
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.C_constructor_1630
            ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3106 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3))
            ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                  ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                     ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                        ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3108 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3))))))
            ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
               ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                  ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                     ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3108 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3)))))
            ((T_IsMonoid_712 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
               ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                  ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3108 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3))))
            ((T_Setoid_46
 -> (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_46
-> (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_560
               ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                  ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                     ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                        ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                           ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3106 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3))))))
               ((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_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
                  ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                     ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                        ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                           ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3106 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3))))))
               ((T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_776
                  ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3108 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3)))
               ((T_IsCommutativeSemiring'737'_3088
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'691'_3110 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3))))
         ((T_Setoid_46
 -> (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_46
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'ze'737''8658'ze_392
            ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                  ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                     ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                        ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3106 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3))))))
            ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1)
            ((T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_776
               ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3108 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3)))
            (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v2) ((T_IsCommutativeSemiring'737'_3088 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> AgdaAny -> AgdaAny
d_zero'737'_3112 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3))))
      ((T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_776
         ((T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3108 (T_IsCommutativeSemiring'737'_3088 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'737'_3088
v3)))
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ
d_IsCommutativeSemiring'691'_3218 :: p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsCommutativeSemiring'691'_3218 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 = ()
data T_IsCommutativeSemiring'691'_3218
  = C_constructor_3338 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
                       MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
                       (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) (AgdaAny -> AgdaAny)
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.+-isCommutativeMonoid
d_'43''45'isCommutativeMonoid_3236 ::
  T_IsCommutativeSemiring'691'_3218 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3236 :: T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3236 T_IsCommutativeSemiring'691'_3218
v0
  = case T_IsCommutativeSemiring'691'_3218
-> T_IsCommutativeSemiring'691'_3218
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v0 of
      C_constructor_3338 T_IsCommutativeMonoid_764
v1 T_IsCommutativeMonoid_764
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v1
      T_IsCommutativeSemiring'691'_3218
_ -> T_IsCommutativeMonoid_764
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.*-isCommutativeMonoid
d_'42''45'isCommutativeMonoid_3238 ::
  T_IsCommutativeSemiring'691'_3218 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3238 :: T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3238 T_IsCommutativeSemiring'691'_3218
v0
  = case T_IsCommutativeSemiring'691'_3218
-> T_IsCommutativeSemiring'691'_3218
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v0 of
      C_constructor_3338 T_IsCommutativeMonoid_764
v1 T_IsCommutativeMonoid_764
v2 AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v3 AgdaAny -> AgdaAny
v4 -> T_IsCommutativeMonoid_764 -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe T_IsCommutativeMonoid_764
v2
      T_IsCommutativeSemiring'691'_3218
_ -> T_IsCommutativeMonoid_764
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.distribˡ
d_distrib'737'_3240 ::
  T_IsCommutativeSemiring'691'_3218 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'737'_3240 :: T_IsCommutativeSemiring'691'_3218
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'737'_3240 T_IsCommutativeSemiring'691'_3218
v0
  = case T_IsCommutativeSemiring'691'_3218
-> T_IsCommutativeSemiring'691'_3218
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v0 of
      C_constructor_3338 T_IsCommutativeMonoid_764
v1 T_IsCommutativeMonoid_764
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'_3218
_ -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.zeroʳ
d_zero'691'_3242 ::
  T_IsCommutativeSemiring'691'_3218 -> AgdaAny -> AgdaAny
d_zero'691'_3242 :: T_IsCommutativeSemiring'691'_3218 -> AgdaAny -> AgdaAny
d_zero'691'_3242 T_IsCommutativeSemiring'691'_3218
v0
  = case T_IsCommutativeSemiring'691'_3218
-> T_IsCommutativeSemiring'691'_3218
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v0 of
      C_constructor_3338 T_IsCommutativeMonoid_764
v1 T_IsCommutativeMonoid_764
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'_3218
_ -> AgdaAny -> AgdaAny
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsCommutativeSemiringʳ.isCommutativeSemiring
d_isCommutativeSemiring_3244 ::
  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'_3218 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_3244 :: 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'_3218
-> T_IsCommutativeSemiring_1750
d_isCommutativeSemiring_3244 ~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'_3218
v8
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeSemiring'691'_3218
-> T_IsCommutativeSemiring_1750
du_isCommutativeSemiring_3244 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny
v6 T_IsCommutativeSemiring'691'_3218
v8
du_isCommutativeSemiring_3244 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsCommutativeSemiring'691'_3218 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1750
du_isCommutativeSemiring_3244 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsCommutativeSemiring'691'_3218
-> T_IsCommutativeSemiring_1750
du_isCommutativeSemiring_3244 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny
v2 T_IsCommutativeSemiring'691'_3218
v3
  = (T_IsSemiring_1640
 -> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeSemiring_1750)
-> AgdaAny -> AgdaAny -> T_IsCommutativeSemiring_1750
forall a b. a -> b
coe
      T_IsSemiring_1640
-> (AgdaAny -> AgdaAny -> AgdaAny) -> T_IsCommutativeSemiring_1750
MAlonzo.Code.Algebra.Structures.C_constructor_1862
      ((T_IsSemiringWithoutAnnihilatingZero_1536
 -> T_Σ_14 -> T_IsSemiring_1640)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemiringWithoutAnnihilatingZero_1536
-> T_Σ_14 -> T_IsSemiring_1640
MAlonzo.Code.Algebra.Structures.C_constructor_1740
         ((T_IsCommutativeMonoid_764
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsSemiringWithoutAnnihilatingZero_1536)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsCommutativeMonoid_764
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsSemiringWithoutAnnihilatingZero_1536
MAlonzo.Code.Algebra.Structures.C_constructor_1630
            ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3236 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3))
            ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                  ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                     ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                        ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3238 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3))))))
            ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
               ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                  ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                     ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3238 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3)))))
            ((T_IsMonoid_712 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
               ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                  ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3238 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3))))
            ((T_Setoid_46
 -> (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_46
-> (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_556
               ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                  ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                     ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                        ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                           ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3236 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3))))))
               ((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_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
                  ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                     ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                        ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                           ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3236 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3))))))
               ((T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_776
                  ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3238 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3)))
               ((T_IsCommutativeSemiring'691'_3218
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_distrib'737'_3240 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3))))
         ((T_Setoid_46
 -> (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_46
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_Σ_14
MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'ze'691''8658'ze_396
            ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
               ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
                  ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
                     ((T_IsCommutativeMonoid_764 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsCommutativeMonoid_764 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_774
                        ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'43''45'isCommutativeMonoid_3236 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3))))))
            ((AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe AgdaAny -> AgdaAny -> AgdaAny
v1)
            ((T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_776
               ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3238 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3)))
            (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v2) ((T_IsCommutativeSemiring'691'_3218 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> AgdaAny -> AgdaAny
d_zero'691'_3242 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3))))
      ((T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_776
         ((T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218 -> T_IsCommutativeMonoid_764
d_'42''45'isCommutativeMonoid_3238 (T_IsCommutativeSemiring'691'_3218 -> AgdaAny
forall a b. a -> b
coe T_IsCommutativeSemiring'691'_3218
v3)))
-- Algebra.Structures.Biased.IsRing*
d_IsRing'42'_3350 :: p -> p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsRing'42'_3350 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 p
a8 = ()
data T_IsRing'42'_3350
  = C_constructor_3420 MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172
                       MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
                       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_3370 ::
  T_IsRing'42'_3350 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3370 :: T_IsRing'42'_3350 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3370 T_IsRing'42'_3350
v0
  = case T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0 of
      C_constructor_3420 T_IsAbelianGroup_1172
v1 T_IsMonoid_712
v2 T_Σ_14
v3 T_Σ_14
v4 -> T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1
      T_IsRing'42'_3350
_ -> T_IsAbelianGroup_1172
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRing*.*-isMonoid
d_'42''45'isMonoid_3372 ::
  T_IsRing'42'_3350 -> MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'42''45'isMonoid_3372 :: T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 T_IsRing'42'_3350
v0
  = case T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0 of
      C_constructor_3420 T_IsAbelianGroup_1172
v1 T_IsMonoid_712
v2 T_Σ_14
v3 T_Σ_14
v4 -> T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2
      T_IsRing'42'_3350
_ -> T_IsMonoid_712
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRing*.distrib
d_distrib_3374 ::
  T_IsRing'42'_3350 -> MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_3374 :: T_IsRing'42'_3350 -> T_Σ_14
d_distrib_3374 T_IsRing'42'_3350
v0
  = case T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0 of
      C_constructor_3420 T_IsAbelianGroup_1172
v1 T_IsMonoid_712
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'_3350
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRing*.zero
d_zero_3376 ::
  T_IsRing'42'_3350 -> MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_3376 :: T_IsRing'42'_3350 -> T_Σ_14
d_zero_3376 T_IsRing'42'_3350
v0
  = case T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0 of
      C_constructor_3420 T_IsAbelianGroup_1172
v1 T_IsMonoid_712
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'_3350
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRing*.isRing
d_isRing_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_IsRing'42'_3350 -> MAlonzo.Code.Algebra.Structures.T_IsRing_2740
d_isRing_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_IsRing'42'_3350
-> T_IsRing_2740
d_isRing_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_IsRing'42'_3350
v9
  = T_IsRing'42'_3350 -> T_IsRing_2740
du_isRing_3378 T_IsRing'42'_3350
v9
du_isRing_3378 ::
  T_IsRing'42'_3350 -> MAlonzo.Code.Algebra.Structures.T_IsRing_2740
du_isRing_3378 :: T_IsRing'42'_3350 -> T_IsRing_2740
du_isRing_3378 T_IsRing'42'_3350
v0
  = (T_IsAbelianGroup_1172
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsRing_2740)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsRing_2740
forall a b. a -> b
coe
      T_IsAbelianGroup_1172
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.C_constructor_2876
      ((T_IsRing'42'_3350 -> T_IsAbelianGroup_1172) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3370 (T_IsRing'42'_3350 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350
v0))
      ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsRing'42'_3350 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350
v0)))))
      ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsRing'42'_3350 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350
v0))))
      ((T_IsMonoid_712 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
         ((T_IsRing'42'_3350 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350
v0)))
      ((T_IsRing'42'_3350 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350 -> T_Σ_14
d_distrib_3374 (T_IsRing'42'_3350 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350
v0))
-- Algebra.Structures.Biased.IsRing*._._.identityʳ
d_identity'691'_3390 ::
  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'_3350 -> AgdaAny -> AgdaAny
d_identity'691'_3390 :: 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'_3350
-> AgdaAny
-> AgdaAny
d_identity'691'_3390 ~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'_3350
v9
  = T_IsRing'42'_3350 -> AgdaAny -> AgdaAny
du_identity'691'_3390 T_IsRing'42'_3350
v9
du_identity'691'_3390 :: T_IsRing'42'_3350 -> AgdaAny -> AgdaAny
du_identity'691'_3390 :: T_IsRing'42'_3350 -> AgdaAny -> AgdaAny
du_identity'691'_3390 T_IsRing'42'_3350
v0
  = (T_IsMonoid_712 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_754
      ((T_IsRing'42'_3350 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350
v0))
-- Algebra.Structures.Biased.IsRing*._._.identityˡ
d_identity'737'_3392 ::
  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'_3350 -> AgdaAny -> AgdaAny
d_identity'737'_3392 :: 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'_3350
-> AgdaAny
-> AgdaAny
d_identity'737'_3392 ~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'_3350
v9
  = T_IsRing'42'_3350 -> AgdaAny -> AgdaAny
du_identity'737'_3392 T_IsRing'42'_3350
v9
du_identity'737'_3392 :: T_IsRing'42'_3350 -> AgdaAny -> AgdaAny
du_identity'737'_3392 :: T_IsRing'42'_3350 -> AgdaAny -> AgdaAny
du_identity'737'_3392 T_IsRing'42'_3350
v0
  = (T_IsMonoid_712 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_752
