content: {-# 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 Data.Text qualified
  import MAlonzo.Code.Agda.Builtin.Equality qualified
  import MAlonzo.Code.Agda.Builtin.Sigma qualified
  import MAlonzo.Code.Agda.Primitive qualified
  import MAlonzo.Code.Algebra.Consequences.Setoid qualified
  import MAlonzo.Code.Algebra.Structures qualified
  import MAlonzo.Code.Relation.Binary.Bundles qualified
  import MAlonzo.Code.Relation.Binary.Structures qualified
  import MAlonzo.RTE (AgdaAny, add64, addInt, coe, eq64, eqInt, erased, geqInt, lt64, ltInt, mul64,
                      mulInt, quot64, quotInt, rem64, remInt, sub64, subInt, word64FromNat,
                      word64ToNat)
  import MAlonzo.RTE qualified
  
  -- 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = erased
  -- Algebra.Structures.Biased._.IsAbelianGroup
  d_IsAbelianGroup_146 a0 a1 a2 a3 a4 a5 a6 = ()
  -- Algebra.Structures.Biased._.IsCommutativeMonoid
  d_IsCommutativeMonoid_158 a0 a1 a2 a3 a4 a5 = ()
  -- Algebra.Structures.Biased._.IsCommutativeSemiring
  d_IsCommutativeSemiring_164 a0 a1 a2 a3 a4 a5 a6 a7 = ()
  -- Algebra.Structures.Biased._.IsMonoid
  d_IsMonoid_196 a0 a1 a2 a3 a4 a5 = ()
  -- Algebra.Structures.Biased._.IsNearSemiring
  d_IsNearSemiring_200 a0 a1 a2 a3 a4 a5 a6 = ()
  -- Algebra.Structures.Biased._.IsRing
  d_IsRing_212 a0 a1 a2 a3 a4 a5 a6 a7 a8 = ()
  -- Algebra.Structures.Biased._.IsSemigroup
  d_IsSemigroup_218 a0 a1 a2 a3 a4 = ()
  -- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero
  d_IsSemiringWithoutAnnihilatingZero_224 a0 a1 a2 a3 a4 a5 a6 a7
    = ()
  -- Algebra.Structures.Biased._.IsSemiringWithoutOne
  d_IsSemiringWithoutOne_226 a0 a1 a2 a3 a4 a5 a6 = ()
  -- Algebra.Structures.Biased._.IsAbelianGroup._//_
  d__'47''47'__234 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny
  d__'47''47'__234 ~v0 ~v1 ~v2 ~v3 v4 ~v5 v6 ~v7
    = du__'47''47'__234 v4 v6
  du__'47''47'__234 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
  du__'47''47'__234 v0 v1
    = coe
        MAlonzo.Code.Algebra.Structures.du__'47''47'__1098 (coe v0)
        (coe v1)
  -- Algebra.Structures.Biased._.IsAbelianGroup.comm
  d_comm_238 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny
  d_comm_238 v0
    = coe MAlonzo.Code.Algebra.Structures.d_comm_1146 (coe v0)
  -- Algebra.Structures.Biased._.IsAbelianGroup.identityʳ
  d_identity'691'_242 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny
  d_identity'691'_242 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_identity'691'_242 v7
  du_identity'691'_242 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny
  du_identity'691'_242 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_identity'691'_728
           (coe MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v1)))
  -- Algebra.Structures.Biased._.IsAbelianGroup.identityˡ
  d_identity'737'_244 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny
  d_identity'737'_244 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_identity'737'_244 v7
  du_identity'737'_244 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny
  du_identity'737'_244 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_identity'737'_726
           (coe MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v1)))
  -- Algebra.Structures.Biased._.IsAbelianGroup.inverseʳ
  d_inverse'691'_248 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny
  d_inverse'691'_248 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_inverse'691'_248 v7
  du_inverse'691'_248 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny
  du_inverse'691'_248 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_inverse'691'_1108
        (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0))
  -- Algebra.Structures.Biased._.IsAbelianGroup.inverseˡ
  d_inverse'737'_250 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny
  d_inverse'737'_250 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_inverse'737'_250 v7
  du_inverse'737'_250 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny
  du_inverse'737'_250 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_inverse'737'_1106
        (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0))
  -- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeMagma
  d_isCommutativeMagma_252 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_212
  d_isCommutativeMagma_252 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_isCommutativeMagma_252 v7
  du_isCommutativeMagma_252 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_212
  du_isCommutativeMagma_252 v0
    = let v1
            = coe
                MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
                (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_586
           (coe
              MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_786
              (coe v1)))
  -- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeMonoid
  d_isCommutativeMonoid_254 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_isCommutativeMonoid_254 ~v0 ~v1 ~v2 ~v3
    = du_isCommutativeMonoid_254
  du_isCommutativeMonoid_254 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  du_isCommutativeMonoid_254 v0 v1 v2 v3
    = coe
        MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204 v3
  -- Algebra.Structures.Biased._.IsAbelianGroup.isCommutativeSemigroup
  d_isCommutativeSemigroup_256 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
  d_isCommutativeSemigroup_256 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_isCommutativeSemigroup_256 v7
  du_isCommutativeSemigroup_256 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
  du_isCommutativeSemigroup_256 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_786
        (coe
           MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
           (coe v0))
  -- Algebra.Structures.Biased._.IsAbelianGroup.isGroup
  d_isGroup_260 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsGroup_1036
  d_isGroup_260 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0)
  -- Algebra.Structures.Biased._.IsAbelianGroup.isInvertibleMagma
  d_isInvertibleMagma_262 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_924
  d_isInvertibleMagma_262 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_isInvertibleMagma_262 v7
  du_isInvertibleMagma_262 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_924
  du_isInvertibleMagma_262 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1122
        (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0))
  -- Algebra.Structures.Biased._.IsAbelianGroup.isInvertibleUnitalMagma
  d_isInvertibleUnitalMagma_264 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_976
  d_isInvertibleUnitalMagma_264 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_isInvertibleUnitalMagma_264 v7
  du_isInvertibleUnitalMagma_264 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_976
  du_isInvertibleUnitalMagma_264 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1124
        (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0))
  -- Algebra.Structures.Biased._.IsAbelianGroup.isPartialEquivalence
  d_isPartialEquivalence_270 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  d_isPartialEquivalence_270 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_isPartialEquivalence_270 v7
  du_isPartialEquivalence_270 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  du_isPartialEquivalence_270 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v2) in
            coe
              (let v4 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v3) in
               coe
                 (coe
                    MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v4))))))
  -- Algebra.Structures.Biased._.IsAbelianGroup.isUnitalMagma
  d_isUnitalMagma_274 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  d_isUnitalMagma_274 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_isUnitalMagma_274 v7
  du_isUnitalMagma_274 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  du_isUnitalMagma_274 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
           (coe MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v1)))
  -- Algebra.Structures.Biased._.IsAbelianGroup.reflexive
  d_reflexive_278 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  d_reflexive_278 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_reflexive_278 v7
  du_reflexive_278 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  du_reflexive_278 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v2) in
            coe
              (let v4 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v3) in
               coe
                 (\ v5 v6 v7 ->
                    coe
                      MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                      (coe MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v4))
                      v5))))
  -- Algebra.Structures.Biased._.IsAbelianGroup.setoid
  d_setoid_280 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  d_setoid_280 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7 = du_setoid_280 v7
  du_setoid_280 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  du_setoid_280 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v2) in
            coe
              (coe
                 MAlonzo.Code.Algebra.Structures.du_setoid_200
                 (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v3)))))
  -- Algebra.Structures.Biased._.IsAbelianGroup.uniqueʳ-⁻¹
  d_unique'691''45''8315''185'_286 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_unique'691''45''8315''185'_286 ~v0 ~v1 ~v2 ~v3 v4 v5 v6 v7
    = du_unique'691''45''8315''185'_286 v4 v5 v6 v7
  du_unique'691''45''8315''185'_286 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_unique'691''45''8315''185'_286 v0 v1 v2 v3
    = coe
        MAlonzo.Code.Algebra.Structures.du_unique'691''45''8315''185'_1120
        (coe v0) (coe v1) (coe v2)
        (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v3))
  -- Algebra.Structures.Biased._.IsAbelianGroup.uniqueˡ-⁻¹
  d_unique'737''45''8315''185'_288 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_unique'737''45''8315''185'_288 ~v0 ~v1 ~v2 ~v3 v4 v5 v6 v7
    = du_unique'737''45''8315''185'_288 v4 v5 v6 v7
  du_unique'737''45''8315''185'_288 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_unique'737''45''8315''185'_288 v0 v1 v2 v3
    = coe
        MAlonzo.Code.Algebra.Structures.du_unique'737''45''8315''185'_1114
        (coe v0) (coe v1) (coe v2)
        (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v3))
  -- Algebra.Structures.Biased._.IsAbelianGroup.∙-congʳ
  d_'8729''45'cong'691'_294 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'691'_294 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_'8729''45'cong'691'_294 v7
  du_'8729''45'cong'691'_294 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'691'_294 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v2) in
            coe
              (coe
                 MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
                 (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v3)))))
  -- Algebra.Structures.Biased._.IsAbelianGroup.∙-congˡ
