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.Data.Nat.Properties where
  
  import Data.Text qualified
  import MAlonzo.Code.Agda.Builtin.Bool qualified
  import MAlonzo.Code.Agda.Builtin.Equality qualified
  import MAlonzo.Code.Agda.Builtin.Nat qualified
  import MAlonzo.Code.Agda.Builtin.Sigma qualified
  import MAlonzo.Code.Agda.Builtin.Unit qualified
  import MAlonzo.Code.Agda.Primitive qualified
  import MAlonzo.Code.Algebra.Bundles qualified
  import MAlonzo.Code.Algebra.Construct.NaturalChoice.Base qualified
  import MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp qualified
  import MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp qualified
  import MAlonzo.Code.Algebra.Definitions.RawMagma qualified
  import MAlonzo.Code.Algebra.Lattice.Bundles qualified
  import MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp qualified
  import MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinOp qualified
  import MAlonzo.Code.Algebra.Lattice.Structures qualified
  import MAlonzo.Code.Algebra.Morphism qualified
  import MAlonzo.Code.Algebra.Structures qualified
  import MAlonzo.Code.Data.Irrelevant qualified
  import MAlonzo.Code.Data.Nat.Base qualified
  import MAlonzo.Code.Data.Sum.Base qualified
  import MAlonzo.Code.Function.Base qualified
  import MAlonzo.Code.Function.Bundles qualified
  import MAlonzo.Code.Function.Metric.Nat.Bundles qualified
  import MAlonzo.Code.Function.Metric.Structures qualified
  import MAlonzo.Code.Relation.Binary.Bundles qualified
  import MAlonzo.Code.Relation.Binary.Consequences qualified
  import MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd qualified
  import MAlonzo.Code.Relation.Binary.Definitions qualified
  import MAlonzo.Code.Relation.Binary.PropositionalEquality.Algebra qualified
  import MAlonzo.Code.Relation.Binary.PropositionalEquality.Core qualified
  import MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties qualified
  import MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple qualified
  import MAlonzo.Code.Relation.Binary.Reasoning.Syntax qualified
  import MAlonzo.Code.Relation.Binary.Structures qualified
  import MAlonzo.Code.Relation.Binary.Structures.Biased qualified
  import MAlonzo.Code.Relation.Nullary.Decidable qualified
  import MAlonzo.Code.Relation.Nullary.Decidable.Core qualified
  import MAlonzo.Code.Relation.Nullary.Negation.Core qualified
  import MAlonzo.Code.Relation.Nullary.Reflects 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
  
  -- Data.Nat.Properties._._DistributesOver_
  d__DistributesOver__10 ::
    (Integer -> Integer -> Integer) ->
    (Integer -> Integer -> Integer) -> ()
  d__DistributesOver__10 = erased
  -- Data.Nat.Properties._._DistributesOverʳ_
  d__DistributesOver'691'__12 ::
    (Integer -> Integer -> Integer) ->
    (Integer -> Integer -> Integer) -> ()
  d__DistributesOver'691'__12 = erased
  -- Data.Nat.Properties._._DistributesOverˡ_
  d__DistributesOver'737'__14 ::
    (Integer -> Integer -> Integer) ->
    (Integer -> Integer -> Integer) -> ()
  d__DistributesOver'737'__14 = erased
  -- Data.Nat.Properties._.Associative
  d_Associative_30 :: (Integer -> Integer -> Integer) -> ()
  d_Associative_30 = erased
  -- Data.Nat.Properties._.Commutative
  d_Commutative_32 :: (Integer -> Integer -> Integer) -> ()
  d_Commutative_32 = erased
  -- Data.Nat.Properties._.Identity
  d_Identity_48 :: Integer -> (Integer -> Integer -> Integer) -> ()
  d_Identity_48 = erased
  -- Data.Nat.Properties._.LeftIdentity
  d_LeftIdentity_72 ::
    Integer -> (Integer -> Integer -> Integer) -> ()
  d_LeftIdentity_72 = erased
  -- Data.Nat.Properties._.LeftZero
  d_LeftZero_80 :: Integer -> (Integer -> Integer -> Integer) -> ()
  d_LeftZero_80 = erased
  -- Data.Nat.Properties._.RightIdentity
  d_RightIdentity_100 ::
    Integer -> (Integer -> Integer -> Integer) -> ()
  d_RightIdentity_100 = erased
  -- Data.Nat.Properties._.RightZero
  d_RightZero_108 :: Integer -> (Integer -> Integer -> Integer) -> ()
  d_RightZero_108 = erased
  -- Data.Nat.Properties._.Zero
  d_Zero_128 :: Integer -> (Integer -> Integer -> Integer) -> ()
  d_Zero_128 = erased
  -- Data.Nat.Properties._.IsCommutativeMonoid
  d_IsCommutativeMonoid_144 a0 a1 = ()
  -- Data.Nat.Properties._.IsCommutativeSemigroup
  d_IsCommutativeSemigroup_148 a0 = ()
  -- Data.Nat.Properties._.IsCommutativeSemiring
  d_IsCommutativeSemiring_150 a0 a1 a2 a3 = ()
  -- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne
  d_IsCommutativeSemiringWithoutOne_152 a0 a1 a2 = ()
  -- Data.Nat.Properties._.IsMagma
  d_IsMagma_176 a0 = ()
  -- Data.Nat.Properties._.IsMonoid
  d_IsMonoid_182 a0 a1 = ()
  -- Data.Nat.Properties._.IsSemigroup
  d_IsSemigroup_204 a0 = ()
  -- Data.Nat.Properties._.IsSemiring
  d_IsSemiring_208 a0 a1 a2 a3 = ()
  -- Data.Nat.Properties._.IsSemiringWithoutOne
  d_IsSemiringWithoutOne_212 a0 a1 a2 = ()
  -- Data.Nat.Properties._.IsCommutativeMonoid.comm
  d_comm_514 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736 ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_comm_514 = erased
  -- Data.Nat.Properties._.IsCommutativeMonoid.isMonoid
  d_isMonoid_530 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736 ->
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_isMonoid_530 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isMonoid_746 (coe v0)
  -- Data.Nat.Properties._.IsCommutativeSemigroup.comm
  d_comm_682 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548 ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_comm_682 = erased
  -- Data.Nat.Properties._.IsCommutativeSemigroup.isSemigroup
  d_isSemigroup_692 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_isSemigroup_692 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isSemigroup_556 (coe v0)
  -- Data.Nat.Properties._.IsCommutativeSemiring.*-comm
  d_'42''45'comm_714 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678 ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'comm_714 = erased
  -- Data.Nat.Properties._.IsCommutativeSemiring.isSemiring
  d_isSemiring_784 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemiring_1570
  d_isSemiring_784 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isSemiring_1692 (coe v0)
  -- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.*-comm
  d_'42''45'comm_814 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1382 ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'comm_814 = erased
  -- Data.Nat.Properties._.IsCommutativeSemiringWithoutOne.isSemiringWithoutOne
  d_isSemiringWithoutOne_850 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1382 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298
  d_isSemiringWithoutOne_850 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutOne_1394
        (coe v0)
  -- Data.Nat.Properties._.IsMagma.isEquivalence
  d_isEquivalence_1476 ::
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176 ->
    MAlonzo.Code.Relation.Binary.Structures.T_IsEquivalence_26
  d_isEquivalence_1476 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isEquivalence_184 (coe v0)
  -- Data.Nat.Properties._.IsMagma.∙-cong
  d_'8729''45'cong_1490 ::
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176 ->
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8729''45'cong_1490 = erased
  -- Data.Nat.Properties._.IsMonoid.identity
  d_identity_1586 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_identity_1586 v0
    = coe MAlonzo.Code.Algebra.Structures.d_identity_698 (coe v0)
  -- Data.Nat.Properties._.IsMonoid.isSemigroup
  d_isSemigroup_1598 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_isSemigroup_1598 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isSemigroup_696 (coe v0)
  -- Data.Nat.Properties._.IsSemigroup.assoc
  d_assoc_2330 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472 ->
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_assoc_2330 = erased
  -- Data.Nat.Properties._.IsSemigroup.isMagma
  d_isMagma_2334 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472 ->
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176
  d_isMagma_2334 v0
    = coe MAlonzo.Code.Algebra.Structures.d_isMagma_480 (coe v0)
  -- Data.Nat.Properties._.IsSemiring.isSemiringWithoutAnnihilatingZero
  d_isSemiringWithoutAnnihilatingZero_2448 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiring_1570 ->
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutAnnihilatingZero_1468
  d_isSemiringWithoutAnnihilatingZero_2448 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_isSemiringWithoutAnnihilatingZero_1584
        (coe v0)
  -- Data.Nat.Properties._.IsSemiring.zero
  d_zero_2462 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiring_1570 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_zero_2462 v0
    = coe MAlonzo.Code.Algebra.Structures.d_zero_1586 (coe v0)
  -- Data.Nat.Properties._.IsSemiringWithoutOne.*-assoc
  d_'42''45'assoc_2546 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'assoc_2546 = erased
  -- Data.Nat.Properties._.IsSemiringWithoutOne.*-cong
  d_'42''45'cong_2548 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'cong_2548 = erased
  -- Data.Nat.Properties._.IsSemiringWithoutOne.+-isCommutativeMonoid
  d_'43''45'isCommutativeMonoid_2562 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'43''45'isCommutativeMonoid_2562 v0
    = coe
        MAlonzo.Code.Algebra.Structures.d_'43''45'isCommutativeMonoid_1316
        (coe v0)
  -- Data.Nat.Properties._.IsSemiringWithoutOne.distrib
  d_distrib_2570 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_distrib_2570 v0
    = coe MAlonzo.Code.Algebra.Structures.d_distrib_1322 (coe v0)
  -- Data.Nat.Properties._.IsSemiringWithoutOne.zero
  d_zero_2590 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_zero_2590 v0
    = coe MAlonzo.Code.Algebra.Structures.d_zero_1324 (coe v0)
  -- Data.Nat.Properties.nonZero?