      ((T_IsRing'42'_3350 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350
v0))
-- Algebra.Structures.Biased.IsRing*._._.isPartialEquivalence
d_isPartialEquivalence_3398 ::
  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'_3350 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_3398 :: 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'_3350
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_3398 ~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'_3350
v9
  = T_IsRing'42'_3350 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3398 T_IsRing'42'_3350
v9
du_isPartialEquivalence_3398 ::
  T_IsRing'42'_3350 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_3398 :: T_IsRing'42'_3350 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3398 T_IsRing'42'_3350
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Algebra.Structures.Biased.IsRing*._._.isUnitalMagma
d_isUnitalMagma_3402 ::
  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'_3350 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_3402 :: 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'_3350
-> T_IsUnitalMagma_666
d_isUnitalMagma_3402 ~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'_3350
v9
  = T_IsRing'42'_3350 -> T_IsUnitalMagma_666
du_isUnitalMagma_3402 T_IsRing'42'_3350
v9
du_isUnitalMagma_3402 ::
  T_IsRing'42'_3350 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_3402 :: T_IsRing'42'_3350 -> T_IsUnitalMagma_666
du_isUnitalMagma_3402 T_IsRing'42'_3350
v0
  = (T_IsMonoid_712 -> T_IsUnitalMagma_666)
-> AgdaAny -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
      ((T_IsRing'42'_3350 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> AgdaAny
forall a b. a -> b
coe T_IsRing'42'_3350
v0))
-- Algebra.Structures.Biased.IsRing*._._.reflexive
d_reflexive_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_IsRing'42'_3350 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_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_IsRing'42'_3350
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_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_IsRing'42'_3350
v9
  = T_IsRing'42'_3350 -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3406 T_IsRing'42'_3350
v9
du_reflexive_3406 ::
  T_IsRing'42'_3350 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_3406 :: T_IsRing'42'_3350 -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3406 T_IsRing'42'_3350
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
            (\ AgdaAny
v4 AgdaAny
v5 AgdaAny
v6 ->
               (T_IsEquivalence_28 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                 T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_42
                 ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3))
                 AgdaAny
v4)))
-- Algebra.Structures.Biased.IsRing*._._.setoid
d_setoid_3408 ::
  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'_3350 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_3408 :: 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'_3350
-> T_Setoid_46
d_setoid_3408 ~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'_3350
v9
  = T_IsRing'42'_3350 -> T_Setoid_46
du_setoid_3408 T_IsRing'42'_3350
v9
du_setoid_3408 ::
  T_IsRing'42'_3350 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_3408 :: T_IsRing'42'_3350 -> T_Setoid_46
du_setoid_3408 T_IsRing'42'_3350
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0) in
    AgdaAny -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488
v2))))
-- Algebra.Structures.Biased.IsRing*._._.∙-congʳ
d_'8729''45'cong'691'_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_IsRing'42'_3350 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_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_IsRing'42'_3350
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_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_IsRing'42'_3350
v9
  = T_IsRing'42'_3350
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3416 T_IsRing'42'_3350
v9
du_'8729''45'cong'691'_3416 ::
  T_IsRing'42'_3350 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3416 :: T_IsRing'42'_3350
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3416 T_IsRing'42'_3350
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: AgdaAny
v4
                   = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5
                      = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v3) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'691'_46
                     ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5)
                     ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                        ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                           (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v4))))))))
-- Algebra.Structures.Biased.IsRing*._._.∙-congˡ
d_'8729''45'cong'737'_3418 ::
  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'_3350 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_3418 :: 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'_3350
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_3418 ~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'_3350
v9
  = T_IsRing'42'_3350
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3418 T_IsRing'42'_3350
v9
du_'8729''45'cong'737'_3418 ::
  T_IsRing'42'_3350 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3418 :: T_IsRing'42'_3350
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3418 T_IsRing'42'_3350
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRing'42'_3350 -> T_IsMonoid_712
d_'42''45'isMonoid_3372 (T_IsRing'42'_3350 -> T_IsRing'42'_3350
forall a b. a -> b
coe T_IsRing'42'_3350
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: AgdaAny
v4
                   = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5
                      = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v3) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'737'_42
                     ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5)
                     ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                        ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                           (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v4))))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero
d_IsRingWithoutAnnihilatingZero_3432 :: p -> p -> p -> p -> p -> p -> p -> p -> p -> T_Level_18
d_IsRingWithoutAnnihilatingZero_3432 p
a0 p
a1 p
a2 p
a3 p
a4 p
a5 p
a6 p
a7 p
a8
  = ()
data T_IsRingWithoutAnnihilatingZero_3432
  = C_constructor_3566 MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172
                       MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
                       MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+-isAbelianGroup
d_'43''45'isAbelianGroup_3450 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 T_IsRingWithoutAnnihilatingZero_3432
v0
  = case T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0 of
      C_constructor_3566 T_IsAbelianGroup_1172
v1 T_IsMonoid_712
v2 T_Σ_14
v3 -> T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1
      T_IsRingWithoutAnnihilatingZero_3432
_ -> T_IsAbelianGroup_1172
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*-isMonoid
d_'42''45'isMonoid_3452 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_'42''45'isMonoid_3452 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 T_IsRingWithoutAnnihilatingZero_3432
v0
  = case T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0 of
      C_constructor_3566 T_IsAbelianGroup_1172
v1 T_IsMonoid_712
v2 T_Σ_14
v3 -> T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v2
      T_IsRingWithoutAnnihilatingZero_3432
_ -> T_IsMonoid_712
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.distrib
d_distrib_3454 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_distrib_3454 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_Σ_14
d_distrib_3454 T_IsRingWithoutAnnihilatingZero_3432
v0
  = case T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0 of
      C_constructor_3566 T_IsAbelianGroup_1172
v1 T_IsMonoid_712
v2 T_Σ_14
v3 -> T_Σ_14 -> T_Σ_14
forall a b. a -> b
coe T_Σ_14
v3
      T_IsRingWithoutAnnihilatingZero_3432
_ -> T_Σ_14
forall a. a
MAlonzo.RTE.mazUnreachableError
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+._//_
d__'47''47'__3458 ::
  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_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny
d__'47''47'__3458 :: 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_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
d__'47''47'__3458 ~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_3432
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__3458 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny
v6
du__'47''47'__3458 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__3458 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
du__'47''47'__3458 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'__1136 ((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_3460 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_3460 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_3460 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
            ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.comm
d_comm_3462 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny
d_comm_3462 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny
d_comm_3462 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsAbelianGroup_1172 -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_comm_1186
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identity
d_identity_3464 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_3464 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_Σ_14
d_identity_3464 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
            ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identityʳ
d_identity'691'_3466 ::
  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_3432 -> AgdaAny -> AgdaAny
d_identity'691'_3466 :: 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_3432
-> AgdaAny
-> AgdaAny
d_identity'691'_3466 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'691'_3466 T_IsRingWithoutAnnihilatingZero_3432
v9
du_identity'691'_3466 ::
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'691'_3466 :: T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'691'_3466 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1074
v2
             = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMonoid_712 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_754
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identityˡ
d_identity'737'_3468 ::
  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_3432 -> AgdaAny -> AgdaAny
d_identity'737'_3468 :: 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_3432
-> AgdaAny
-> AgdaAny
d_identity'737'_3468 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'737'_3468 T_IsRingWithoutAnnihilatingZero_3432
v9
du_identity'737'_3468 ::
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'737'_3468 :: T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'737'_3468 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1074
v2
             = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMonoid_712 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_752
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverse
d_inverse_3470 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_inverse_3470 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_Σ_14
d_inverse_3470 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsGroup_1074 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_inverse_1090
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
         ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverseʳ
d_inverse'691'_3472 ::
  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_3432 -> AgdaAny -> AgdaAny
d_inverse'691'_3472 :: 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_3432
-> AgdaAny
-> AgdaAny
d_inverse'691'_3472 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_inverse'691'_3472 T_IsRingWithoutAnnihilatingZero_3432
v9
du_inverse'691'_3472 ::
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_inverse'691'_3472 :: T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_inverse'691'_3472 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsGroup_1074 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1074 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'691'_1146
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverseˡ
d_inverse'737'_3474 ::
  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_3432 -> AgdaAny -> AgdaAny
d_inverse'737'_3474 :: 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_3432
-> AgdaAny
-> AgdaAny
d_inverse'737'_3474 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_inverse'737'_3474 T_IsRingWithoutAnnihilatingZero_3432
v9
du_inverse'737'_3474 ::
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_inverse'737'_3474 :: T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_inverse'737'_3474 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      ((T_IsGroup_1074 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1074 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'737'_1144
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeMagma
d_isCommutativeMagma_3476 ::
  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_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
d_isCommutativeMagma_3476 :: 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_3432
-> T_IsCommutativeMagma_214
d_isCommutativeMagma_3476 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_3476 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isCommutativeMagma_3476 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_214
du_isCommutativeMagma_3476 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsCommutativeMagma_214
du_isCommutativeMagma_3476 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_IsCommutativeMagma_214
forall a b. a -> b
coe
      (let v2 :: AgdaAny
v2
             = (T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                 T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
                 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsCommutativeSemigroup_568 -> T_IsCommutativeMagma_214
MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_606
            ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
               (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeMonoid
d_isCommutativeMonoid_3478 ::
  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_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
d_isCommutativeMonoid_3478 :: 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_3432
-> T_IsCommutativeMonoid_764
d_isCommutativeMonoid_3478 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_3478 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isCommutativeMonoid_3478 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_764
du_isCommutativeMonoid_3478 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsCommutativeMonoid_764
du_isCommutativeMonoid_3478 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> T_IsCommutativeMonoid_764
forall a b. a -> b
coe
      T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeSemigroup
d_isCommutativeSemigroup_3480 ::
  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_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_3480 :: 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_3432
-> T_IsCommutativeSemigroup_568
d_isCommutativeSemigroup_3480 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_3480 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isCommutativeSemigroup_3480 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_3480 :: T_IsRingWithoutAnnihilatingZero_3432
-> T_IsCommutativeSemigroup_568
du_isCommutativeSemigroup_3480 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_IsCommutativeSemigroup_568
forall a b. a -> b
coe
      ((T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsCommutativeMonoid_764 -> T_IsCommutativeSemigroup_568
MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_814
         ((T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsAbelianGroup_1172 -> T_IsCommutativeMonoid_764
MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1244
            (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isEquivalence
d_isEquivalence_3482 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_3482 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsEquivalence_28
d_isEquivalence_3482 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28)
-> AgdaAny -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
               ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                  ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isGroup
d_isGroup_3484 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsGroup_1074
d_isGroup_3484 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsGroup_1074
d_isGroup_3484 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsAbelianGroup_1172 -> T_IsGroup_1074)
-> AgdaAny -> T_IsGroup_1074
forall a b. a -> b
coe
      T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isInvertibleMagma
d_isInvertibleMagma_3486 ::
  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_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
d_isInvertibleMagma_3486 :: 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_3432
-> T_IsInvertibleMagma_958
d_isInvertibleMagma_3486 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_3486 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isInvertibleMagma_3486 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_958
du_isInvertibleMagma_3486 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsInvertibleMagma_958
du_isInvertibleMagma_3486 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_IsInvertibleMagma_958
forall a b. a -> b
coe
      ((T_IsGroup_1074 -> T_IsInvertibleMagma_958) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1074 -> T_IsInvertibleMagma_958
MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1160
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isInvertibleUnitalMagma
d_isInvertibleUnitalMagma_3488 ::
  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_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_3488 :: 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_3432
-> T_IsInvertibleUnitalMagma_1012
d_isInvertibleUnitalMagma_3488 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432
-> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_3488 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isInvertibleUnitalMagma_3488 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_3488 :: T_IsRingWithoutAnnihilatingZero_3432