  d_'8729''45'cong'737'_296 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    (AgdaAny -> AgdaAny) ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'737'_296 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_'8729''45'cong'737'_296 v7
  du_'8729''45'cong'737'_296 ::
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'737'_296 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v2) in
            coe
              (coe
                 MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
                 (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v3)))))
  -- Algebra.Structures.Biased._.IsCommutativeMonoid.comm
  d_comm_528 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736 ->
    AgdaAny -> AgdaAny -> AgdaAny
  d_comm_528 v0
    = coe MAlonzo.Code.Algebra.Structures.d_comm_748 (coe v0)
  -- Algebra.Structures.Biased._.IsCommutativeMonoid.isMonoid
  d_isMonoid_544 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736 ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_isMonoid_544 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (coe v0)
  -- Algebra.Structures.Biased._.IsCommutativeSemiring.*-comm
  d_'42''45'comm_728 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678 ->
    AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'comm_728 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'comm_1694 (coe v0)
  -- Algebra.Structures.Biased._.IsCommutativeSemiring.isSemiring
  d_isSemiring_798 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemiring_1570
  d_isSemiring_798 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isSemiring_1692 (coe v0)
  -- Algebra.Structures.Biased._.IsMonoid.identity
  d_identity_1600 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_identity_1600 v0
    = coe MAlonzo.Code.Algebra.Structures.d_identity_698 (coe v0)
  -- Algebra.Structures.Biased._.IsMonoid.identityʳ
  d_identity'691'_1602 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny -> AgdaAny
  d_identity'691'_1602 ~v0 ~v1 ~v2 ~v3 = du_identity'691'_1602
  du_identity'691'_1602 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny -> AgdaAny
  du_identity'691'_1602 v0 v1 v2
    = coe MAlonzo.Code.Algebra.Structures.du_identity'691'_728 v2
  -- Algebra.Structures.Biased._.IsMonoid.identityˡ
  d_identity'737'_1604 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny -> AgdaAny
  d_identity'737'_1604 ~v0 ~v1 ~v2 ~v3 = du_identity'737'_1604
  du_identity'737'_1604 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny -> AgdaAny
  du_identity'737'_1604 v0 v1 v2
    = coe MAlonzo.Code.Algebra.Structures.du_identity'737'_726 v2
  -- Algebra.Structures.Biased._.IsMonoid.isPartialEquivalence
  d_isPartialEquivalence_1610 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  d_isPartialEquivalence_1610 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 v6
    = du_isPartialEquivalence_1610 v6
  du_isPartialEquivalence_1610 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  du_isPartialEquivalence_1610 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v0) in
      coe
        (let v2 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v2))))
  -- Algebra.Structures.Biased._.IsMonoid.isSemigroup
  d_isSemigroup_1612 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_isSemigroup_1612 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v0)
  -- Algebra.Structures.Biased._.IsMonoid.isUnitalMagma
  d_isUnitalMagma_1614 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  d_isUnitalMagma_1614 ~v0 ~v1 ~v2 ~v3 = du_isUnitalMagma_1614
  du_isUnitalMagma_1614 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  du_isUnitalMagma_1614 v0 v1 v2
    = coe MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730 v2
  -- Algebra.Structures.Biased._.IsMonoid.reflexive
  d_reflexive_1618 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  d_reflexive_1618 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 v6 = du_reflexive_1618 v6
  du_reflexive_1618 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  du_reflexive_1618 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v0) in
      coe
        (let v2 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v1) in
         coe
           (\ v3 v4 v5 ->
              coe
                MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                (coe MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v2))
                v3))
  -- Algebra.Structures.Biased._.IsMonoid.setoid
  d_setoid_1620 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  d_setoid_1620 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 v6 = du_setoid_1620 v6
  du_setoid_1620 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  du_setoid_1620 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_setoid_200
           (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v1)))
  -- Algebra.Structures.Biased._.IsMonoid.∙-congʳ
  d_'8729''45'cong'691'_1628 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'691'_1628 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 v6
    = du_'8729''45'cong'691'_1628 v6
  du_'8729''45'cong'691'_1628 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'691'_1628 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
           (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v1)))
  -- Algebra.Structures.Biased._.IsMonoid.∙-congˡ
  d_'8729''45'cong'737'_1630 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'737'_1630 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 v6
    = du_'8729''45'cong'737'_1630 v6
  du_'8729''45'cong'737'_1630 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'737'_1630 v0
    = let v1
            = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
           (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v1)))
  -- Algebra.Structures.Biased._.IsNearSemiring.*-assoc
  d_'42''45'assoc_1700 ::
    MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'assoc_1700 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1240 (coe v0)
  -- Algebra.Structures.Biased._.IsNearSemiring.*-cong
  d_'42''45'cong_1702 ::
    MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
    AgdaAny ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'cong_1702 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1238 (coe v0)
  -- Algebra.Structures.Biased._.IsNearSemiring.+-isMonoid
  d_'43''45'isMonoid_1728 ::
    MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_'43''45'isMonoid_1728 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'43''45'isMonoid_1236 (coe v0)
  -- Algebra.Structures.Biased._.IsNearSemiring.distribʳ
  d_distrib'691'_1734 ::
    MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_distrib'691'_1734 v0
    = coe MAlonzo.Code.Algebra.Structures.d_distrib'691'_1242 (coe v0)
  -- Algebra.Structures.Biased._.IsNearSemiring.zeroˡ
  d_zero'737'_1750 ::
    MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218 ->
    AgdaAny -> AgdaAny
  d_zero'737'_1750 v0
    = coe MAlonzo.Code.Algebra.Structures.d_zero'737'_1244 (coe v0)
  -- Algebra.Structures.Biased._.IsRing.*-assoc
  d_'42''45'assoc_2118 ::
    MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'assoc_2118 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_2676 (coe v0)
  -- Algebra.Structures.Biased._.IsRing.*-cong
  d_'42''45'cong_2120 ::
    MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
    AgdaAny ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'cong_2120 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'cong_2674 (coe v0)
  -- Algebra.Structures.Biased._.IsRing.*-identity
  d_'42''45'identity_2126 ::
    MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'identity_2126 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'42''45'identity_2678 (coe v0)
  -- Algebra.Structures.Biased._.IsRing.+-isAbelianGroup
  d_'43''45'isAbelianGroup_2154 ::
    MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
  d_'43''45'isAbelianGroup_2154 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'43''45'isAbelianGroup_2672
        (coe v0)
  -- Algebra.Structures.Biased._.IsRing.distrib
  d_distrib_2184 ::
    MAlonzo.Code.Algebra.Structures.T_IsRing_2650 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_distrib_2184 v0
    = coe MAlonzo.Code.Algebra.Structures.d_distrib_2680 (coe v0)
  -- Algebra.Structures.Biased._.IsSemigroup.assoc
  d_assoc_2344 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_assoc_2344 v0
    = coe MAlonzo.Code.Algebra.Structures.d_assoc_482 (coe v0)
  -- Algebra.Structures.Biased._.IsSemigroup.isMagma
  d_isMagma_2348 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472 ->
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176
  d_isMagma_2348 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-assoc
  d_'42''45'assoc_2484 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'assoc_2484 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1492 (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-cong
  d_'42''45'cong_2486 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
    AgdaAny ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'cong_2486 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1490 (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.*-identity
  d_'42''45'identity_2492 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'identity_2492 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'42''45'identity_1494 (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.+-isCommutativeMonoid
  d_'43''45'isCommutativeMonoid_2522 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'43''45'isCommutativeMonoid_2522 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1488
        (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutAnnihilatingZero.distrib
  d_distrib_2534 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_distrib_2534 v0
    = coe MAlonzo.Code.Algebra.Structures.d_distrib_1496 (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutOne.*-assoc
  d_'42''45'assoc_2560 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'assoc_2560 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'assoc_1320 (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutOne.*-cong
  d_'42''45'cong_2562 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    AgdaAny ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'42''45'cong_2562 v0
    = coe MAlonzo.Code.Algebra.Structures.d_'42''45'cong_1318 (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutOne.+-isCommutativeMonoid
  d_'43''45'isCommutativeMonoid_2576 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'43''45'isCommutativeMonoid_2576 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1316
        (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutOne.distrib
  d_distrib_2584 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_distrib_2584 v0
    = coe MAlonzo.Code.Algebra.Structures.d_distrib_1322 (coe v0)
  -- Algebra.Structures.Biased._.IsSemiringWithoutOne.zero
  d_zero_2604 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_zero_2604 v0
    = coe MAlonzo.Code.Algebra.Structures.d_zero_1324 (coe v0)
  -- Algebra.Structures.Biased.IsCommutativeMonoidˡ
  d_IsCommutativeMonoid'737'_2662 a0 a1 a2 a3 a4 a5 = ()
  data T_IsCommutativeMonoid'737'_2662
    = C_IsCommutativeMonoid'737''46'constructor_34899 MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