  d_nonZero'63'_2652 ::
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d_nonZero'63'_2652 v0
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_false_8)
               (coe MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
        _ -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_true_10)
               (coe
                  MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22
                  (coe
                     MAlonzo.Code.Data.Nat.Base.C_NonZero'46'constructor_3575
                     (coe MAlonzo.Code.Agda.Builtin.Unit.C_tt_8)))
  -- Data.Nat.Properties.nonTrivial?
  d_nonTrivial'63'_2656 ::
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d_nonTrivial'63'_2656 v0
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_false_8)
               (coe MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
        1 -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_false_8)
               (coe MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
        _ -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_true_10)
               (coe
                  MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22
                  (coe
                     MAlonzo.Code.Data.Nat.Base.C_NonTrivial'46'constructor_5661
                     (coe MAlonzo.Code.Agda.Builtin.Unit.C_tt_8)))
  -- Data.Nat.Properties.suc-injective
  d_suc'45'injective_2660 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_suc'45'injective_2660 = erased
  -- Data.Nat.Properties.≡ᵇ⇒≡
  d_'8801''7495''8658''8801'_2666 ::
    Integer ->
    Integer ->
    AgdaAny -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8801''7495''8658''8801'_2666 = erased
  -- Data.Nat.Properties.≡⇒≡ᵇ
  d_'8801''8658''8801''7495'_2678 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 -> AgdaAny
  d_'8801''8658''8801''7495'_2678 v0 ~v1 ~v2
    = du_'8801''8658''8801''7495'_2678 v0
  du_'8801''8658''8801''7495'_2678 :: Integer -> AgdaAny
  du_'8801''8658''8801''7495'_2678 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Agda.Builtin.Unit.C_tt_8
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe (coe du_'8801''8658''8801''7495'_2678 (coe v1))
  -- Data.Nat.Properties._≟_
  d__'8799'__2688 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8799'__2688 v0 v1
    = coe
        MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
        erased (\ v2 -> coe du_'8801''8658''8801''7495'_2678 (coe v0))
        (coe
           MAlonzo.Code.Relation.Nullary.Decidable.Core.d_T'63'_66
           (coe eqInt (coe v0) (coe v1)))
  -- Data.Nat.Properties.≡-irrelevant
  d_'8801''45'irrelevant_2694 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8801''45'irrelevant_2694 = erased
  -- Data.Nat.Properties.≟-diag
  d_'8799''45'diag_2698 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8799''45'diag_2698 = erased
  -- Data.Nat.Properties.≡-isDecEquivalence
  d_'8801''45'isDecEquivalence_2700 ::
    MAlonzo.Code.Relation.Binary.Structures.T_IsDecEquivalence_44
  d_'8801''45'isDecEquivalence_2700
    = coe
        MAlonzo.Code.Relation.Binary.Structures.C_IsDecEquivalence'46'constructor_3083
        (coe
           MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_396)
        (coe d__'8799'__2688)
  -- Data.Nat.Properties.≡-decSetoid
  d_'8801''45'decSetoid_2702 ::
    MAlonzo.Code.Relation.Binary.Bundles.T_DecSetoid_84
  d_'8801''45'decSetoid_2702
    = coe
        MAlonzo.Code.Relation.Binary.Bundles.C_DecSetoid'46'constructor_1389
        d_'8801''45'isDecEquivalence_2700
  -- Data.Nat.Properties.0≢1+n
  d_0'8802'1'43'n_2704 ::
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_0'8802'1'43'n_2704 = erased
  -- Data.Nat.Properties.1+n≢0
  d_1'43'n'8802'0_2706 ::
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_1'43'n'8802'0_2706 = erased
  -- Data.Nat.Properties.1+n≢n
  d_1'43'n'8802'n_2708 ::
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_1'43'n'8802'n_2708 = erased
  -- Data.Nat.Properties.<ᵇ⇒<
  d_'60''7495''8658''60'_2716 ::
    Integer ->
    Integer -> AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''7495''8658''60'_2716 v0 ~v1 ~v2
    = du_'60''7495''8658''60'_2716 v0
  du_'60''7495''8658''60'_2716 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'60''7495''8658''60'_2716 v0
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
               (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (coe du_'60''7495''8658''60'_2716 (coe v1)))
  -- Data.Nat.Properties.<⇒<ᵇ
  d_'60''8658''60''7495'_2728 ::
    Integer ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22 -> AgdaAny
  d_'60''8658''60''7495'_2728 ~v0 ~v1 v2
    = du_'60''8658''60''7495'_2728 v2
  du_'60''8658''60''7495'_2728 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 -> AgdaAny
  du_'60''8658''60''7495'_2728 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3
          -> case coe v3 of
               MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                 -> coe MAlonzo.Code.Agda.Builtin.Unit.C_tt_8
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
                 -> coe
                      du_'60''8658''60''7495'_2728
                      (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6)
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.<ᵇ-reflects-<
  d_'60''7495''45'reflects'45''60'_2736 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Reflects.T_Reflects_16
  d_'60''7495''45'reflects'45''60'_2736 v0 v1
    = coe
        MAlonzo.Code.Relation.Nullary.Reflects.du_fromEquivalence_132
        (coe ltInt (coe v0) (coe v1))
        (\ v2 -> coe du_'60''7495''8658''60'_2716 (coe v0))
        (coe du_'60''8658''60''7495'_2728)
  -- Data.Nat.Properties.≤ᵇ⇒≤
  d_'8804''7495''8658''8804'_2746 ::
    Integer ->
    Integer -> AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''7495''8658''8804'_2746 v0 ~v1 ~v2
    = du_'8804''7495''8658''8804'_2746 v0
  du_'8804''7495''8658''8804'_2746 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''7495''8658''8804'_2746 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe (coe du_'60''7495''8658''60'_2716 (coe v1))
  -- Data.Nat.Properties.≤⇒≤ᵇ
  d_'8804''8658''8804''7495'_2758 ::
    Integer ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22 -> AgdaAny
  d_'8804''8658''8804''7495'_2758 ~v0 ~v1 v2
    = du_'8804''8658''8804''7495'_2758 v2
  du_'8804''8658''8804''7495'_2758 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 -> AgdaAny
  du_'8804''8658''8804''7495'_2758 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe MAlonzo.Code.Agda.Builtin.Unit.C_tt_8
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3
          -> coe
               du_'60''8658''60''7495'_2728
               (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3)
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.≤ᵇ-reflects-≤
  d_'8804''7495''45'reflects'45''8804'_2766 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Reflects.T_Reflects_16
  d_'8804''7495''45'reflects'45''8804'_2766 v0 v1
    = coe
        MAlonzo.Code.Relation.Nullary.Reflects.du_fromEquivalence_132
        (coe
           MAlonzo.Code.Data.Nat.Base.d__'8804''7495'__14 (coe v0) (coe v1))
        (\ v2 -> coe du_'8804''7495''8658''8804'_2746 (coe v0))
        (coe du_'8804''8658''8804''7495'_2758)
  -- Data.Nat.Properties.≤-reflexive
  d_'8804''45'reflexive_2772 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''45'reflexive_2772 v0 ~v1 ~v2
    = du_'8804''45'reflexive_2772 v0
  du_'8804''45'reflexive_2772 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''45'reflexive_2772 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (coe du_'8804''45'reflexive_2772 (coe v1)))
  -- Data.Nat.Properties.≤-refl
  d_'8804''45'refl_2776 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''45'refl_2776 v0 = coe du_'8804''45'reflexive_2772 (coe v0)
  -- Data.Nat.Properties.≤-antisym
  d_'8804''45'antisym_2778 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8804''45'antisym_2778 = erased
  -- Data.Nat.Properties.≤-trans
  d_'8804''45'trans_2784 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''45'trans_2784 ~v0 ~v1 ~v2 v3 v4
    = du_'8804''45'trans_2784 v3 v4
  du_'8804''45'trans_2784 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''45'trans_2784 v0 v1
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> case coe v1 of
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v7
                 -> coe
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                      (coe du_'8804''45'trans_2784 (coe v4) (coe v7))
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.≤-total
  d_'8804''45'total_2790 ::
    Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_'8804''45'total_2790 v0 v1
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
               (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe
                         MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                         (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
                  _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Sum.Base.du_map_84
                            (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34)
                            (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34)
                            (d_'8804''45'total_2790 (coe v2) (coe v3))))
  -- Data.Nat.Properties.≤-irrelevant
  d_'8804''45'irrelevant_2796 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8804''45'irrelevant_2796 = erased
  -- Data.Nat.Properties._≤?_