-> T_IsInvertibleUnitalMagma_1012
du_isInvertibleUnitalMagma_3488 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_IsInvertibleUnitalMagma_1012
forall a b. a -> b
coe
      ((T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1074 -> T_IsInvertibleUnitalMagma_1012
MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1162
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isMagma
d_isMagma_3490 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_3490 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMagma_178
d_isMagma_3490 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
            ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isMonoid
d_isMonoid_3492 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsMonoid_712
d_isMonoid_3492 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_isMonoid_3492 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> T_IsMonoid_712
forall a b. a -> b
coe
      T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
         ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isPartialEquivalence
d_isPartialEquivalence_3494 ::
  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_3432 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_3494 :: 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_3432
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_3494 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3494 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isPartialEquivalence_3494 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_3494 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3494 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1074
v2
             = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_712
v3
                = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_488
v4
                   = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: T_IsMagma_178
v5 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v4) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
                     ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v5)))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isSemigroup
d_isSemigroup_3496 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_3496 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsSemigroup_488
d_isSemigroup_3496 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488)
-> AgdaAny -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
            ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isUnitalMagma
d_isUnitalMagma_3498 ::
  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_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_3498 :: 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_3432
-> T_IsUnitalMagma_666
d_isUnitalMagma_3498 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsUnitalMagma_666
du_isUnitalMagma_3498 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isUnitalMagma_3498 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_3498 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsUnitalMagma_666
du_isUnitalMagma_3498 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1074
v2
             = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMonoid_712 -> T_IsUnitalMagma_666) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> AgdaAny
forall a b. a -> b
coe T_IsGroup_1074
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.refl
d_refl_3500 ::
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
d_refl_3500 :: T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
d_refl_3500 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsEquivalence_28 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
      ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
                  ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                     ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.reflexive
d_reflexive_3502 ::
  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_3432 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_3502 :: 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_3432
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_3502 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3502 T_IsRingWithoutAnnihilatingZero_3432
v9
du_reflexive_3502 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_3502 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3502 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1074
v2
             = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_712
v3
                = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_488
v4
                   = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: T_IsMagma_178
v5 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v4) in
                (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
                  (\ AgdaAny
v6 AgdaAny
v7 AgdaAny
v8 ->
                     (T_IsEquivalence_28 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                       T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_42
                       ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v5))
                       AgdaAny
v6)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.setoid
d_setoid_3504 ::
  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_3432 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_3504 :: 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_3432
-> T_Setoid_46
d_setoid_3504 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_Setoid_46
du_setoid_3504 T_IsRingWithoutAnnihilatingZero_3432
v9
du_setoid_3504 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_3504 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_Setoid_46
du_setoid_3504 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1074
v2
             = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_712
v3
                = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_488
v4
                   = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
                  ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488
v4))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.sym
d_sym_3506 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_sym_3506 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_sym_3506 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsEquivalence_28 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_28 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_sym_38
      ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
                  ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                     ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.trans
d_trans_3508 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_trans_3508 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_trans_3508 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsEquivalence_28
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_28
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_trans_40
      ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
                  ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                     ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.uniqueʳ-⁻¹
d_unique'691''45''8315''185'_3510 ::
  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_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_unique'691''45''8315''185'_3510 :: 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_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_unique'691''45''8315''185'_3510 ~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_3432
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'691''45''8315''185'_3510 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3432
v9
du_unique'691''45''8315''185'_3510 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_unique'691''45''8315''185'_3510 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'691''45''8315''185'_3510 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny
v1 AgdaAny
v2 T_IsRingWithoutAnnihilatingZero_3432
v3
  = let v4 :: T_IsAbelianGroup_1172
v4 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v3) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (((AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_IsGroup_1074
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsGroup_1074
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.du_unique'691''45''8315''185'_1158
         ((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_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v4)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.uniqueˡ-⁻¹
d_unique'737''45''8315''185'_3512 ::
  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_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_unique'737''45''8315''185'_3512 :: 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_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_unique'737''45''8315''185'_3512 ~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_3432
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'737''45''8315''185'_3512 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3432
v9
du_unique'737''45''8315''185'_3512 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_unique'737''45''8315''185'_3512 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
du_unique'737''45''8315''185'_3512 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny
v1 AgdaAny
v2 T_IsRingWithoutAnnihilatingZero_3432
v3
  = let v4 :: T_IsAbelianGroup_1172
v4 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v3) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (((AgdaAny -> AgdaAny -> AgdaAny)
 -> AgdaAny
 -> (AgdaAny -> AgdaAny)
 -> T_IsGroup_1074
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> (AgdaAny -> AgdaAny)
-> T_IsGroup_1074
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.du_unique'737''45''8315''185'_1152
         ((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_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> AgdaAny
forall a b. a -> b
coe T_IsAbelianGroup_1172
v4)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.⁻¹-cong
d_'8315''185''45'cong_3514 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8315''185''45'cong_3514 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8315''185''45'cong_3514 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsGroup_1074 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsGroup_1074 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8315''185''45'cong_1092
      ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
         ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-cong
d_'8729''45'cong_3516 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong_3516 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong_3516 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
               ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                  ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-congʳ
d_'8729''45'cong'691'_3518 ::
  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_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_3518 :: 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_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_3518 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3518 T_IsRingWithoutAnnihilatingZero_3432
v9
du_'8729''45'cong'691'_3518 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3518 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3518 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1074
v2
             = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_712
v3
                = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_488
v4
                   = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: T_IsMagma_178
v5 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v4) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (let v6 :: AgdaAny
v6
                         = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v5) in
                   AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     (let v7 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v7
                            = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v5) in
                      AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'691'_46
                           ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v7)
                           ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                              T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                              ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                                 T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                                 (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v6))))))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-congˡ
d_'8729''45'cong'737'_3520 ::
  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_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_3520 :: 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_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_3520 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3520 T_IsRingWithoutAnnihilatingZero_3432
v9
du_'8729''45'cong'737'_3520 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3520 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3520 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsAbelianGroup_1172
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsGroup_1074
v2
             = T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184 (T_IsAbelianGroup_1172 -> T_IsAbelianGroup_1172
forall a b. a -> b
coe T_IsAbelianGroup_1172
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMonoid_712
v3
                = T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088 (T_IsGroup_1074 -> T_IsGroup_1074
forall a b. a -> b
coe T_IsGroup_1074
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: T_IsSemigroup_488
v4
                   = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: T_IsMagma_178
v5 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v4) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (let v6 :: AgdaAny
v6
                         = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v5) in
                   AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     (let v7 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v7
                            = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v5) in
                      AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'737'_42
                           ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v7)
                           ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                              T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                              ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                                 T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                                 (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v6))))))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.assoc
d_assoc_3524 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_3524 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_assoc_3524 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identity
d_identity_3526 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_identity_3526 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_Σ_14
d_identity_3526 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMonoid_712 -> T_Σ_14) -> AgdaAny -> T_Σ_14
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identityʳ
d_identity'691'_3528 ::
  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_3432 -> AgdaAny -> AgdaAny
d_identity'691'_3528 :: 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_3432
-> AgdaAny
-> AgdaAny
d_identity'691'_3528 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'691'_3528 T_IsRingWithoutAnnihilatingZero_3432
v9
du_identity'691'_3528 ::
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'691'_3528 :: T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'691'_3528 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMonoid_712 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_754
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identityˡ
d_identity'737'_3530 ::
  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_3432 -> AgdaAny -> AgdaAny
d_identity'737'_3530 :: 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_3432
-> AgdaAny
-> AgdaAny
d_identity'737'_3530 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'737'_3530 T_IsRingWithoutAnnihilatingZero_3432
v9
du_identity'737'_3530 ::
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'737'_3530 :: T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_identity'737'_3530 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMonoid_712 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'737'_752
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isEquivalence
d_isEquivalence_3532 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_28
d_isEquivalence_3532 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsEquivalence_28
d_isEquivalence_3532 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMagma_178 -> T_IsEquivalence_28)
-> AgdaAny -> T_IsEquivalence_28
forall a b. a -> b
coe
      T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isMagma
d_isMagma_3534 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsMagma_178
d_isMagma_3534 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMagma_178
d_isMagma_3534 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> T_IsMagma_178
forall a b. a -> b
coe
      T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
      ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
         ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isPartialEquivalence
d_isPartialEquivalence_3536 ::
  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_3432 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
d_isPartialEquivalence_3536 :: 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_3432
-> T_IsPartialEquivalence_16
d_isPartialEquivalence_3536 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3536 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isPartialEquivalence_3536 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
du_isPartialEquivalence_3536 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsPartialEquivalence_16
du_isPartialEquivalence_3536 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_IsPartialEquivalence_16
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            ((T_IsEquivalence_28 -> T_IsPartialEquivalence_16)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsEquivalence_28 -> T_IsPartialEquivalence_16
MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_44
               ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isSemigroup
d_isSemigroup_3538 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsSemigroup_488
d_isSemigroup_3538 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsSemigroup_488
d_isSemigroup_3538 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMonoid_712 -> T_IsSemigroup_488)
-> AgdaAny -> T_IsSemigroup_488
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isUnitalMagma
d_isUnitalMagma_3540 ::
  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_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
d_isUnitalMagma_3540 :: 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_3432
-> T_IsUnitalMagma_666
d_isUnitalMagma_3540 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsUnitalMagma_666
du_isUnitalMagma_3540 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isUnitalMagma_3540 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_666
du_isUnitalMagma_3540 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsUnitalMagma_666
du_isUnitalMagma_3540 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMonoid_712 -> T_IsUnitalMagma_666)
-> AgdaAny -> T_IsUnitalMagma_666
forall a b. a -> b
coe
      T_IsMonoid_712 -> T_IsUnitalMagma_666
MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_756
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.refl
d_refl_3542 ::
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
d_refl_3542 :: T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
d_refl_3542 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsEquivalence_28 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
      ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.reflexive
d_reflexive_3544 ::
  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_3432 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
d_reflexive_3544 :: 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_3432
-> AgdaAny
-> AgdaAny
-> T__'8801'__12
-> AgdaAny
d_reflexive_3544 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3544 T_IsRingWithoutAnnihilatingZero_3432
v9
du_reflexive_3544 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny ->
  AgdaAny ->
  MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
du_reflexive_3544 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
du_reflexive_3544 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> T__'8801'__12 -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny) -> AgdaAny
forall a b. a -> b
coe
            (\ AgdaAny
v4 AgdaAny
v5 AgdaAny
v6 ->
               (T_IsEquivalence_28 -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                 T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.du_reflexive_42
                 ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3))
                 AgdaAny
v4)))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.setoid
d_setoid_3546 ::
  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_3432 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
d_setoid_3546 :: 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_3432
-> T_Setoid_46
d_setoid_3546 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_Setoid_46
du_setoid_3546 T_IsRingWithoutAnnihilatingZero_3432
v9
du_setoid_3546 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_46
du_setoid_3546 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_Setoid_46
du_setoid_3546 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> T_Setoid_46
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
            ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> AgdaAny
forall a b. a -> b
coe T_IsSemigroup_488
v2))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.sym
d_sym_3548 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_sym_3548 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_sym_3548 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsEquivalence_28 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_28 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_sym_38
      ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.trans
d_trans_3550 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_trans_3550 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_trans_3550 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsEquivalence_28
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_IsEquivalence_28
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_trans_40
      ((T_IsMagma_178 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_IsEquivalence_28
MAlonzo.Code.Algebra.Structures.d_isEquivalence_186
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-cong
d_'8729''45'cong_3552 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong_3552 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong_3552 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
forall a b. a -> b
coe
      T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
      ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-congʳ
d_'8729''45'cong'691'_3554 ::
  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_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'691'_3554 :: 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_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'691'_3554 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3554 T_IsRingWithoutAnnihilatingZero_3432
v9
du_'8729''45'cong'691'_3554 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3554 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'691'_3554 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: AgdaAny
v4
                   = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5
                      = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v3) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'691'_46
                     ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5)
                     ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                        ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                           (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v4))))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-congˡ
d_'8729''45'cong'737'_3556 ::
  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_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
d_'8729''45'cong'737'_3556 :: 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_3432
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
d_'8729''45'cong'737'_3556 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3556 T_IsRingWithoutAnnihilatingZero_3432
v9
du_'8729''45'cong'737'_3556 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3556 :: T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
du_'8729''45'cong'737'_3556 T_IsRingWithoutAnnihilatingZero_3432
v0
  = let v1 :: T_IsMonoid_712
v1 = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432
-> T_IsRingWithoutAnnihilatingZero_3432
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0) in
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
      (let v2 :: T_IsSemigroup_488
v2
             = T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722 (T_IsMonoid_712 -> T_IsMonoid_712
forall a b. a -> b
coe T_IsMonoid_712
v1) in
       AgdaAny -> AgdaAny
forall a b. a -> b
coe
         (let v3 :: T_IsMagma_178
v3 = T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496 (T_IsSemigroup_488 -> T_IsSemigroup_488
forall a b. a -> b
coe T_IsSemigroup_488
v2) in
          AgdaAny -> AgdaAny
forall a b. a -> b
coe
            (let v4 :: AgdaAny
v4
                   = (T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202 (T_IsMagma_178 -> AgdaAny
forall a b. a -> b
coe T_IsMagma_178
v3) in
             AgdaAny -> AgdaAny
forall a b. a -> b
coe
               (let v5 :: AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5
                      = T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188 (T_IsMagma_178 -> T_IsMagma_178
forall a b. a -> b
coe T_IsMagma_178
v3) in
                AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  (((AgdaAny
  -> AgdaAny -> AgdaAny -> AgdaAny -> 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 -> AgdaAny)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Consequences.Base.du_'8729''45'cong'737'_42
                     ((AgdaAny
 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny
forall a b. a -> b
coe AgdaAny
-> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
v5)
                     ((T_IsEquivalence_28 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                        T_IsEquivalence_28 -> AgdaAny -> AgdaAny
MAlonzo.Code.Relation.Binary.Structures.d_refl_36
                        ((T_Setoid_46 -> T_IsEquivalence_28) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                           T_Setoid_46 -> T_IsEquivalence_28
MAlonzo.Code.Relation.Binary.Bundles.d_isEquivalence_62
                           (AgdaAny -> AgdaAny
forall a b. a -> b
coe AgdaAny
v4))))))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zeroˡ
d_zero'737'_3558 ::
  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_3432 -> AgdaAny -> AgdaAny
d_zero'737'_3558 :: 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_3432
-> AgdaAny
-> AgdaAny
d_zero'737'_3558 ~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_3432
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