                                                      (AgdaAny -> AgdaAny)
                                                      (AgdaAny -> AgdaAny -> AgdaAny)
  -- Algebra.Structures.Biased.IsCommutativeMonoidˡ.isSemigroup
  d_isSemigroup_2674 ::
    T_IsCommutativeMonoid'737'_2662 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_isSemigroup_2674 v0
    = case coe v0 of
        C_IsCommutativeMonoid'737''46'constructor_34899 v1 v2 v3 -> coe v1
        _                                                        -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeMonoidˡ.identityˡ
  d_identity'737'_2676 ::
    T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny
  d_identity'737'_2676 v0
    = case coe v0 of
        C_IsCommutativeMonoid'737''46'constructor_34899 v1 v2 v3 -> coe v2
        _                                                        -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeMonoidˡ.comm
  d_comm_2678 ::
    T_IsCommutativeMonoid'737'_2662 -> AgdaAny -> AgdaAny -> AgdaAny
  d_comm_2678 v0
    = case coe v0 of
        C_IsCommutativeMonoid'737''46'constructor_34899 v1 v2 v3 -> coe v3
        _                                                        -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeMonoidˡ.isCommutativeMonoid
  d_isCommutativeMonoid_2680 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsCommutativeMonoid'737'_2662 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_isCommutativeMonoid_2680 ~v0 ~v1 ~v2 ~v3 v4 v5 v6
    = du_isCommutativeMonoid_2680 v4 v5 v6
  du_isCommutativeMonoid_2680 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsCommutativeMonoid'737'_2662 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  du_isCommutativeMonoid_2680 v0 v1 v2
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_17695
        (coe
           MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_15873
           (coe d_isSemigroup_2674 (coe v2))
           (coe
              MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'id'737''8658'id_352
              (let v3 = d_isSemigroup_2674 (coe v2) in
               coe
                 (coe
                    MAlonzo.Code.Algebra.Structures.du_setoid_200
                    (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v3))))
              (coe v0) (coe d_comm_2678 (coe v2)) (coe v1)
              (coe d_identity'737'_2676 (coe v2))))
        (coe d_comm_2678 (coe v2))
  -- Algebra.Structures.Biased.IsCommutativeMonoidʳ
  d_IsCommutativeMonoid'691'_2716 a0 a1 a2 a3 a4 a5 = ()
  data T_IsCommutativeMonoid'691'_2716
    = C_IsCommutativeMonoid'691''46'constructor_36341 MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
                                                      (AgdaAny -> AgdaAny)
                                                      (AgdaAny -> AgdaAny -> AgdaAny)
  -- Algebra.Structures.Biased.IsCommutativeMonoidʳ.isSemigroup
  d_isSemigroup_2728 ::
    T_IsCommutativeMonoid'691'_2716 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_isSemigroup_2728 v0
    = case coe v0 of
        C_IsCommutativeMonoid'691''46'constructor_36341 v1 v2 v3 -> coe v1
        _                                                        -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeMonoidʳ.identityʳ
  d_identity'691'_2730 ::
    T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny
  d_identity'691'_2730 v0
    = case coe v0 of
        C_IsCommutativeMonoid'691''46'constructor_36341 v1 v2 v3 -> coe v2
        _                                                        -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeMonoidʳ.comm
  d_comm_2732 ::
    T_IsCommutativeMonoid'691'_2716 -> AgdaAny -> AgdaAny -> AgdaAny
  d_comm_2732 v0
    = case coe v0 of
        C_IsCommutativeMonoid'691''46'constructor_36341 v1 v2 v3 -> coe v3
        _                                                        -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeMonoidʳ.isCommutativeMonoid
  d_isCommutativeMonoid_2734 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsCommutativeMonoid'691'_2716 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_isCommutativeMonoid_2734 ~v0 ~v1 ~v2 ~v3 v4 v5 v6
    = du_isCommutativeMonoid_2734 v4 v5 v6
  du_isCommutativeMonoid_2734 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsCommutativeMonoid'691'_2716 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  du_isCommutativeMonoid_2734 v0 v1 v2
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_17695
        (coe
           MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_15873
           (coe d_isSemigroup_2728 (coe v2))
           (coe
              MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'id'691''8658'id_356
              (let v3 = d_isSemigroup_2728 (coe v2) in
               coe
                 (coe
                    MAlonzo.Code.Algebra.Structures.du_setoid_200
                    (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v3))))
              (coe v0) (coe d_comm_2732 (coe v2)) (coe v1)
              (coe d_identity'691'_2730 (coe v2))))
        (coe d_comm_2732 (coe v2))
  -- Algebra.Structures.Biased.IsSemiringWithoutOne*
  d_IsSemiringWithoutOne'42'_2772 a0 a1 a2 a3 a4 a5 a6 = ()
  data T_IsSemiringWithoutOne'42'_2772
    = C_IsSemiringWithoutOne'42''46'constructor_37821 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                      MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
                                                      MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
                                                      MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  -- Algebra.Structures.Biased.IsSemiringWithoutOne*.+-isCommutativeMonoid
  d_'43''45'isCommutativeMonoid_2788 ::
    T_IsSemiringWithoutOne'42'_2772 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'43''45'isCommutativeMonoid_2788 v0
    = case coe v0 of
        C_IsSemiringWithoutOne'42''46'constructor_37821 v1 v2 v3 v4
          -> coe v1
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsSemiringWithoutOne*.*-isSemigroup
  d_'42''45'isSemigroup_2790 ::
    T_IsSemiringWithoutOne'42'_2772 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_'42''45'isSemigroup_2790 v0
    = case coe v0 of
        C_IsSemiringWithoutOne'42''46'constructor_37821 v1 v2 v3 v4
          -> coe v2
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsSemiringWithoutOne*.distrib
  d_distrib_2792 ::
    T_IsSemiringWithoutOne'42'_2772 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_distrib_2792 v0
    = case coe v0 of
        C_IsSemiringWithoutOne'42''46'constructor_37821 v1 v2 v3 v4
          -> coe v3
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsSemiringWithoutOne*.zero
  d_zero_2794 ::
    T_IsSemiringWithoutOne'42'_2772 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_zero_2794 v0
    = case coe v0 of
        C_IsSemiringWithoutOne'42''46'constructor_37821 v1 v2 v3 v4
          -> coe v4
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsSemiringWithoutOne*.isSemiringWithoutOne
  d_isSemiringWithoutOne_2796 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsSemiringWithoutOne'42'_2772 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298
  d_isSemiringWithoutOne_2796 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_isSemiringWithoutOne_2796 v7
  du_isSemiringWithoutOne_2796 ::
    T_IsSemiringWithoutOne'42'_2772 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298
  du_isSemiringWithoutOne_2796 v0
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutOne'46'constructor_37629
        (coe d_'43''45'isCommutativeMonoid_2788 (coe v0))
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe d_'42''45'isSemigroup_2790 (coe v0))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_assoc_482
           (coe d_'42''45'isSemigroup_2790 (coe v0)))
        (coe d_distrib_2792 (coe v0)) (coe d_zero_2794 (coe v0))
  -- Algebra.Structures.Biased.IsNearSemiring*
  d_IsNearSemiring'42'_2834 a0 a1 a2 a3 a4 a5 a6 = ()
  data T_IsNearSemiring'42'_2834
    = C_IsNearSemiring'42''46'constructor_39635 MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
                                                MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
                                                (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
                                                (AgdaAny -> AgdaAny)
  -- Algebra.Structures.Biased.IsNearSemiring*.+-isMonoid
  d_'43''45'isMonoid_2850 ::
    T_IsNearSemiring'42'_2834 ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_'43''45'isMonoid_2850 v0
    = case coe v0 of
        C_IsNearSemiring'42''46'constructor_39635 v1 v2 v3 v4 -> coe v1
        _                                                     -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsNearSemiring*.*-isSemigroup
  d_'42''45'isSemigroup_2852 ::
    T_IsNearSemiring'42'_2834 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_'42''45'isSemigroup_2852 v0
    = case coe v0 of
        C_IsNearSemiring'42''46'constructor_39635 v1 v2 v3 v4 -> coe v2
        _                                                     -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsNearSemiring*.distribʳ
  d_distrib'691'_2854 ::
    T_IsNearSemiring'42'_2834 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_distrib'691'_2854 v0
    = case coe v0 of
        C_IsNearSemiring'42''46'constructor_39635 v1 v2 v3 v4 -> coe v3
        _                                                     -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsNearSemiring*.zeroˡ
  d_zero'737'_2856 :: T_IsNearSemiring'42'_2834 -> AgdaAny -> AgdaAny
  d_zero'737'_2856 v0
    = case coe v0 of
        C_IsNearSemiring'42''46'constructor_39635 v1 v2 v3 v4 -> coe v4
        _                                                     -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsNearSemiring*.isNearSemiring
  d_isNearSemiring_2858 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsNearSemiring'42'_2834 ->
    MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218
  d_isNearSemiring_2858 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 v7
    = du_isNearSemiring_2858 v7
  du_isNearSemiring_2858 ::
    T_IsNearSemiring'42'_2834 ->
    MAlonzo.Code.Algebra.Structures.T_IsNearSemiring_1218
  du_isNearSemiring_2858 v0
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsNearSemiring'46'constructor_35025
        (coe d_'43''45'isMonoid_2850 (coe v0))
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe d_'42''45'isSemigroup_2852 (coe v0))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_assoc_482
           (coe d_'42''45'isSemigroup_2852 (coe v0)))
        (coe d_distrib'691'_2854 (coe v0)) (coe d_zero'737'_2856 (coe v0))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*
  d_IsSemiringWithoutAnnihilatingZero'42'_2898 a0 a1 a2 a3 a4 a5 a6
                                               a7
    = ()
  data T_IsSemiringWithoutAnnihilatingZero'42'_2898
    = C_IsSemiringWithoutAnnihilatingZero'42''46'constructor_41443 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                                   MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