  d__'8804''63'__2802 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8804''63'__2802 v0 v1
    = coe
        MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
        (\ v2 -> coe du_'8804''7495''8658''8804'_2746 (coe v0))
        (coe du_'8804''8658''8804''7495'_2758)
        (coe
           MAlonzo.Code.Relation.Nullary.Decidable.Core.d_T'63'_66
           (coe
              MAlonzo.Code.Data.Nat.Base.d__'8804''7495'__14 (coe v0) (coe v1)))
  -- Data.Nat.Properties._≥?_
  d__'8805''63'__2808 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8805''63'__2808 v0 v1
    = coe d__'8804''63'__2802 (coe v1) (coe v0)
  -- Data.Nat.Properties.≤-isPreorder
  d_'8804''45'isPreorder_2810 ::
    MAlonzo.Code.Relation.Binary.Structures.T_IsPreorder_70
  d_'8804''45'isPreorder_2810
    = coe
        MAlonzo.Code.Relation.Binary.Structures.C_IsPreorder'46'constructor_4003
        (coe
           MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_396)
        (\ v0 v1 v2 -> coe du_'8804''45'reflexive_2772 v0)
        (\ v0 v1 v2 v3 v4 -> coe du_'8804''45'trans_2784 v3 v4)
  -- Data.Nat.Properties.≤-isTotalPreorder
  d_'8804''45'isTotalPreorder_2812 ::
    MAlonzo.Code.Relation.Binary.Structures.T_IsTotalPreorder_124
  d_'8804''45'isTotalPreorder_2812
    = coe
        MAlonzo.Code.Relation.Binary.Structures.C_IsTotalPreorder'46'constructor_8325
        (coe d_'8804''45'isPreorder_2810) (coe d_'8804''45'total_2790)
  -- Data.Nat.Properties.≤-isPartialOrder
  d_'8804''45'isPartialOrder_2814 ::
    MAlonzo.Code.Relation.Binary.Structures.T_IsPartialOrder_174
  d_'8804''45'isPartialOrder_2814
    = coe
        MAlonzo.Code.Relation.Binary.Structures.C_IsPartialOrder'46'constructor_9853
        (coe d_'8804''45'isPreorder_2810) erased
  -- Data.Nat.Properties.≤-isTotalOrder
  d_'8804''45'isTotalOrder_2816 ::
    MAlonzo.Code.Relation.Binary.Structures.T_IsTotalOrder_404
  d_'8804''45'isTotalOrder_2816
    = coe
        MAlonzo.Code.Relation.Binary.Structures.C_IsTotalOrder'46'constructor_20555
        (coe d_'8804''45'isPartialOrder_2814) (coe d_'8804''45'total_2790)
  -- Data.Nat.Properties.≤-isDecTotalOrder
  d_'8804''45'isDecTotalOrder_2818 ::
    MAlonzo.Code.Relation.Binary.Structures.T_IsDecTotalOrder_460
  d_'8804''45'isDecTotalOrder_2818
    = coe
        MAlonzo.Code.Relation.Binary.Structures.C_IsDecTotalOrder'46'constructor_22695
        (coe d_'8804''45'isTotalOrder_2816) (coe d__'8799'__2688)
        (coe d__'8804''63'__2802)
  -- Data.Nat.Properties.≤-preorder
  d_'8804''45'preorder_2820 ::
    MAlonzo.Code.Relation.Binary.Bundles.T_Preorder_132
  d_'8804''45'preorder_2820
    = coe
        MAlonzo.Code.Relation.Binary.Bundles.C_Preorder'46'constructor_2267
        d_'8804''45'isPreorder_2810
  -- Data.Nat.Properties.≤-totalPreorder
  d_'8804''45'totalPreorder_2822 ::
    MAlonzo.Code.Relation.Binary.Bundles.T_TotalPreorder_222
  d_'8804''45'totalPreorder_2822
    = coe
        MAlonzo.Code.Relation.Binary.Bundles.C_TotalPreorder'46'constructor_4573
        d_'8804''45'isTotalPreorder_2812
  -- Data.Nat.Properties.≤-poset
  d_'8804''45'poset_2824 ::
    MAlonzo.Code.Relation.Binary.Bundles.T_Poset_314
  d_'8804''45'poset_2824
    = coe
        MAlonzo.Code.Relation.Binary.Bundles.C_Poset'46'constructor_6389
        d_'8804''45'isPartialOrder_2814
  -- Data.Nat.Properties.≤-totalOrder
  d_'8804''45'totalOrder_2826 ::
    MAlonzo.Code.Relation.Binary.Bundles.T_TotalOrder_764
  d_'8804''45'totalOrder_2826
    = coe
        MAlonzo.Code.Relation.Binary.Bundles.C_TotalOrder'46'constructor_15747
        d_'8804''45'isTotalOrder_2816
  -- Data.Nat.Properties.≤-decTotalOrder
  d_'8804''45'decTotalOrder_2828 ::
    MAlonzo.Code.Relation.Binary.Bundles.T_DecTotalOrder_866
  d_'8804''45'decTotalOrder_2828
    = coe
        MAlonzo.Code.Relation.Binary.Bundles.C_DecTotalOrder'46'constructor_17849
        d_'8804''45'isDecTotalOrder_2818
  -- Data.Nat.Properties.s≤s-injective
  d_s'8804's'45'injective_2834 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_s'8804's'45'injective_2834 = erased
  -- Data.Nat.Properties.≤-pred
  d_'8804''45'pred_2836 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''45'pred_2836 ~v0 ~v1 = du_'8804''45'pred_2836
  du_'8804''45'pred_2836 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''45'pred_2836
    = coe MAlonzo.Code.Data.Nat.Base.du_s'8804's'8315''185'_62
  -- Data.Nat.Properties.m≤n⇒m≤1+n
  d_m'8804'n'8658'm'8804'1'43'n_2838 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'8658'm'8804'1'43'n_2838 ~v0 ~v1 v2
    = du_m'8804'n'8658'm'8804'1'43'n_2838 v2
  du_m'8804'n'8658'm'8804'1'43'n_2838 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'n'8658'm'8804'1'43'n_2838 v0 = coe v0
  -- Data.Nat.Properties.n≤1+n
  d_n'8804'1'43'n_2844 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_n'8804'1'43'n_2844 v0 = coe d_'8804''45'refl_2776 (coe v0)
  -- Data.Nat.Properties.1+n≰n
  d_1'43'n'8816'n_2846 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_1'43'n'8816'n_2846 = erased
  -- Data.Nat.Properties.n≤0⇒n≡0
  d_n'8804'0'8658'n'8801'0_2850 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_n'8804'0'8658'n'8801'0_2850 = erased
  -- Data.Nat.Properties.n≤1⇒n≡0∨n≡1
  d_n'8804'1'8658'n'8801'0'8744'n'8801'1_2852 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_n'8804'1'8658'n'8801'0'8744'n'8801'1_2852 ~v0 v1
    = du_n'8804'1'8658'n'8801'0'8744'n'8801'1_2852 v1
  du_n'8804'1'8658'n'8801'0'8744'n'8801'1_2852 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  du_n'8804'1'8658'n'8801'0'8744'n'8801'1_2852 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 erased
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3
          -> coe
               seq (coe v3) (coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased)
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.<⇒≤
  d_'60''8658''8804'_2854 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''8658''8804'_2854 ~v0 ~v1 v2 = du_'60''8658''8804'_2854 v2
  du_'60''8658''8804'_2854 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'60''8658''8804'_2854 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3
          -> case coe v3 of
               MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
                 -> coe
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                      (coe
                         du_'60''8658''8804'_2854
                         (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6))
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.<⇒≢
  d_'60''8658''8802'_2858 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'60''8658''8802'_2858 = erased
  -- Data.Nat.Properties.>⇒≢
  d_'62''8658''8802'_2862 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'62''8658''8802'_2862 = erased
  -- Data.Nat.Properties.≤⇒≯
  d_'8804''8658''8815'_2864 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'8804''8658''8815'_2864 = erased
  -- Data.Nat.Properties.<⇒≱
  d_'60''8658''8817'_2870 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'60''8658''8817'_2870 = erased
  -- Data.Nat.Properties.<⇒≯
  d_'60''8658''8815'_2876 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'60''8658''8815'_2876 = erased
  -- Data.Nat.Properties.≰⇒≮
  d_'8816''8658''8814'_2882 ::
    Integer ->
    Integer ->
    (MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'8816''8658''8814'_2882 = erased
  -- Data.Nat.Properties.≰⇒>
  d_'8816''8658''62'_2888 ::
    Integer ->
    Integer ->
    (MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8816''8658''62'_2888 v0 v1 ~v2 = du_'8816''8658''62'_2888 v0 v1
  du_'8816''8658''62'_2888 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8816''8658''62'_2888 v0 v1
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
               erased
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                         (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
                  _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                            (coe du_'8816''8658''62'_2888 (coe v2) (coe v3))))
  -- Data.Nat.Properties.≰⇒≥
  d_'8816''8658''8805'_2900 ::
    Integer ->
    Integer ->
    (MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8816''8658''8805'_2900 v0 v1 ~v2
    = du_'8816''8658''8805'_2900 v0 v1
  du_'8816''8658''8805'_2900 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8816''8658''8805'_2900 v0 v1
    = coe
        du_'60''8658''8804'_2854
        (coe du_'8816''8658''62'_2888 (coe v0) (coe v1))
  -- Data.Nat.Properties.≮⇒≥
  d_'8814''8658''8805'_2902 ::
    Integer ->
    Integer ->
    (MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8814''8658''8805'_2902 v0 v1 ~v2
    = du_'8814''8658''8805'_2902 v0 v1
  du_'8814''8658''8805'_2902 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8814''8658''8805'_2902 v0 v1
    = case coe v1 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v2 = subInt (coe v1) (coe (1 :: Integer)) in
             coe
               (case coe v0 of
                  0 -> coe
                         MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
                         erased
                  _ -> let v3 = subInt (coe v0) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                            (coe du_'8814''8658''8805'_2902 (coe v3) (coe v2))))
  -- Data.Nat.Properties.≤∧≢⇒<
  d_'8804''8743''8802''8658''60'_2918 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''8743''8802''8658''60'_2918 ~v0 v1 v2 ~v3
    = du_'8804''8743''8802''8658''60'_2918 v1 v2
  du_'8804''8743''8802''8658''60'_2918 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''8743''8802''8658''60'_2918 v0 v1