du_zero'737'_3558 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3432
v9
du_zero'737'_3558 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_zero'737'_3558 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
du_zero'737'_3558 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny -> AgdaAny
v2 AgdaAny
v3 T_IsRingWithoutAnnihilatingZero_3432
v4
  = (T_Setoid_46
 -> (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_46
-> (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'_644
      ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
                  ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                     ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))))))
      ((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_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
                  ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                     ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))))))
      ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))))
      ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
               ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                  ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
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_3432 -> T_Σ_14)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_Σ_14
d_distrib_3454 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))
      ((T_IsMonoid_712 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_754
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
            ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))))
      ((T_IsGroup_1074 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1074 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'691'_1146
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
            ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zeroʳ
d_zero'691'_3560 ::
  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_3432 -> AgdaAny -> AgdaAny
d_zero'691'_3560 :: 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_3432
-> AgdaAny
-> AgdaAny
d_zero'691'_3560 ~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_3432
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
du_zero'691'_3560 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3432
v9
du_zero'691'_3560 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny -> AgdaAny
du_zero'691'_3560 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> AgdaAny
-> AgdaAny
du_zero'691'_3560 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny -> AgdaAny
v2 AgdaAny
v3 T_IsRingWithoutAnnihilatingZero_3432
v4
  = (T_Setoid_46
 -> (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_46
-> (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'_656
      ((T_IsMagma_178 -> T_Setoid_46) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178 -> T_Setoid_46
MAlonzo.Code.Algebra.Structures.du_setoid_202
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
                  ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                     ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))))))
      ((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_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
                  ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                     T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                     ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))))))
      ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))))
      ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
               ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
                  T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
                  ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
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_3432 -> T_Σ_14)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_Σ_14
d_distrib_3454 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))
      ((T_IsMonoid_712 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_identity'691'_754
         ((T_IsGroup_1074 -> T_IsMonoid_712) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsGroup_1074 -> T_IsMonoid_712
MAlonzo.Code.Algebra.Structures.d_isMonoid_1088
            ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4)))))
      ((T_IsGroup_1074 -> AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsGroup_1074 -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.du_inverse'691'_1146
         ((T_IsAbelianGroup_1172 -> T_IsGroup_1074) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsAbelianGroup_1172 -> T_IsGroup_1074
MAlonzo.Code.Algebra.Structures.d_isGroup_1184
            ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4))))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zero
d_zero_3562 ::
  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_3432 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
d_zero_3562 :: 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_3432
-> T_Σ_14
d_zero_3562 ~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_3432
v9
  = (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> T_Σ_14
du_zero_3562 AgdaAny -> AgdaAny -> AgdaAny
v4 AgdaAny -> AgdaAny -> AgdaAny
v5 AgdaAny -> AgdaAny
v6 AgdaAny
v7 T_IsRingWithoutAnnihilatingZero_3432
v9
du_zero_3562 ::
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny -> AgdaAny) ->
  (AgdaAny -> AgdaAny) ->
  AgdaAny ->
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
du_zero_3562 :: (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny)
-> AgdaAny
-> T_IsRingWithoutAnnihilatingZero_3432
-> T_Σ_14
du_zero_3562 AgdaAny -> AgdaAny -> AgdaAny
v0 AgdaAny -> AgdaAny -> AgdaAny
v1 AgdaAny -> AgdaAny
v2 AgdaAny
v3 T_IsRingWithoutAnnihilatingZero_3432
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_3432
 -> 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_3432
-> AgdaAny
-> AgdaAny
du_zero'737'_3558 ((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_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4))
      (((AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny)
 -> AgdaAny
 -> T_IsRingWithoutAnnihilatingZero_3432
 -> 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_3432
-> AgdaAny
-> AgdaAny
du_zero'691'_3560 ((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_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v4))
-- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.isRing
d_isRing_3564 ::
  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_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740
d_isRing_3564 :: 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_3432
-> T_IsRing_2740
d_isRing_3564 ~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_3432
v9
  = T_IsRingWithoutAnnihilatingZero_3432 -> T_IsRing_2740
du_isRing_3564 T_IsRingWithoutAnnihilatingZero_3432
v9
du_isRing_3564 ::
  T_IsRingWithoutAnnihilatingZero_3432 ->
  MAlonzo.Code.Algebra.Structures.T_IsRing_2740
du_isRing_3564 :: T_IsRingWithoutAnnihilatingZero_3432 -> T_IsRing_2740
du_isRing_3564 T_IsRingWithoutAnnihilatingZero_3432
v0
  = (T_IsAbelianGroup_1172
 -> (AgdaAny
     -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
 -> T_Σ_14
 -> T_Σ_14
 -> T_IsRing_2740)
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> T_IsRing_2740
forall a b. a -> b
coe
      T_IsAbelianGroup_1172
-> (AgdaAny
    -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> T_Σ_14
-> T_Σ_14
-> T_IsRing_2740
MAlonzo.Code.Algebra.Structures.C_constructor_2876
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsAbelianGroup_1172
d_'43''45'isAbelianGroup_3450 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))
      ((T_IsMagma_178
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny
 -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMagma_178
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
-> AgdaAny
MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_188
         ((T_IsSemigroup_488 -> T_IsMagma_178) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsSemigroup_488 -> T_IsMagma_178
MAlonzo.Code.Algebra.Structures.d_isMagma_496
            ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
               T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
               ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))))
      ((T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsSemigroup_488 -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
MAlonzo.Code.Algebra.Structures.d_assoc_498
         ((T_IsMonoid_712 -> T_IsSemigroup_488) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
            T_IsMonoid_712 -> T_IsSemigroup_488
MAlonzo.Code.Algebra.Structures.d_isSemigroup_722
            ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))))
      ((T_IsMonoid_712 -> T_Σ_14) -> AgdaAny -> AgdaAny
forall a b. a -> b
coe
         T_IsMonoid_712 -> T_Σ_14
MAlonzo.Code.Algebra.Structures.d_identity_724
         ((T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_IsMonoid_712
d_'42''45'isMonoid_3452 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0)))
      ((T_IsRingWithoutAnnihilatingZero_3432 -> T_Σ_14)
-> AgdaAny -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432 -> T_Σ_14
d_distrib_3454 (T_IsRingWithoutAnnihilatingZero_3432 -> AgdaAny
forall a b. a -> b
coe T_IsRingWithoutAnnihilatingZero_3432
v0))