                                                                   MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.+-isCommutativeMonoid
  d_'43''45'isCommutativeMonoid_2914 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'43''45'isCommutativeMonoid_2914 v0
    = case coe v0 of
        C_IsSemiringWithoutAnnihilatingZero'42''46'constructor_41443 v1 v2 v3
          -> coe v1
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.*-isMonoid
  d_'42''45'isMonoid_2916 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_'42''45'isMonoid_2916 v0
    = case coe v0 of
        C_IsSemiringWithoutAnnihilatingZero'42''46'constructor_41443 v1 v2 v3
          -> coe v2
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.distrib
  d_distrib_2918 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_distrib_2918 v0
    = case coe v0 of
        C_IsSemiringWithoutAnnihilatingZero'42''46'constructor_41443 v1 v2 v3
          -> coe v3
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*.isSemiringWithoutAnnihilatingZero
  d_isSemiringWithoutAnnihilatingZero_2920 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468
  d_isSemiringWithoutAnnihilatingZero_2920 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5
                                           ~v6 ~v7 v8
    = du_isSemiringWithoutAnnihilatingZero_2920 v8
  du_isSemiringWithoutAnnihilatingZero_2920 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468
  du_isSemiringWithoutAnnihilatingZero_2920 v0
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutAnnihilatingZero'46'constructor_43811
        (coe d_'43''45'isCommutativeMonoid_2914 (coe v0))
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe d_'42''45'isMonoid_2916 (coe v0)))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_assoc_482
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe d_'42''45'isMonoid_2916 (coe v0))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_identity_698
           (coe d_'42''45'isMonoid_2916 (coe v0)))
        (coe d_distrib_2918 (coe v0))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.identityʳ
  d_identity'691'_2932 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
  d_identity'691'_2932 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 v8
    = du_identity'691'_2932 v8
  du_identity'691'_2932 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
  du_identity'691'_2932 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_identity'691'_728
        (coe d_'42''45'isMonoid_2916 (coe v0))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.identityˡ
  d_identity'737'_2934 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
  d_identity'737'_2934 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 v8
    = du_identity'737'_2934 v8
  du_identity'737'_2934 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 -> AgdaAny -> AgdaAny
  du_identity'737'_2934 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_identity'737'_726
        (coe d_'42''45'isMonoid_2916 (coe v0))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.isPartialEquivalence
  d_isPartialEquivalence_2940 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  d_isPartialEquivalence_2940 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 v8
    = du_isPartialEquivalence_2940 v8
  du_isPartialEquivalence_2940 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  du_isPartialEquivalence_2940 v0
    = let v1 = d_'42''45'isMonoid_2916 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (let v3 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2) in
            coe
              (coe
                 MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v3)))))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.isUnitalMagma
  d_isUnitalMagma_2944 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  d_isUnitalMagma_2944 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 v8
    = du_isUnitalMagma_2944 v8
  du_isUnitalMagma_2944 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  du_isUnitalMagma_2944 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
        (coe d_'42''45'isMonoid_2916 (coe v0))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.reflexive
  d_reflexive_2948 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  d_reflexive_2948 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 v8
    = du_reflexive_2948 v8
  du_reflexive_2948 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  du_reflexive_2948 v0
    = let v1 = d_'42''45'isMonoid_2916 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (let v3 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2) in
            coe
              (\ v4 v5 v6 ->
                 coe
                   MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                   (coe MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v3))
                   v4)))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.setoid
  d_setoid_2950 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  d_setoid_2950 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 v8
    = du_setoid_2950 v8
  du_setoid_2950 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  du_setoid_2950 v0
    = let v1 = d_'42''45'isMonoid_2916 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_setoid_200
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.∙-congʳ
  d_'8729''45'cong'691'_2958 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'691'_2958 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 v8
    = du_'8729''45'cong'691'_2958 v8
  du_'8729''45'cong'691'_2958 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'691'_2958 v0
    = let v1 = d_'42''45'isMonoid_2916 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsSemiringWithoutAnnihilatingZero*._._.∙-congˡ
  d_'8729''45'cong'737'_2960 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'737'_2960 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 v8
    = du_'8729''45'cong'737'_2960 v8
  du_'8729''45'cong'737'_2960 ::
    T_IsSemiringWithoutAnnihilatingZero'42'_2898 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'737'_2960 v0
    = let v1 = d_'42''45'isMonoid_2916 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsCommutativeSemiringˡ
  d_IsCommutativeSemiring'737'_2970 a0 a1 a2 a3 a4 a5 a6 a7 = ()
  data T_IsCommutativeSemiring'737'_2970
    = C_IsCommutativeSemiring'737''46'constructor_43731 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                        MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                        (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
                                                        (AgdaAny -> AgdaAny)
  -- Algebra.Structures.Biased.IsCommutativeSemiringˡ.+-isCommutativeMonoid
  d_'43''45'isCommutativeMonoid_2988 ::
    T_IsCommutativeSemiring'737'_2970 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'43''45'isCommutativeMonoid_2988 v0
    = case coe v0 of
        C_IsCommutativeSemiring'737''46'constructor_43731 v1 v2 v3 v4
          -> coe v1
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeSemiringˡ.*-isCommutativeMonoid
  d_'42''45'isCommutativeMonoid_2990 ::
    T_IsCommutativeSemiring'737'_2970 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'42''45'isCommutativeMonoid_2990 v0
    = case coe v0 of
        C_IsCommutativeSemiring'737''46'constructor_43731 v1 v2 v3 v4
          -> coe v2
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeSemiringˡ.distribʳ
  d_distrib'691'_2992 ::
    T_IsCommutativeSemiring'737'_2970 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_distrib'691'_2992 v0
    = case coe v0 of
        C_IsCommutativeSemiring'737''46'constructor_43731 v1 v2 v3 v4
          -> coe v3
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeSemiringˡ.zeroˡ
  d_zero'737'_2994 ::
    T_IsCommutativeSemiring'737'_2970 -> AgdaAny -> AgdaAny
  d_zero'737'_2994 v0
    = case coe v0 of
        C_IsCommutativeSemiring'737''46'constructor_43731 v1 v2 v3 v4
          -> coe v4
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeSemiringˡ.isCommutativeSemiring
  d_isCommutativeSemiring_2996 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsCommutativeSemiring'737'_2970 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
  d_isCommutativeSemiring_2996 ~v0 ~v1 ~v2 ~v3 v4 v5 v6 ~v7 v8
    = du_isCommutativeSemiring_2996 v4 v5 v6 v8
  du_isCommutativeSemiring_2996 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsCommutativeSemiring'737'_2970 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
  du_isCommutativeSemiring_2996 v0 v1 v2 v3
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemiring'46'constructor_51895
        (coe
           MAlonzo.Code.Algebra.Structures.C_IsSemiring'46'constructor_48071
           (coe
              MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutAnnihilatingZero'46'constructor_43811
              (coe d_'43''45'isCommutativeMonoid_2988 (coe v3))
              (coe
                 MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMagma_480
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                       (coe
                          MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                          (coe d_'42''45'isCommutativeMonoid_2990 (coe v3))))))
              (coe
                 MAlonzo.Code.Algebra.Structures.d_assoc_482
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                       (coe d_'42''45'isCommutativeMonoid_2990 (coe v3)))))
              (coe
                 MAlonzo.Code.Algebra.Structures.d_identity_698
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                    (coe d_'42''45'isCommutativeMonoid_2990 (coe v3))))
              (coe
                 MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'distr'691''8658'distr_528
                 (let v4 = d_'43''45'isCommutativeMonoid_2988 (coe v3) in
                  coe
                    (let v5
                           = MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (coe v4) in
                     coe
                       (let v6
                              = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v5) in
                        coe
                          (coe
                             MAlonzo.Code.Algebra.Structures.du_setoid_200
                             (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v6))))))
                 (coe v1) (coe v0)
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isMagma_480
                       (coe
                          MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                          (coe
                             MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                             (coe d_'43''45'isCommutativeMonoid_2988 (coe v3))))))
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_comm_748
                    (coe d_'42''45'isCommutativeMonoid_2990 (coe v3)))
                 (coe d_distrib'691'_2992 (coe v3))))
           (coe
              MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'ze'737''8658'ze_372
              (let v4 = d_'43''45'isCommutativeMonoid_2988 (coe v3) in
               coe
                 (let v5
                        = MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (coe v4) in
                  coe
                    (let v6