    = case coe v0 of
        0 -> coe
               seq (coe v1)
               (coe
                  MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
                  erased)
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                    -> coe
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                         (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
                    -> coe
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                         (coe du_'8804''8743''8802''8658''60'_2918 (coe v2) (coe v5))
                  _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.≤∧≮⇒≡
  d_'8804''8743''8814''8658''8801'_2936 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    (MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8804''8743''8814''8658''8801'_2936 = erased
  -- Data.Nat.Properties.≤-<-connex
  d_'8804''45''60''45'connex_2942 ::
    Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_'8804''45''60''45'connex_2942 v0 v1
    = let v2
            = coe
                MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
                (\ v2 -> coe du_'8804''7495''8658''8804'_2746 (coe v0))
                (coe du_'8804''8658''8804''7495'_2758)
                (coe
                   MAlonzo.Code.Relation.Nullary.Decidable.Core.d_T'63'_66
                   (coe
                      MAlonzo.Code.Data.Nat.Base.d__'8804''7495'__14 (coe v0)
                      (coe v1))) in
      coe
        (case coe v2 of
           MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v3 v4
             -> if coe v3
                  then case coe v4 of
                         MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v5
                           -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 (coe v5)
                         _ -> MAlonzo.RTE.mazUnreachableError
                  else coe
                         seq (coe v4)
                         (coe
                            MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                            (coe du_'8816''8658''62'_2888 (coe v0) (coe v1)))
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.≥->-connex
  d_'8805''45''62''45'connex_2964 ::
    Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_'8805''45''62''45'connex_2964 v0 v1
    = coe d_'8804''45''60''45'connex_2942 (coe v1) (coe v0)
  -- Data.Nat.Properties.<-≤-connex
  d_'60''45''8804''45'connex_2966 ::
    Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_'60''45''8804''45'connex_2966
    = coe
        MAlonzo.Code.Relation.Binary.Consequences.du_flip'45'Connex_854
        (coe d_'8804''45''60''45'connex_2942)
  -- Data.Nat.Properties.>-≥-connex
  d_'62''45''8805''45'connex_2968 ::
    Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_'62''45''8805''45'connex_2968
    = coe
        MAlonzo.Code.Relation.Binary.Consequences.du_flip'45'Connex_854
        (coe d_'8805''45''62''45'connex_2964)
  -- Data.Nat.Properties.<-irrefl
  d_'60''45'irrefl_2970 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'60''45'irrefl_2970 = erased
  -- Data.Nat.Properties.<-asym
  d_'60''45'asym_2974 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'60''45'asym_2974 = erased
  -- Data.Nat.Properties.<-trans
  d_'60''45'trans_2980 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''45'trans_2980 ~v0 v1 ~v2 v3 v4
    = du_'60''45'trans_2980 v1 v3 v4
  du_'60''45'trans_2980 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'60''45'trans_2980 v0 v1 v2
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
          -> let v6 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v2 of
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9
                    -> coe
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                         (coe
                            du_'8804''45'trans_2784 (coe v5)
                            (coe
                               du_'8804''45'trans_2784 (coe d_n'8804'1'43'n_2844 (coe v6))
                               (coe v9)))
                  _ -> MAlonzo.RTE.mazUnreachableError)
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.≤-<-trans
  d_'8804''45''60''45'trans_2986 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''45''60''45'trans_2986 ~v0 ~v1 ~v2 v3 v4
    = du_'8804''45''60''45'trans_2986 v3 v4
  du_'8804''45''60''45'trans_2986 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''45''60''45'trans_2986 v0 v1
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> coe
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
               (coe du_'8804''45'trans_2784 (coe v0) (coe v4))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.<-≤-trans
  d_'60''45''8804''45'trans_2992 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''45''8804''45'trans_2992 ~v0 ~v1 ~v2 v3 v4
    = du_'60''45''8804''45'trans_2992 v3 v4
  du_'60''45''8804''45'trans_2992 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'60''45''8804''45'trans_2992 v0 v1
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> case coe v1 of
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v7
                 -> coe
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                      (coe du_'8804''45'trans_2784 (coe v4) (coe v7))
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.<-cmp
  d_'60''45'cmp_2998 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Binary.Definitions.T_Tri_158
  d_'60''45'cmp_2998 v0 v1
    = let v2
            = coe
                MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
                erased (\ v2 -> coe du_'8801''8658''8801''7495'_2678 (coe v0))
                (coe
                   MAlonzo.Code.Relation.Nullary.Decidable.Core.d_T'63'_66
                   (coe eqInt (coe v0) (coe v1))) in
      coe
        (case coe v2 of
           MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v3 v4
             -> if coe v3
                  then case coe v4 of
                         MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v5
                           -> coe MAlonzo.Code.Relation.Binary.Definitions.C_tri'8776'_180 v5
                         _ -> MAlonzo.RTE.mazUnreachableError
                  else coe
                         seq (coe v4)
                         (let v5 = ltInt (coe v0) (coe v1) in
                          coe
                            (if coe v5
                               then coe
                                      seq
                                      (coe
                                         MAlonzo.Code.Relation.Nullary.Reflects.d_T'45'reflects_66
                                         (coe v5))
                                      (coe
                                         MAlonzo.Code.Relation.Binary.Definitions.C_tri'60'_172
                                         (coe du_'60''7495''8658''60'_2716 (coe v0)))
                               else coe
                                      seq
                                      (coe
                                         MAlonzo.Code.Relation.Nullary.Reflects.d_T'45'reflects_66
                                         (coe v5))
                                      (coe
                                         MAlonzo.Code.Relation.Binary.Definitions.C_tri'62'_188
                                         (coe
                                            du_'8804''8743''8802''8658''60'_2918 (coe v0)
                                            (coe du_'8814''8658''8805'_2902 (coe v0) (coe v1))))))
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties._<?_
  d__'60''63'__3030 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'60''63'__3030 v0 v1
    = coe
        d__'8804''63'__2802 (coe addInt (coe (1 :: Integer)) (coe v0))
        (coe v1)
  -- Data.Nat.Properties._>?_
  d__'62''63'__3036 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'62''63'__3036 v0 v1 = coe d__'60''63'__3030 (coe v1) (coe v0)
  -- Data.Nat.Properties.<-irrelevant
  d_'60''45'irrelevant_3038 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'60''45'irrelevant_3038 = erased
  -- Data.Nat.Properties.<-resp₂-≡
  d_'60''45'resp'8322''45''8801'_3040 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'60''45'resp'8322''45''8801'_3040
    = coe
        MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
        (coe (\ v0 v1 v2 v3 v4 -> v4)) (coe (\ v0 v1 v2 v3 v4 -> v4))
  -- Data.Nat.Properties.<-isStrictPartialOrder
  d_'60''45'isStrictPartialOrder_3046 ::
    MAlonzo.Code.Relation.Binary.Structures.T_IsStrictPartialOrder_290
  d_'60''45'isStrictPartialOrder_3046
    = coe
        MAlonzo.Code.Relation.Binary.Structures.C_IsStrictPartialOrder'46'constructor_14045
        (coe
           MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_396)
        (\ v0 v1 v2 v3 v4 -> coe du_'60''45'trans_2980 v1 v3 v4)
        d_'60''45'resp'8322''45''8801'_3040
  -- Data.Nat.Properties.<-isStrictTotalOrder
  d_'60''45'isStrictTotalOrder_3048 ::
    MAlonzo.Code.Relation.Binary.Structures.T_IsStrictTotalOrder_534
  d_'60''45'isStrictTotalOrder_3048
    = coe
        MAlonzo.Code.Relation.Binary.Structures.Biased.du_isStrictTotalOrder'7580'_538
        (coe
           MAlonzo.Code.Relation.Binary.Structures.Biased.C_IsStrictTotalOrder'7580''46'constructor_6029
           (coe
              MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_396)
           (\ v0 v1 v2 v3 v4 -> coe du_'60''45'trans_2980 v1 v3 v4)
           (coe d_'60''45'cmp_2998))
  -- Data.Nat.Properties.<-strictPartialOrder
  d_'60''45'strictPartialOrder_3050 ::
    MAlonzo.Code.Relation.Binary.Bundles.T_StrictPartialOrder_556
  d_'60''45'strictPartialOrder_3050
    = coe
        MAlonzo.Code.Relation.Binary.Bundles.C_StrictPartialOrder'46'constructor_11097
        d_'60''45'isStrictPartialOrder_3046
  -- Data.Nat.Properties.<-strictTotalOrder
  d_'60''45'strictTotalOrder_3052 ::
    MAlonzo.Code.Relation.Binary.Bundles.T_StrictTotalOrder_1036
  d_'60''45'strictTotalOrder_3052
    = coe
        MAlonzo.Code.Relation.Binary.Bundles.C_StrictTotalOrder'46'constructor_21059
        d_'60''45'isStrictTotalOrder_3048
  -- Data.Nat.Properties.s<s-injective
  d_s'60's'45'injective_3058 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_s'60's'45'injective_3058 = erased
  -- Data.Nat.Properties.<-pred
  d_'60''45'pred_3060 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''45'pred_3060 ~v0 ~v1 = du_'60''45'pred_3060
  du_'60''45'pred_3060 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'60''45'pred_3060
    = coe MAlonzo.Code.Data.Nat.Base.du_s'60's'8315''185'_70
  -- Data.Nat.Properties.m<n⇒m<1+n
  d_m'60'n'8658'm'60'1'43'n_3062 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'm'60'1'43'n_3062 ~v0 ~v1 v2
    = du_m'60'n'8658'm'60'1'43'n_3062 v2
  du_m'60'n'8658'm'60'1'43'n_3062 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'8658'm'60'1'43'n_3062 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3
          -> case coe v3 of
               MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                 -> coe
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                      (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
                 -> coe
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                      (coe
                         du_m'60'n'8658'm'60'1'43'n_3062
                         (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6))
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.n≮0
  d_n'8814'0_3066 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_n'8814'0_3066 = erased
  -- Data.Nat.Properties.n≮n
  d_n'8814'n_3070 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_n'8814'n_3070 = erased
  -- Data.Nat.Properties.0<1+n
  d_0'60'1'43'n_3074 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_0'60'1'43'n_3074 ~v0 = du_0'60'1'43'n_3074
  du_0'60'1'43'n_3074 :: MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_0'60'1'43'n_3074
    = coe
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
        (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
  -- Data.Nat.Properties.n<1+n
  d_n'60'1'43'n_3078 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_n'60'1'43'n_3078 v0
    = coe
        d_'8804''45'refl_2776 (coe addInt (coe (1 :: Integer)) (coe v0))
  -- Data.Nat.Properties.n<1⇒n≡0
  d_n'60'1'8658'n'8801'0_3082 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_n'60'1'8658'n'8801'0_3082 = erased
  -- Data.Nat.Properties.n>0⇒n≢0
  d_n'62'0'8658'n'8802'0_3086 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_n'62'0'8658'n'8802'0_3086 = erased
  -- Data.Nat.Properties.n≢0⇒n>0
  d_n'8802'0'8658'n'62'0_3090 ::
    Integer ->
    (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_n'8802'0'8658'n'62'0_3090 v0 ~v1
    = du_n'8802'0'8658'n'62'0_3090 v0
  du_n'8802'0'8658'n'62'0_3090 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_n'8802'0'8658'n'62'0_3090 v0
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
               erased
        _ -> coe du_0'60'1'43'n_3074
  -- Data.Nat.Properties.m<n⇒0<n
  d_m'60'n'8658'0'60'n_3096 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'0'60'n_3096 ~v0 ~v1 = du_m'60'n'8658'0'60'n_3096
  du_m'60'n'8658'0'60'n_3096 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'8658'0'60'n_3096
    = coe du_'8804''45'trans_2784 (coe du_0'60'1'43'n_3074)
  -- Data.Nat.Properties.m<n⇒n≢0
  d_m'60'n'8658'n'8802'0_3098 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'60'n'8658'n'8802'0_3098 = erased
  -- Data.Nat.Properties.m<n⇒m≤1+n
  d_m'60'n'8658'm'8804'1'43'n_3102 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'm'8804'1'43'n_3102 ~v0 ~v1 v2
    = du_m'60'n'8658'm'8804'1'43'n_3102 v2
  du_m'60'n'8658'm'8804'1'43'n_3102 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'8658'm'8804'1'43'n_3102 v0
    = coe du_'60''8658''8804'_2854 (coe v0)
  -- Data.Nat.Properties.m<1+n⇒m<n∨m≡n
  d_m'60'1'43'n'8658'm'60'n'8744'm'8801'n_3108 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_m'60'1'43'n'8658'm'60'n'8744'm'8801'n_3108 v0 v1 v2
    = case coe v0 of
        0 -> case coe v1 of
               0 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased
               _ -> coe
                      MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 (coe du_0'60'1'43'n_3074)
        _ -> let v3 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v2 of
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
                    -> let v7 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Sum.Base.du_map_84
                            (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34) erased
                            (d_m'60'1'43'n'8658'm'60'n'8744'm'8801'n_3108
                               (coe v3) (coe v7) (coe v6)))
                  _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.m≤n⇒m<n∨m≡n
  d_m'8804'n'8658'm'60'n'8744'm'8801'n_3118 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_m'8804'n'8658'm'60'n'8744'm'8801'n_3118 v0 v1 v2
    = coe
        d_m'60'1'43'n'8658'm'60'n'8744'm'8801'n_3108 (coe v0) (coe v1)
        (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v2)
  -- Data.Nat.Properties.m<1+n⇒m≤n
  d_m'60'1'43'n'8658'm'8804'n_3122 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'1'43'n'8658'm'8804'n_3122 ~v0 ~v1 v2
    = du_m'60'1'43'n'8658'm'8804'n_3122 v2
  du_m'60'1'43'n'8658'm'8804'n_3122 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'1'43'n'8658'm'8804'n_3122 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3 -> coe v3
        _                                           -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.∀[m≤n⇒m≢o]⇒n<o
  d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_3132 ::
    Integer ->
    Integer ->
    (Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_3132 v0 v1 ~v2
    = du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_3132 v0 v1
  du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_3132 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_3132 v0 v1
    = case coe v1 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
               erased
        _ -> let v2 = subInt (coe v1) (coe (1 :: Integer)) in
             coe
               (case coe v0 of
                  0 -> coe du_0'60'1'43'n_3074
                  _ -> let v3 = subInt (coe v0) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                            (coe
                               du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_3132 (coe v3)
                               (coe v2))))
  -- Data.Nat.Properties._.rec
  d_rec_3150 ::
    Integer ->
    Integer ->
    (Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_rec_3150 = erased
  -- Data.Nat.Properties.∀[m<n⇒m≢o]⇒n≤o
  d_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_3160 ::
    Integer ->
    Integer ->
    (Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_3160 v0 v1 ~v2
    = du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_3160 v0 v1
  du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_3160 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_3160 v0 v1
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe
                         MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
                         erased
                  _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                            (coe
                               du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_3160 (coe v2)
                               (coe v3))))
  -- Data.Nat.Properties._.rec
  d_rec_3180 ::
    Integer ->
    Integer ->
    (Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_rec_3180 = erased
  -- Data.Nat.Properties.≤-Reasoning._._IsRelatedTo_
  d__IsRelatedTo__3188 a0 a1 = ()
  -- Data.Nat.Properties.≤-Reasoning._._∎
  d__'8718'_3190 ::
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d__'8718'_3190
    = let v0 = d_'8804''45'isPreorder_2810 in
      coe
        (coe
           MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du__'8718'_492
           (coe
              MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
              (coe v0)))
  -- Data.Nat.Properties.≤-Reasoning._.<-go
  d_'60''45'go_3192 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_'60''45'go_3192
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'60''45'go_152
        (\ v0 v1 v2 v3 v4 -> coe du_'60''45'trans_2980 v1 v3 v4)
        (coe
           MAlonzo.Code.Relation.Binary.PropositionalEquality.Core.du_resp'8322'_160)
        (\ v0 v1 v2 v3 v4 -> coe du_'60''45''8804''45'trans_2992 v3 v4)
  -- Data.Nat.Properties.≤-Reasoning._.IsEquality
  d_IsEquality_3194 a0 a1 a2 = ()
  -- Data.Nat.Properties.≤-Reasoning._.IsEquality?
  d_IsEquality'63'_3196 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d_IsEquality'63'_3196 v0 v1 v2
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_IsEquality'63'_224
        v2
  -- Data.Nat.Properties.≤-Reasoning._.IsStrict
  d_IsStrict_3198 a0 a1 a2 = ()
  -- Data.Nat.Properties.≤-Reasoning._.IsStrict?