                           = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v5) in
                     coe
                       (coe
                          MAlonzo.Code.Algebra.Structures.du_setoid_200
                          (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v6))))))
              (coe v1)
              (coe
                 MAlonzo.Code.Algebra.Structures.d_comm_748
                 (coe d_'42''45'isCommutativeMonoid_2990 (coe v3)))
              (coe v2) (coe d_zero'737'_2994 (coe v3))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_comm_748
           (coe d_'42''45'isCommutativeMonoid_2990 (coe v3)))
  -- Algebra.Structures.Biased.IsCommutativeSemiringʳ
  d_IsCommutativeSemiring'691'_3098 a0 a1 a2 a3 a4 a5 a6 a7 = ()
  data T_IsCommutativeSemiring'691'_3098
    = C_IsCommutativeSemiring'691''46'constructor_48791 MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                        MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
                                                        (AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny)
                                                        (AgdaAny -> AgdaAny)
  -- Algebra.Structures.Biased.IsCommutativeSemiringʳ.+-isCommutativeMonoid
  d_'43''45'isCommutativeMonoid_3116 ::
    T_IsCommutativeSemiring'691'_3098 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'43''45'isCommutativeMonoid_3116 v0
    = case coe v0 of
        C_IsCommutativeSemiring'691''46'constructor_48791 v1 v2 v3 v4
          -> coe v1
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeSemiringʳ.*-isCommutativeMonoid
  d_'42''45'isCommutativeMonoid_3118 ::
    T_IsCommutativeSemiring'691'_3098 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'42''45'isCommutativeMonoid_3118 v0
    = case coe v0 of
        C_IsCommutativeSemiring'691''46'constructor_48791 v1 v2 v3 v4
          -> coe v2
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeSemiringʳ.distribˡ
  d_distrib'737'_3120 ::
    T_IsCommutativeSemiring'691'_3098 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_distrib'737'_3120 v0
    = case coe v0 of
        C_IsCommutativeSemiring'691''46'constructor_48791 v1 v2 v3 v4
          -> coe v3
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeSemiringʳ.zeroʳ
  d_zero'691'_3122 ::
    T_IsCommutativeSemiring'691'_3098 -> AgdaAny -> AgdaAny
  d_zero'691'_3122 v0
    = case coe v0 of
        C_IsCommutativeSemiring'691''46'constructor_48791 v1 v2 v3 v4
          -> coe v4
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsCommutativeSemiringʳ.isCommutativeSemiring
  d_isCommutativeSemiring_3124 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsCommutativeSemiring'691'_3098 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
  d_isCommutativeSemiring_3124 ~v0 ~v1 ~v2 ~v3 v4 v5 v6 ~v7 v8
    = du_isCommutativeSemiring_3124 v4 v5 v6 v8
  du_isCommutativeSemiring_3124 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsCommutativeSemiring'691'_3098 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
  du_isCommutativeSemiring_3124 v0 v1 v2 v3
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemiring'46'constructor_51895
        (coe
           MAlonzo.Code.Algebra.Structures.C_IsSemiring'46'constructor_48071
           (coe
              MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutAnnihilatingZero'46'constructor_43811
              (coe d_'43''45'isCommutativeMonoid_3116 (coe v3))
              (coe
                 MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMagma_480
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                       (coe
                          MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                          (coe d_'42''45'isCommutativeMonoid_3118 (coe v3))))))
              (coe
                 MAlonzo.Code.Algebra.Structures.d_assoc_482
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                       (coe d_'42''45'isCommutativeMonoid_3118 (coe v3)))))
              (coe
                 MAlonzo.Code.Algebra.Structures.d_identity_698
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                    (coe d_'42''45'isCommutativeMonoid_3118 (coe v3))))
              (coe
                 MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'distr'737''8658'distr_524
                 (let v4 = d_'43''45'isCommutativeMonoid_3116 (coe v3) in
                  coe
                    (let v5
                           = MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (coe v4) in
                     coe
                       (let v6
                              = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v5) in
                        coe
                          (coe
                             MAlonzo.Code.Algebra.Structures.du_setoid_200
                             (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v6))))))
                 (coe v1) (coe v0)
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isMagma_480
                       (coe
                          MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                          (coe
                             MAlonzo.Code.Algebra.Structures.d_isMonoid_746
                             (coe d_'43''45'isCommutativeMonoid_3116 (coe v3))))))
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_comm_748
                    (coe d_'42''45'isCommutativeMonoid_3118 (coe v3)))
                 (coe d_distrib'737'_3120 (coe v3))))
           (coe
              MAlonzo.Code.Algebra.Consequences.Setoid.du_comm'8743'ze'691''8658'ze_376
              (let v4 = d_'43''45'isCommutativeMonoid_3116 (coe v3) in
               coe
                 (let v5
                        = MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (coe v4) in
                  coe
                    (let v6
                           = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v5) in
                     coe
                       (coe
                          MAlonzo.Code.Algebra.Structures.du_setoid_200
                          (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v6))))))
              (coe v1)
              (coe
                 MAlonzo.Code.Algebra.Structures.d_comm_748
                 (coe d_'42''45'isCommutativeMonoid_3118 (coe v3)))
              (coe v2) (coe d_zero'691'_3122 (coe v3))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_comm_748
           (coe d_'42''45'isCommutativeMonoid_3118 (coe v3)))
  -- Algebra.Structures.Biased.IsRing*
  d_IsRing'42'_3228 a0 a1 a2 a3 a4 a5 a6 a7 a8 = ()
  data T_IsRing'42'_3228
    = C_IsRing'42''46'constructor_53915 MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
                                        MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
                                        MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
                                        MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  -- Algebra.Structures.Biased.IsRing*.+-isAbelianGroup
  d_'43''45'isAbelianGroup_3248 ::
    T_IsRing'42'_3228 ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
  d_'43''45'isAbelianGroup_3248 v0
    = case coe v0 of
        C_IsRing'42''46'constructor_53915 v1 v2 v3 v4 -> coe v1
        _                                             -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsRing*.*-isMonoid
  d_'42''45'isMonoid_3250 ::
    T_IsRing'42'_3228 -> MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_'42''45'isMonoid_3250 v0
    = case coe v0 of
        C_IsRing'42''46'constructor_53915 v1 v2 v3 v4 -> coe v2
        _                                             -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsRing*.distrib
  d_distrib_3252 ::
    T_IsRing'42'_3228 -> MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_distrib_3252 v0
    = case coe v0 of
        C_IsRing'42''46'constructor_53915 v1 v2 v3 v4 -> coe v3
        _                                             -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsRing*.zero
  d_zero_3254 ::
    T_IsRing'42'_3228 -> MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_zero_3254 v0
    = case coe v0 of
        C_IsRing'42''46'constructor_53915 v1 v2 v3 v4 -> coe v4
        _                                             -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsRing*.isRing
  d_isRing_3256 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRing'42'_3228 -> MAlonzo.Code.Algebra.Structures.T_IsRing_2650
  d_isRing_3256 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isRing_3256 v9
  du_isRing_3256 ::
    T_IsRing'42'_3228 -> MAlonzo.Code.Algebra.Structures.T_IsRing_2650
  du_isRing_3256 v0
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsRing'46'constructor_95033
        (coe d_'43''45'isAbelianGroup_3248 (coe v0))
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe d_'42''45'isMonoid_3250 (coe v0)))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_assoc_482
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe d_'42''45'isMonoid_3250 (coe v0))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_identity_698
           (coe d_'42''45'isMonoid_3250 (coe v0)))
        (coe d_distrib_3252 (coe v0))
  -- Algebra.Structures.Biased.IsRing*._._.identityʳ
  d_identity'691'_3268 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny -> AgdaAny -> T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
  d_identity'691'_3268 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_identity'691'_3268 v9
  du_identity'691'_3268 :: T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
  du_identity'691'_3268 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_identity'691'_728
        (coe d_'42''45'isMonoid_3250 (coe v0))
  -- Algebra.Structures.Biased.IsRing*._._.identityˡ
  d_identity'737'_3270 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny -> AgdaAny -> T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
  d_identity'737'_3270 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_identity'737'_3270 v9
  du_identity'737'_3270 :: T_IsRing'42'_3228 -> AgdaAny -> AgdaAny
  du_identity'737'_3270 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_identity'737'_726
        (coe d_'42''45'isMonoid_3250 (coe v0))
  -- Algebra.Structures.Biased.IsRing*._._.isPartialEquivalence
  d_isPartialEquivalence_3276 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRing'42'_3228 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  d_isPartialEquivalence_3276 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isPartialEquivalence_3276 v9
  du_isPartialEquivalence_3276 ::
    T_IsRing'42'_3228 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  du_isPartialEquivalence_3276 v0
    = let v1 = d_'42''45'isMonoid_3250 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (let v3 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2) in
            coe
              (coe
                 MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v3)))))
  -- Algebra.Structures.Biased.IsRing*._._.isUnitalMagma
  d_isUnitalMagma_3280 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRing'42'_3228 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  d_isUnitalMagma_3280 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isUnitalMagma_3280 v9