  d_IsStrict'63'_3200 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d_IsStrict'63'_3200 v0 v1 v2
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_IsStrict'63'_188
        v2
  -- Data.Nat.Properties.≤-Reasoning._.begin_
  d_begin__3202 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_begin__3202
    = let v0 = d_'8804''45'isPreorder_2810 in
      coe
        (let v1 = \ v1 v2 v3 -> coe du_'60''8658''8804'_2854 v3 in
         coe
           (coe
              MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__46
              (coe
                 MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_start_96
                 (coe v0) (coe v1))))
  -- Data.Nat.Properties.≤-Reasoning._.begin-contradiction_
  d_begin'45'contradiction__3204 ::
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    AgdaAny -> MAlonzo.Code.Agda.Primitive.T_Level_18 -> () -> AgdaAny
  d_begin'45'contradiction__3204 v0 v1 v2 v3 v4
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin'45'contradiction__246
  -- Data.Nat.Properties.≤-Reasoning._.begin_
  d_begin__3206 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    AgdaAny -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_begin__3206 = erased
  -- Data.Nat.Properties.≤-Reasoning._.begin_
  d_begin__3208 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_begin__3208
    = let v0
            = coe
                MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_strictRelation_202 in
      coe
        (\ v1 v2 v3 v4 ->
           coe
             MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__126
             (coe v0) v1 v2 v3)
  -- Data.Nat.Properties.≤-Reasoning._.eqRelation
  d_eqRelation_3210 ::
    MAlonzo.Code.Relation.Binary.Reasoning.Syntax.T_SubRelation_60
  d_eqRelation_3210
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_eqRelation_238
  -- Data.Nat.Properties.≤-Reasoning._.extractEquality
  d_extractEquality_3214 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T_IsEquality_208 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_extractEquality_3214 = erased
  -- Data.Nat.Properties.≤-Reasoning._.extractStrict
  d_extractStrict_3216 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T_IsStrict_172 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_extractStrict_3216 v0 v1 v2 v3
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_extractStrict_198
        v2 v3
  -- Data.Nat.Properties.≤-Reasoning._.start
  d_start_3224 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_start_3224
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_start_96
        (coe d_'8804''45'isPreorder_2810)
        (\ v0 v1 v2 -> coe du_'60''8658''8804'_2854 v2)
  -- Data.Nat.Properties.≤-Reasoning._.step-<
  d_step'45''60'_3226 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_step'45''60'_3226
    = let v0
            = \ v0 v1 v2 v3 v4 -> coe du_'60''45'trans_2980 v1 v3 v4 in
      coe
        (let v1
               = coe
                   MAlonzo.Code.Relation.Binary.PropositionalEquality.Core.du_resp'8322'_160 in
         coe
           (let v2
                  = \ v2 v3 v4 v5 v6 -> coe du_'60''45''8804''45'trans_2992 v5 v6 in
            coe
              (coe
                 MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''60'_312
                 (coe
                    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'60''45'go_152
                    (coe v0) (coe v1) (coe v2)))))
  -- Data.Nat.Properties.≤-Reasoning._.step-≡
  d_step'45''8801'_3228 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_step'45''8801'_3228
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801'_450
        (coe (\ v0 v1 v2 v3 v4 -> v4))
  -- Data.Nat.Properties.≤-Reasoning._.step-≡-∣
  d_step'45''8801''45''8739'_3230 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_step'45''8801''45''8739'_3230 ~v0 ~v1 v2
    = du_step'45''8801''45''8739'_3230 v2
  du_step'45''8801''45''8739'_3230 ::
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  du_step'45''8801''45''8739'_3230 v0 = coe v0
  -- Data.Nat.Properties.≤-Reasoning._.step-≡-⟨
  d_step'45''8801''45''10216'_3232 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_step'45''8801''45''10216'_3232
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10216'_448
        (coe (\ v0 v1 v2 v3 v4 -> v4))
  -- Data.Nat.Properties.≤-Reasoning._.step-≡-⟩
  d_step'45''8801''45''10217'_3234 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_step'45''8801''45''10217'_3234
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
        (coe (\ v0 v1 v2 v3 v4 -> v4))
  -- Data.Nat.Properties.≤-Reasoning._.step-≡˘
  d_step'45''8801''728'_3236 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_step'45''8801''728'_3236
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''728'_452
        (coe (\ v0 v1 v2 v3 v4 -> v4))
  -- Data.Nat.Properties.≤-Reasoning._.step-≤
  d_step'45''8804'_3238 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_step'45''8804'_3238
    = let v0 = d_'8804''45'isPreorder_2810 in
      coe
        (let v1
               = \ v1 v2 v3 v4 v5 -> coe du_'8804''45''60''45'trans_2986 v4 v5 in
         coe
           (coe
              MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8804'_308
              (coe
                 MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8804''45'go_138
                 (coe v0) (coe v1))))
  -- Data.Nat.Properties.≤-Reasoning._.stop
  d_stop_3240 ::
    Integer ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_stop_3240
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
        (coe d_'8804''45'isPreorder_2810)
  -- Data.Nat.Properties.≤-Reasoning._.strictRelation
  d_strictRelation_3244 ::
    MAlonzo.Code.Relation.Binary.Reasoning.Syntax.T_SubRelation_60
  d_strictRelation_3244
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_strictRelation_202
  -- Data.Nat.Properties.≤-Reasoning._.≈-go
  d_'8776''45'go_3246 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_'8776''45'go_3246
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8776''45'go_124
        (coe d_'8804''45'isPreorder_2810)
        (coe
           MAlonzo.Code.Relation.Binary.PropositionalEquality.Core.du_resp'8322'_160)
  -- Data.Nat.Properties.≤-Reasoning._.≡-go
  d_'8801''45'go_3248 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_'8801''45'go_3248 ~v0 ~v1 ~v2 ~v3 v4 = du_'8801''45'go_3248 v4
  du_'8801''45'go_3248 ::
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  du_'8801''45'go_3248 v0 = coe v0
  -- Data.Nat.Properties.≤-Reasoning._.≤-go
  d_'8804''45'go_3250 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78 ->
    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.T__IsRelatedTo__78
  d_'8804''45'go_3250
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8804''45'go_138
        (coe d_'8804''45'isPreorder_2810)
        (\ v0 v1 v2 v3 v4 -> coe du_'8804''45''60''45'trans_2986 v3 v4)
  -- Data.Nat.Properties.+-suc
  d_'43''45'suc_3272 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'suc_3272 = erased
  -- Data.Nat.Properties.+-assoc
  d_'43''45'assoc_3280 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'assoc_3280 = erased
  -- Data.Nat.Properties.+-identityˡ
  d_'43''45'identity'737'_3288 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'identity'737'_3288 = erased
  -- Data.Nat.Properties.+-identityʳ
  d_'43''45'identity'691'_3290 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'identity'691'_3290 = erased
  -- Data.Nat.Properties.+-identity
  d_'43''45'identity_3294 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'43''45'identity_3294
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.+-comm
  d_'43''45'comm_3296 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'comm_3296 = erased
  -- Data.Nat.Properties.+-cancelˡ-≡
  d_'43''45'cancel'737''45''8801'_3304 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'cancel'737''45''8801'_3304 = erased
  -- Data.Nat.Properties.+-cancelʳ-≡
  d_'43''45'cancel'691''45''8801'_3312 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'cancel'691''45''8801'_3312 = erased
  -- Data.Nat.Properties.+-cancel-≡
  d_'43''45'cancel'45''8801'_3314 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'43''45'cancel'45''8801'_3314
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.+-isMagma
  d_'43''45'isMagma_3316 ::
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176
  d_'43''45'isMagma_3316
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsMagma'46'constructor_1867
        (coe
           MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_396)
        erased
  -- Data.Nat.Properties.+-isSemigroup
  d_'43''45'isSemigroup_3318 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_'43''45'isSemigroup_3318
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsSemigroup'46'constructor_10417
        (coe d_'43''45'isMagma_3316) erased
  -- Data.Nat.Properties.+-isCommutativeSemigroup
  d_'43''45'isCommutativeSemigroup_3320 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
  d_'43''45'isCommutativeSemigroup_3320
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemigroup'46'constructor_12093
        (coe d_'43''45'isSemigroup_3318) erased
  -- Data.Nat.Properties.+-0-isMonoid
  d_'43''45'0'45'isMonoid_3322 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_'43''45'0'45'isMonoid_3322
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_15873
        (coe d_'43''45'isSemigroup_3318) (coe d_'43''45'identity_3294)
  -- Data.Nat.Properties.+-0-isCommutativeMonoid
  d_'43''45'0'45'isCommutativeMonoid_3324 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'43''45'0'45'isCommutativeMonoid_3324
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_17695
        (coe d_'43''45'0'45'isMonoid_3322) erased
  -- Data.Nat.Properties.+-magma
  d_'43''45'magma_3326 :: MAlonzo.Code.Algebra.Bundles.T_Magma_68
  d_'43''45'magma_3326
    = coe
        MAlonzo.Code.Algebra.Bundles.C_Magma'46'constructor_1279 addInt
        d_'43''45'isMagma_3316
  -- Data.Nat.Properties.+-semigroup
  d_'43''45'semigroup_3328 ::
    MAlonzo.Code.Algebra.Bundles.T_Semigroup_536
  d_'43''45'semigroup_3328
    = coe
        MAlonzo.Code.Algebra.Bundles.C_Semigroup'46'constructor_9793 addInt
        d_'43''45'isSemigroup_3318
  -- Data.Nat.Properties.+-commutativeSemigroup
  d_'43''45'commutativeSemigroup_3330 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeSemigroup_662
  d_'43''45'commutativeSemigroup_3330
    = coe
        MAlonzo.Code.Algebra.Bundles.C_CommutativeSemigroup'46'constructor_12035
        addInt d_'43''45'isCommutativeSemigroup_3320
  -- Data.Nat.Properties.+-0-monoid
  d_'43''45'0'45'monoid_3332 ::
    MAlonzo.Code.Algebra.Bundles.T_Monoid_882
  d_'43''45'0'45'monoid_3332
    = coe
        MAlonzo.Code.Algebra.Bundles.C_Monoid'46'constructor_16157 addInt
        (0 :: Integer) d_'43''45'0'45'isMonoid_3322
  -- Data.Nat.Properties.+-0-commutativeMonoid
  d_'43''45'0'45'commutativeMonoid_3334 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeMonoid_962
  d_'43''45'0'45'commutativeMonoid_3334
    = coe
        MAlonzo.Code.Algebra.Bundles.C_CommutativeMonoid'46'constructor_17931
        addInt (0 :: Integer) d_'43''45'0'45'isCommutativeMonoid_3324
  -- Data.Nat.Properties.∸-magma
  d_'8760''45'magma_3336 :: MAlonzo.Code.Algebra.Bundles.T_Magma_68
  d_'8760''45'magma_3336
    = coe
        MAlonzo.Code.Relation.Binary.PropositionalEquality.Algebra.du_magma_20
        (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22)
  -- Data.Nat.Properties.m≢1+m+n
  d_m'8802'1'43'm'43'n_3342 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'8802'1'43'm'43'n_3342 = erased
  -- Data.Nat.Properties.m≢1+n+m
  d_m'8802'1'43'n'43'm_3352 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'8802'1'43'n'43'm_3352 = erased
  -- Data.Nat.Properties.m+1+n≢m
  d_m'43'1'43'n'8802'm_3362 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'43'1'43'n'8802'm_3362 = erased
  -- Data.Nat.Properties.m+1+n≢n
  d_m'43'1'43'n'8802'n_3370 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'43'1'43'n'8802'n_3370 = erased
  -- Data.Nat.Properties.m+1+n≢0
  d_m'43'1'43'n'8802'0_3384 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'43'1'43'n'8802'0_3384 = erased
  -- Data.Nat.Properties.m+n≡0⇒m≡0
  d_m'43'n'8801'0'8658'm'8801'0_3398 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'43'n'8801'0'8658'm'8801'0_3398 = erased
  -- Data.Nat.Properties.m+n≡0⇒n≡0
  d_m'43'n'8801'0'8658'n'8801'0_3406 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'43'n'8801'0'8658'n'8801'0_3406 = erased
  -- Data.Nat.Properties.+-cancelˡ-≤
  d_'43''45'cancel'737''45''8804'_3414 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'cancel'737''45''8804'_3414 v0 ~v1 ~v2 v3
    = du_'43''45'cancel'737''45''8804'_3414 v0 v3
  du_'43''45'cancel'737''45''8804'_3414 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'cancel'737''45''8804'_3414 v0 v1
    = case coe v0 of
        0 -> coe v1
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
                    -> coe du_'43''45'cancel'737''45''8804'_3414 (coe v2) (coe v5)
                  _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.+-cancelʳ-≤
  d_'43''45'cancel'691''45''8804'_3422 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'cancel'691''45''8804'_3422 v0 ~v1 ~v2 v3
    = du_'43''45'cancel'691''45''8804'_3422 v0 v3
  du_'43''45'cancel'691''45''8804'_3422 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'cancel'691''45''8804'_3422 v0 v1
    = coe du_'43''45'cancel'737''45''8804'_3414 (coe v0) (coe v1)
  -- Data.Nat.Properties.+-cancel-≤
  d_'43''45'cancel'45''8804'_3432 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'43''45'cancel'45''8804'_3432
    = coe
        MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
        (\ v0 v1 v2 v3 -> coe du_'43''45'cancel'737''45''8804'_3414 v0 v3)
        (\ v0 v1 v2 v3 -> coe du_'43''45'cancel'691''45''8804'_3422 v0 v3)
  -- Data.Nat.Properties.+-cancelˡ-<
  d_'43''45'cancel'737''45''60'_3434 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'cancel'737''45''60'_3434 v0 ~v1 ~v2 v3
    = du_'43''45'cancel'737''45''60'_3434 v0 v3
  du_'43''45'cancel'737''45''60'_3434 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'cancel'737''45''60'_3434 v0 v1
    = coe du_'43''45'cancel'737''45''8804'_3414 (coe v0) (coe v1)
  -- Data.Nat.Properties.+-cancelʳ-<
  d_'43''45'cancel'691''45''60'_3444 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'cancel'691''45''60'_3444 v0 ~v1 ~v2 v3
    = du_'43''45'cancel'691''45''60'_3444 v0 v3
  du_'43''45'cancel'691''45''60'_3444 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'cancel'691''45''60'_3444 v0 v1
    = coe du_'43''45'cancel'691''45''8804'_3422 (coe v0) (coe v1)
  -- Data.Nat.Properties.+-cancel-<
  d_'43''45'cancel'45''60'_3454 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'43''45'cancel'45''60'_3454
    = coe
        MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
        (\ v0 v1 v2 v3 -> coe du_'43''45'cancel'737''45''60'_3434 v0 v3)
        (\ v0 v1 v2 v3 -> coe du_'43''45'cancel'691''45''60'_3444 v0 v3)
  -- Data.Nat.Properties.m≤n⇒m≤o+n
  d_m'8804'n'8658'm'8804'o'43'n_3458 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'8658'm'8804'o'43'n_3458 ~v0 ~v1 ~v2 v3
    = du_m'8804'n'8658'm'8804'o'43'n_3458 v3
  du_m'8804'n'8658'm'8804'o'43'n_3458 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'n'8658'm'8804'o'43'n_3458 v0 = coe v0
  -- Data.Nat.Properties.m≤n⇒m≤n+o
  d_m'8804'n'8658'm'8804'n'43'o_3468 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'8658'm'8804'n'43'o_3468 ~v0 ~v1 ~v2 v3
    = du_m'8804'n'8658'm'8804'n'43'o_3468 v3
  du_m'8804'n'8658'm'8804'n'43'o_3468 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'n'8658'm'8804'n'43'o_3468 v0 = coe v0
  -- Data.Nat.Properties.m≤m+n
  d_m'8804'm'43'n_3482 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'm'43'n_3482 v0 ~v1 = du_m'8804'm'43'n_3482 v0
  du_m'8804'm'43'n_3482 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'm'43'n_3482 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (coe du_m'8804'm'43'n_3482 (coe v1)))
  -- Data.Nat.Properties.m≤n+m
  d_m'8804'n'43'm_3494 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'43'm_3494 v0 ~v1 = du_m'8804'n'43'm_3494 v0
  du_m'8804'n'43'm_3494 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'n'43'm_3494 v0 = coe du_m'8804'm'43'n_3482 (coe v0)
  -- Data.Nat.Properties.m+n≤o⇒m≤o
  d_m'43'n'8804'o'8658'm'8804'o_3508 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'43'n'8804'o'8658'm'8804'o_3508 v0 ~v1 ~v2 v3
    = du_m'43'n'8804'o'8658'm'8804'o_3508 v0 v3
  du_m'43'n'8804'o'8658'm'8804'o_3508 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'43'n'8804'o'8658'm'8804'o_3508 v0 v1
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
                    -> coe
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                         (coe du_m'43'n'8804'o'8658'm'8804'o_3508 (coe v2) (coe v5))
                  _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.m+n≤o⇒n≤o
  d_m'43'n'8804'o'8658'n'8804'o_3522 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'43'n'8804'o'8658'n'8804'o_3522 v0 ~v1 ~v2 v3
    = du_m'43'n'8804'o'8658'n'8804'o_3522 v0 v3
  du_m'43'n'8804'o'8658'n'8804'o_3522 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'43'n'8804'o'8658'n'8804'o_3522 v0 v1
    = case coe v0 of
        0 -> coe v1
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  du_m'43'n'8804'o'8658'n'8804'o_3522 (coe v2)
                  (coe du_'60''8658''8804'_2854 (coe v1)))
  -- Data.Nat.Properties.+-mono-≤
  d_'43''45'mono'45''8804'_3530 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'mono'45''8804'_3530 ~v0 ~v1 ~v2 v3 v4 v5
    = du_'43''45'mono'45''8804'_3530 v3 v4 v5
  du_'43''45'mono'45''8804'_3530 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'mono'45''8804'_3530 v0 v1 v2
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe
               du_'8804''45'trans_2784 (coe v2)
               (coe du_m'8804'n'43'm_3494 (coe v0))
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
          -> coe
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
               (coe du_'43''45'mono'45''8804'_3530 (coe v0) (coe v5) (coe v2))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.+-monoˡ-≤
  d_'43''45'mono'737''45''8804'_3544 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'mono'737''45''8804'_3544 v0 ~v1 ~v2 v3
    = du_'43''45'mono'737''45''8804'_3544 v0 v3
  du_'43''45'mono'737''45''8804'_3544 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'mono'737''45''8804'_3544 v0 v1
    = coe
        du_'43''45'mono'45''8804'_3530 (coe v0) (coe v1)
        (coe d_'8804''45'refl_2776 (coe v0))
  -- Data.Nat.Properties.+-monoʳ-≤
  d_'43''45'mono'691''45''8804'_3554 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'mono'691''45''8804'_3554 v0 ~v1 v2 v3