  du_isUnitalMagma_3280 ::
    T_IsRing'42'_3228 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  du_isUnitalMagma_3280 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
        (coe d_'42''45'isMonoid_3250 (coe v0))
  -- Algebra.Structures.Biased.IsRing*._._.reflexive
  d_reflexive_3284 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRing'42'_3228 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  d_reflexive_3284 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_reflexive_3284 v9
  du_reflexive_3284 ::
    T_IsRing'42'_3228 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  du_reflexive_3284 v0
    = let v1 = d_'42''45'isMonoid_3250 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (let v3 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2) in
            coe
              (\ v4 v5 v6 ->
                 coe
                   MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                   (coe MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v3))
                   v4)))
  -- Algebra.Structures.Biased.IsRing*._._.setoid
  d_setoid_3286 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRing'42'_3228 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  d_setoid_3286 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_setoid_3286 v9
  du_setoid_3286 ::
    T_IsRing'42'_3228 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  du_setoid_3286 v0
    = let v1 = d_'42''45'isMonoid_3250 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_setoid_200
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsRing*._._.∙-congʳ
  d_'8729''45'cong'691'_3294 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRing'42'_3228 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'691'_3294 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_'8729''45'cong'691'_3294 v9
  du_'8729''45'cong'691'_3294 ::
    T_IsRing'42'_3228 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'691'_3294 v0
    = let v1 = d_'42''45'isMonoid_3250 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsRing*._._.∙-congˡ
  d_'8729''45'cong'737'_3296 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRing'42'_3228 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'737'_3296 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_'8729''45'cong'737'_3296 v9
  du_'8729''45'cong'737'_3296 ::
    T_IsRing'42'_3228 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'737'_3296 v0
    = let v1 = d_'42''45'isMonoid_3250 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero
  d_IsRingWithoutAnnihilatingZero_3308 a0 a1 a2 a3 a4 a5 a6 a7 a8
    = ()
  data T_IsRingWithoutAnnihilatingZero_3308
    = C_IsRingWithoutAnnihilatingZero'46'constructor_56523 MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
                                                           MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
                                                           MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+-isAbelianGroup
  d_'43''45'isAbelianGroup_3326 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsAbelianGroup_1132
  d_'43''45'isAbelianGroup_3326 v0
    = case coe v0 of
        C_IsRingWithoutAnnihilatingZero'46'constructor_56523 v1 v2 v3
          -> coe v1
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*-isMonoid
  d_'42''45'isMonoid_3328 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_'42''45'isMonoid_3328 v0
    = case coe v0 of
        C_IsRingWithoutAnnihilatingZero'46'constructor_56523 v1 v2 v3
          -> coe v2
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.distrib
  d_distrib_3330 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_distrib_3330 v0
    = case coe v0 of
        C_IsRingWithoutAnnihilatingZero'46'constructor_56523 v1 v2 v3
          -> coe v3
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+._//_
  d__'47''47'__3334 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny
  d__'47''47'__3334 ~v0 ~v1 ~v2 ~v3 v4 ~v5 v6 ~v7 ~v8 ~v9
    = du__'47''47'__3334 v4 v6
  du__'47''47'__3334 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) -> AgdaAny -> AgdaAny -> AgdaAny
  du__'47''47'__3334 v0 v1
    = coe
        MAlonzo.Code.Algebra.Structures.du__'47''47'__1098 (coe v0)
        (coe v1)
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.assoc
  d_assoc_3336 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_assoc_3336 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_assoc_482
        (coe
           MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                 (coe d_'43''45'isAbelianGroup_3326 (coe v0)))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.comm
  d_comm_3338 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny
  d_comm_3338 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_comm_1146
        (coe d_'43''45'isAbelianGroup_3326 (coe v0))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identity
  d_identity_3340 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_identity_3340 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_identity_698
        (coe
           MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
           (coe
              MAlonzo.Code.Algebra.Structures.d_isGroup_1144
              (coe d_'43''45'isAbelianGroup_3326 (coe v0))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identityʳ
  d_identity'691'_3342 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_identity'691'_3342 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_identity'691'_3342 v9
  du_identity'691'_3342 ::
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  du_identity'691'_3342 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_identity'691'_728
              (coe MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v2))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.identityˡ
  d_identity'737'_3344 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_identity'737'_3344 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_identity'737'_3344 v9
  du_identity'737'_3344 ::
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  du_identity'737'_3344 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_identity'737'_726
              (coe MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v2))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverse
  d_inverse_3346 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_inverse_3346 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_inverse_1052
        (coe
           MAlonzo.Code.Algebra.Structures.d_isGroup_1144
           (coe d_'43''45'isAbelianGroup_3326 (coe v0)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverseʳ
  d_inverse'691'_3348 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_inverse'691'_3348 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_inverse'691'_3348 v9
  du_inverse'691'_3348 ::
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  du_inverse'691'_3348 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_inverse'691'_1108
           (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.inverseˡ
  d_inverse'737'_3350 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_inverse'737'_3350 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_inverse'737'_3350 v9
  du_inverse'737'_3350 ::
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  du_inverse'737'_3350 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_inverse'737'_1106
           (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeMagma
  d_isCommutativeMagma_3352 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_212
  d_isCommutativeMagma_3352 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isCommutativeMagma_3352 v9
  du_isCommutativeMagma_3352 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMagma_212
  du_isCommutativeMagma_3352 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = coe
                   MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
                   (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_isCommutativeMagma_586
              (coe
                 MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_786
                 (coe v2))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeMonoid
  d_isCommutativeMonoid_3354 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_isCommutativeMonoid_3354 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isCommutativeMonoid_3354 v9
  du_isCommutativeMonoid_3354 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  du_isCommutativeMonoid_3354 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
        (coe d_'43''45'isAbelianGroup_3326 (coe v0))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isCommutativeSemigroup
  d_isCommutativeSemigroup_3356 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
  d_isCommutativeSemigroup_3356 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8
                                v9
    = du_isCommutativeSemigroup_3356 v9
  du_isCommutativeSemigroup_3356 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
  du_isCommutativeSemigroup_3356 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_isCommutativeSemigroup_786
           (coe
              MAlonzo.Code.Algebra.Structures.du_isCommutativeMonoid_1204
              (coe v1)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isEquivalence
  d_isEquivalence_3358 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
  d_isEquivalence_3358 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
        (coe
           MAlonzo.Code.Algebra.Structures.d_isMagma_480
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                    (coe d_'43''45'isAbelianGroup_3326 (coe v0))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isGroup
  d_isGroup_3360 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsGroup_1036
  d_isGroup_3360 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isGroup_1144
        (coe d_'43''45'isAbelianGroup_3326 (coe v0))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isInvertibleMagma
  d_isInvertibleMagma_3362 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_924
  d_isInvertibleMagma_3362 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isInvertibleMagma_3362 v9
  du_isInvertibleMagma_3362 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsInvertibleMagma_924
  du_isInvertibleMagma_3362 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_isInvertibleMagma_1122
           (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isInvertibleUnitalMagma
  d_isInvertibleUnitalMagma_3364 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_976
  d_isInvertibleUnitalMagma_3364 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8
                                 v9
    = du_isInvertibleUnitalMagma_3364 v9
  du_isInvertibleUnitalMagma_3364 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsInvertibleUnitalMagma_976
  du_isInvertibleUnitalMagma_3364 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_isInvertibleUnitalMagma_1124
           (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isMagma
  d_isMagma_3366 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176
  d_isMagma_3366 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isMagma_480
        (coe
           MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                 (coe d_'43''45'isAbelianGroup_3326 (coe v0)))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isMonoid
  d_isMonoid_3368 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_isMonoid_3368 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
        (coe
           MAlonzo.Code.Algebra.Structures.d_isGroup_1144
           (coe d_'43''45'isAbelianGroup_3326 (coe v0)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isPartialEquivalence
  d_isPartialEquivalence_3370 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  d_isPartialEquivalence_3370 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isPartialEquivalence_3370 v9
  du_isPartialEquivalence_3370 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  du_isPartialEquivalence_3370 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v2) in
            coe
              (let v4
                     = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v3) in
               coe
                 (let v5 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v4) in
                  coe
                    (coe
                       MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
                       (coe
                          MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v5)))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isSemigroup
  d_isSemigroup_3372 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_isSemigroup_3372 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
        (coe
           MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
           (coe
              MAlonzo.Code.Algebra.Structures.d_isGroup_1144
              (coe d_'43''45'isAbelianGroup_3326 (coe v0))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.isUnitalMagma
  d_isUnitalMagma_3374 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  d_isUnitalMagma_3374 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isUnitalMagma_3374 v9
  du_isUnitalMagma_3374 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  du_isUnitalMagma_3374 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
              (coe MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v2))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.refl
  d_refl_3376 ::
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_refl_3376 v0
    = coe
        MAlonzo.Code.Relation.Binary.Structures.d_refl_34
        (coe
           MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                       (coe d_'43''45'isAbelianGroup_3326 (coe v0)))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.reflexive
  d_reflexive_3378 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  d_reflexive_3378 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_reflexive_3378 v9
  du_reflexive_3378 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  du_reflexive_3378 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v2) in
            coe
              (let v4
                     = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v3) in
               coe
                 (let v5 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v4) in
                  coe
                    (\ v6 v7 v8 ->
                       coe
                         MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                         (coe MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v5))
                         v6)))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.setoid
  d_setoid_3380 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  d_setoid_3380 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_setoid_3380 v9
  du_setoid_3380 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  du_setoid_3380 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v2) in
            coe
              (let v4
                     = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v3) in
               coe
                 (coe
                    MAlonzo.Code.Algebra.Structures.du_setoid_200
                    (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v4))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.sym
  d_sym_3382 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_sym_3382 v0
    = coe
        MAlonzo.Code.Relation.Binary.Structures.d_sym_36
        (coe
           MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                       (coe d_'43''45'isAbelianGroup_3326 (coe v0)))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.trans
  d_trans_3384 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_trans_3384 v0
    = coe
        MAlonzo.Code.Relation.Binary.Structures.d_trans_38
        (coe
           MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                       (coe d_'43''45'isAbelianGroup_3326 (coe v0)))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.uniqueʳ-⁻¹
  d_unique'691''45''8315''185'_3386 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_unique'691''45''8315''185'_3386 ~v0 ~v1 ~v2 ~v3 v4 ~v5 v6 v7 ~v8
                                    v9
    = du_unique'691''45''8315''185'_3386 v4 v6 v7 v9
  du_unique'691''45''8315''185'_3386 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_unique'691''45''8315''185'_3386 v0 v1 v2 v3
    = let v4 = d_'43''45'isAbelianGroup_3326 (coe v3) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_unique'691''45''8315''185'_1120
           (coe v0) (coe v2) (coe v1)
           (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v4)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.uniqueˡ-⁻¹
  d_unique'737''45''8315''185'_3388 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_unique'737''45''8315''185'_3388 ~v0 ~v1 ~v2 ~v3 v4 ~v5 v6 v7 ~v8
                                    v9
    = du_unique'737''45''8315''185'_3388 v4 v6 v7 v9
  du_unique'737''45''8315''185'_3388 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_unique'737''45''8315''185'_3388 v0 v1 v2 v3
    = let v4 = d_'43''45'isAbelianGroup_3326 (coe v3) in
      coe
        (coe
           MAlonzo.Code.Algebra.Structures.du_unique'737''45''8315''185'_1114
           (coe v0) (coe v2) (coe v1)
           (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v4)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.⁻¹-cong
  d_'8315''185''45'cong_3390 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8315''185''45'cong_3390 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'8315''185''45'cong_1054
        (coe
           MAlonzo.Code.Algebra.Structures.d_isGroup_1144
           (coe d_'43''45'isAbelianGroup_3326 (coe v0)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-cong
  d_'8729''45'cong_3392 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong_3392 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
        (coe
           MAlonzo.Code.Algebra.Structures.d_isMagma_480
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                    (coe d_'43''45'isAbelianGroup_3326 (coe v0))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-congʳ
  d_'8729''45'cong'691'_3394 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'691'_3394 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_'8729''45'cong'691'_3394 v9
  du_'8729''45'cong'691'_3394 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'691'_3394 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v2) in
            coe
              (let v4
                     = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v3) in
               coe
                 (coe
                    MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
                    (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v4))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.+.∙-congˡ
  d_'8729''45'cong'737'_3396 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'737'_3396 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_'8729''45'cong'737'_3396 v9
  du_'8729''45'cong'737'_3396 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'737'_3396 v0
    = let v1 = d_'43''45'isAbelianGroup_3326 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v1) in
         coe
           (let v3
                  = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v2) in
            coe
              (let v4
                     = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v3) in
               coe
                 (coe
                    MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
                    (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v4))))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.assoc
  d_assoc_3400 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_assoc_3400 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_assoc_482
        (coe
           MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
           (coe d_'42''45'isMonoid_3328 (coe v0)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identity
  d_identity_3402 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_identity_3402 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_identity_698
        (coe d_'42''45'isMonoid_3328 (coe v0))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identityʳ
  d_identity'691'_3404 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_identity'691'_3404 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_identity'691'_3404 v9
  du_identity'691'_3404 ::
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  du_identity'691'_3404 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_identity'691'_728
        (coe d_'42''45'isMonoid_3328 (coe v0))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.identityˡ
  d_identity'737'_3406 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_identity'737'_3406 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_identity'737'_3406 v9
  du_identity'737'_3406 ::
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  du_identity'737'_3406 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_identity'737'_726
        (coe d_'42''45'isMonoid_3328 (coe v0))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isEquivalence
  d_isEquivalence_3408 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
  d_isEquivalence_3408 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
        (coe
           MAlonzo.Code.Algebra.Structures.d_isMagma_480
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe d_'42''45'isMonoid_3328 (coe v0))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isMagma
  d_isMagma_3410 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176
  d_isMagma_3410 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isMagma_480
        (coe
           MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
           (coe d_'42''45'isMonoid_3328 (coe v0)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isPartialEquivalence
  d_isPartialEquivalence_3412 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  d_isPartialEquivalence_3412 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isPartialEquivalence_3412 v9
  du_isPartialEquivalence_3412 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialEquivalence_16
  du_isPartialEquivalence_3412 v0
    = let v1 = d_'42''45'isMonoid_3328 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (let v3 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2) in
            coe
              (coe
                 MAlonzo.Code.Relation.Binary.Structures.du_isPartialEquivalence_42
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v3)))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isSemigroup
  d_isSemigroup_3414 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_isSemigroup_3414 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
        (coe d_'42''45'isMonoid_3328 (coe v0))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.isUnitalMagma
  d_isUnitalMagma_3416 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  d_isUnitalMagma_3416 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isUnitalMagma_3416 v9
  du_isUnitalMagma_3416 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsUnitalMagma_642
  du_isUnitalMagma_3416 v0
    = coe
        MAlonzo.Code.Algebra.Structures.du_isUnitalMagma_730
        (coe d_'42''45'isMonoid_3328 (coe v0))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.refl
  d_refl_3418 ::
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_refl_3418 v0
    = coe
        MAlonzo.Code.Relation.Binary.Structures.d_refl_34
        (coe
           MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe d_'42''45'isMonoid_3328 (coe v0)))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.reflexive
  d_reflexive_3420 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  d_reflexive_3420 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_reflexive_3420 v9
  du_reflexive_3420 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny ->
    AgdaAny ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  du_reflexive_3420 v0
    = let v1 = d_'42''45'isMonoid_3328 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (let v3 = MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2) in
            coe
              (\ v4 v5 v6 ->
                 coe
                   MAlonzo.Code.Relation.Binary.Structures.du_reflexive_40
                   (coe MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v3))
                   v4)))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.setoid
  d_setoid_3422 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  d_setoid_3422 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_setoid_3422 v9
  du_setoid_3422 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Relation.Binary.Bundles.T_Setoid_44
  du_setoid_3422 v0
    = let v1 = d_'42''45'isMonoid_3328 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_setoid_200
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.sym
  d_sym_3424 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_sym_3424 v0
    = coe
        MAlonzo.Code.Relation.Binary.Structures.d_sym_36
        (coe
           MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe d_'42''45'isMonoid_3328 (coe v0)))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.trans
  d_trans_3426 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_trans_3426 v0
    = coe
        MAlonzo.Code.Relation.Binary.Structures.d_trans_38
        (coe
           MAlonzo.Code.Algebra.Structures.d_isEquivalence_184
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe d_'42''45'isMonoid_3328 (coe v0)))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-cong
  d_'8729''45'cong_3428 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong_3428 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
        (coe
           MAlonzo.Code.Algebra.Structures.d_isMagma_480
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe d_'42''45'isMonoid_3328 (coe v0))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-congʳ
  d_'8729''45'cong'691'_3430 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'691'_3430 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_'8729''45'cong'691'_3430 v9
  du_'8729''45'cong'691'_3430 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'691'_3430 v0
    = let v1 = d_'42''45'isMonoid_3328 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'691'_206
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.*.∙-congˡ
  d_'8729''45'cong'737'_3432 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  d_'8729''45'cong'737'_3432 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_'8729''45'cong'737'_3432 v9
  du_'8729''45'cong'737'_3432 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny -> AgdaAny
  du_'8729''45'cong'737'_3432 v0
    = let v1 = d_'42''45'isMonoid_3328 (coe v0) in
      coe
        (let v2
               = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v1) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_'8729''45'cong'737'_202
              (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v2))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zeroˡ
  d_zero'737'_3434 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_zero'737'_3434 ~v0 ~v1 ~v2 ~v3 v4 v5 v6 v7 ~v8 v9
    = du_zero'737'_3434 v4 v5 v6 v7 v9
  du_zero'737'_3434 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  du_zero'737'_3434 v0 v1 v2 v3 v4
    = coe
        MAlonzo.Code.Algebra.Consequences.Setoid.du_assoc'8743'distrib'691''8743'id'691''8743'inv'691''8658'ze'737'_594
        (let v5 = d_'43''45'isAbelianGroup_3326 (coe v4) in
         coe
           (let v6
                  = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v5) in
            coe
              (let v7
                     = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v6) in
               coe
                 (let v8
                        = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v7) in
                  coe
                    (coe
                       MAlonzo.Code.Algebra.Structures.du_setoid_200
                       (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v8)))))))
        (coe v0) (coe v1) (coe v2) (coe v3)
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                       (coe d_'43''45'isAbelianGroup_3326 (coe v4)))))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe d_'42''45'isMonoid_3328 (coe v4)))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_assoc_482
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                    (coe d_'43''45'isAbelianGroup_3326 (coe v4))))))
        (coe
           MAlonzo.Code.Agda.Builtin.Sigma.d_snd_30
           (coe d_distrib_3330 (coe v4)))
        (let v5 = d_'43''45'isAbelianGroup_3326 (coe v4) in
         coe
           (let v6
                  = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v5) in
            coe
              (coe
                 MAlonzo.Code.Algebra.Structures.du_identity'691'_728
                 (coe MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v6)))))
        (let v5 = d_'43''45'isAbelianGroup_3326 (coe v4) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_inverse'691'_1108
              (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v5))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zeroʳ
  d_zero'691'_3436 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  d_zero'691'_3436 ~v0 ~v1 ~v2 ~v3 v4 v5 v6 v7 ~v8 v9
    = du_zero'691'_3436 v4 v5 v6 v7 v9
  du_zero'691'_3436 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 -> AgdaAny -> AgdaAny
  du_zero'691'_3436 v0 v1 v2 v3 v4
    = coe
        MAlonzo.Code.Algebra.Consequences.Setoid.du_assoc'8743'distrib'737''8743'id'691''8743'inv'691''8658'ze'691'_606
        (let v5 = d_'43''45'isAbelianGroup_3326 (coe v4) in
         coe
           (let v6
                  = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v5) in
            coe
              (let v7
                     = MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v6) in
               coe
                 (let v8
                        = MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v7) in
                  coe
                    (coe
                       MAlonzo.Code.Algebra.Structures.du_setoid_200
                       (coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v8)))))))
        (coe v0) (coe v1) (coe v2) (coe v3)
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                    (coe
                       MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                       (coe d_'43''45'isAbelianGroup_3326 (coe v4)))))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe d_'42''45'isMonoid_3328 (coe v4)))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_assoc_482
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isMonoid_1050
                 (coe
                    MAlonzo.Code.Algebra.Structures.d_isGroup_1144
                    (coe d_'43''45'isAbelianGroup_3326 (coe v4))))))
        (coe
           MAlonzo.Code.Agda.Builtin.Sigma.d_fst_28
           (coe d_distrib_3330 (coe v4)))
        (let v5 = d_'43''45'isAbelianGroup_3326 (coe v4) in
         coe
           (let v6
                  = MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v5) in
            coe
              (coe
                 MAlonzo.Code.Algebra.Structures.du_identity'691'_728
                 (coe MAlonzo.Code.Algebra.Structures.d_isMonoid_1050 (coe v6)))))
        (let v5 = d_'43''45'isAbelianGroup_3326 (coe v4) in
         coe
           (coe
              MAlonzo.Code.Algebra.Structures.du_inverse'691'_1108
              (coe MAlonzo.Code.Algebra.Structures.d_isGroup_1144 (coe v5))))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.zero
  d_zero_3438 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_zero_3438 ~v0 ~v1 ~v2 ~v3 v4 v5 v6 v7 ~v8 v9
    = du_zero_3438 v4 v5 v6 v7 v9
  du_zero_3438 ::
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  du_zero_3438 v0 v1 v2 v3 v4
    = coe
        MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
        (coe
           du_zero'737'_3434 (coe v0) (coe v1) (coe v2) (coe v3) (coe v4))
        (coe
           du_zero'691'_3436 (coe v0) (coe v1) (coe v2) (coe v3) (coe v4))
  -- Algebra.Structures.Biased.IsRingWithoutAnnihilatingZero.isRing
  d_isRing_3440 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    (AgdaAny -> AgdaAny -> ()) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny -> AgdaAny) ->
    (AgdaAny -> AgdaAny) ->
    AgdaAny ->
    AgdaAny ->
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsRing_2650
  d_isRing_3440 ~v0 ~v1 ~v2 ~v3 ~v4 ~v5 ~v6 ~v7 ~v8 v9
    = du_isRing_3440 v9
  du_isRing_3440 ::
    T_IsRingWithoutAnnihilatingZero_3308 ->
    MAlonzo.Code.Algebra.Structures.T_IsRing_2650
  du_isRing_3440 v0
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsRing'46'constructor_95033
        (coe d_'43''45'isAbelianGroup_3326 (coe v0))
        (coe
           MAlonzo.Code.Algebra.Structures.d_'8729''45'cong_186
           (coe
              MAlonzo.Code.Algebra.Structures.d_isMagma_480
              (coe
                 MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
                 (coe d_'42''45'isMonoid_3328 (coe v0)))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_assoc_482
           (coe
              MAlonzo.Code.Algebra.Structures.d_isSemigroup_696
              (coe d_'42''45'isMonoid_3328 (coe v0))))
        (coe
           MAlonzo.Code.Algebra.Structures.d_identity_698
           (coe d_'42''45'isMonoid_3328 (coe v0)))
        (coe d_distrib_3330 (coe v0))
  