    = du_'43''45'mono'691''45''8804'_3554 v0 v2 v3
  du_'43''45'mono'691''45''8804'_3554 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'mono'691''45''8804'_3554 v0 v1 v2
    = coe
        du_'43''45'mono'45''8804'_3530 (coe v1)
        (coe d_'8804''45'refl_2776 (coe v0)) (coe v2)
  -- Data.Nat.Properties.+-mono-<-≤
  d_'43''45'mono'45''60''45''8804'_3560 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'mono'45''60''45''8804'_3560 ~v0 ~v1 ~v2 ~v3 v4 v5
    = du_'43''45'mono'45''60''45''8804'_3560 v4 v5
  du_'43''45'mono'45''60''45''8804'_3560 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'mono'45''60''45''8804'_3560 v0 v1
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> case coe v4 of
               MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                 -> coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v1
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v7
                 -> coe
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                      (coe
                         du_'43''45'mono'45''60''45''8804'_3560
                         (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v7) (coe v1))
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.+-mono-≤-<
  d_'43''45'mono'45''8804''45''60'_3570 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'mono'45''8804''45''60'_3570 ~v0 ~v1 ~v2 v3 v4 v5
    = du_'43''45'mono'45''8804''45''60'_3570 v3 v4 v5
  du_'43''45'mono'45''8804''45''60'_3570 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'mono'45''8804''45''60'_3570 v0 v1 v2
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe
               du_'8804''45'trans_2784 (coe v2)
               (coe du_m'8804'n'43'm_3494 (coe v0))
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
          -> coe
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
               (coe
                  du_'43''45'mono'45''8804''45''60'_3570 (coe v0) (coe v5) (coe v2))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.+-mono-<
  d_'43''45'mono'45''60'_3580 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'mono'45''60'_3580 ~v0 ~v1 ~v2 v3 v4
    = du_'43''45'mono'45''60'_3580 v3 v4
  du_'43''45'mono'45''60'_3580 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'mono'45''60'_3580 v0 v1
    = coe
        du_'43''45'mono'45''8804''45''60'_3570 (coe v0)
        (coe du_'60''8658''8804'_2854 (coe v1))
  -- Data.Nat.Properties.+-monoˡ-<
  d_'43''45'mono'737''45''60'_3588 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'mono'737''45''60'_3588 v0 ~v1 ~v2
    = du_'43''45'mono'737''45''60'_3588 v0
  du_'43''45'mono'737''45''60'_3588 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'mono'737''45''60'_3588 v0
    = coe du_'43''45'mono'737''45''8804'_3544 (coe v0)
  -- Data.Nat.Properties.+-monoʳ-<
  d_'43''45'mono'691''45''60'_3596 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'43''45'mono'691''45''60'_3596 v0 ~v1 ~v2 v3
    = du_'43''45'mono'691''45''60'_3596 v0 v3
  du_'43''45'mono'691''45''60'_3596 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'43''45'mono'691''45''60'_3596 v0 v1
    = case coe v0 of
        0 -> coe v1
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (coe du_'43''45'mono'691''45''60'_3596 (coe v2) (coe v1)))
  -- Data.Nat.Properties.m+1+n≰m
  d_m'43'1'43'n'8816'm_3608 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'43'1'43'n'8816'm_3608 = erased
  -- Data.Nat.Properties.m<m+n
  d_m'60'm'43'n_3618 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'm'43'n_3618 v0 ~v1 v2 = du_m'60'm'43'n_3618 v0 v2
  du_m'60'm'43'n_3618 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'm'43'n_3618 v0 v1
    = case coe v0 of
        0 -> coe v1
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (coe du_m'60'm'43'n_3618 (coe v2) (coe v1)))
  -- Data.Nat.Properties.m<n+m
  d_m'60'n'43'm_3630 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'43'm_3630 v0 ~v1 v2 = du_m'60'n'43'm_3630 v0 v2
  du_m'60'n'43'm_3630 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'43'm_3630 v0 v1
    = coe du_m'60'm'43'n_3618 (coe v0) (coe v1)
  -- Data.Nat.Properties.m+n≮n
  d_m'43'n'8814'n_3646 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'43'n'8814'n_3646 = erased
  -- Data.Nat.Properties.m+n≮m
  d_m'43'n'8814'm_3660 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'43'n'8814'm_3660 = erased
  -- Data.Nat.Properties.*-suc
  d_'42''45'suc_3672 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'suc_3672 = erased
  -- Data.Nat.Properties.*-identityˡ
  d_'42''45'identity'737'_3684 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'identity'737'_3684 = erased
  -- Data.Nat.Properties.*-identityʳ
  d_'42''45'identity'691'_3688 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'identity'691'_3688 = erased
  -- Data.Nat.Properties.*-identity
  d_'42''45'identity_3692 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'identity_3692
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.*-zeroˡ
  d_'42''45'zero'737'_3694 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'zero'737'_3694 = erased
  -- Data.Nat.Properties.*-zeroʳ
  d_'42''45'zero'691'_3696 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'zero'691'_3696 = erased
  -- Data.Nat.Properties.*-zero
  d_'42''45'zero_3700 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'zero_3700
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.*-comm
  d_'42''45'comm_3702 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'comm_3702 = erased
  -- Data.Nat.Properties.*-distribʳ-+
  d_'42''45'distrib'691''45''43'_3712 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'691''45''43'_3712 = erased
  -- Data.Nat.Properties.*-distribˡ-+
  d_'42''45'distrib'737''45''43'_3726 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'737''45''43'_3726 = erased
  -- Data.Nat.Properties.*-distrib-+
  d_'42''45'distrib'45''43'_3728 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'distrib'45''43'_3728
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.*-assoc
  d_'42''45'assoc_3730 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'assoc_3730 = erased
  -- Data.Nat.Properties.*-isMagma
  d_'42''45'isMagma_3744 ::
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176
  d_'42''45'isMagma_3744
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsMagma'46'constructor_1867
        (coe
           MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_396)
        erased
  -- Data.Nat.Properties.*-isSemigroup
  d_'42''45'isSemigroup_3746 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_'42''45'isSemigroup_3746
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsSemigroup'46'constructor_10417
        (coe d_'42''45'isMagma_3744) erased
  -- Data.Nat.Properties.*-isCommutativeSemigroup
  d_'42''45'isCommutativeSemigroup_3748 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
  d_'42''45'isCommutativeSemigroup_3748
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemigroup'46'constructor_12093
        (coe d_'42''45'isSemigroup_3746) erased
  -- Data.Nat.Properties.*-1-isMonoid
  d_'42''45'1'45'isMonoid_3750 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_'42''45'1'45'isMonoid_3750
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_15873
        (coe d_'42''45'isSemigroup_3746) (coe d_'42''45'identity_3692)
  -- Data.Nat.Properties.*-1-isCommutativeMonoid
  d_'42''45'1'45'isCommutativeMonoid_3752 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'42''45'1'45'isCommutativeMonoid_3752
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_17695
        (coe d_'42''45'1'45'isMonoid_3750) erased
  -- Data.Nat.Properties.+-*-isSemiring
  d_'43''45''42''45'isSemiring_3754 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiring_1570
  d_'43''45''42''45'isSemiring_3754
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsSemiring'46'constructor_48071
        (coe
           MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutAnnihilatingZero'46'constructor_43811
           (coe d_'43''45'0'45'isCommutativeMonoid_3324) erased erased
           (coe d_'42''45'identity_3692) (coe d_'42''45'distrib'45''43'_3728))
        (coe d_'42''45'zero_3700)
  -- Data.Nat.Properties.+-*-isCommutativeSemiring
  d_'43''45''42''45'isCommutativeSemiring_3756 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
  d_'43''45''42''45'isCommutativeSemiring_3756
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemiring'46'constructor_51895
        (coe d_'43''45''42''45'isSemiring_3754) erased
  -- Data.Nat.Properties.*-magma
  d_'42''45'magma_3758 :: MAlonzo.Code.Algebra.Bundles.T_Magma_68
  d_'42''45'magma_3758
    = coe
        MAlonzo.Code.Algebra.Bundles.C_Magma'46'constructor_1279 mulInt
        d_'42''45'isMagma_3744
  -- Data.Nat.Properties.*-semigroup
  d_'42''45'semigroup_3760 ::
    MAlonzo.Code.Algebra.Bundles.T_Semigroup_536
  d_'42''45'semigroup_3760
    = coe
        MAlonzo.Code.Algebra.Bundles.C_Semigroup'46'constructor_9793 mulInt
        d_'42''45'isSemigroup_3746
  -- Data.Nat.Properties.*-commutativeSemigroup
  d_'42''45'commutativeSemigroup_3762 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeSemigroup_662
  d_'42''45'commutativeSemigroup_3762
    = coe
        MAlonzo.Code.Algebra.Bundles.C_CommutativeSemigroup'46'constructor_12035
        mulInt d_'42''45'isCommutativeSemigroup_3748
  -- Data.Nat.Properties.*-1-monoid
  d_'42''45'1'45'monoid_3764 ::
    MAlonzo.Code.Algebra.Bundles.T_Monoid_882
  d_'42''45'1'45'monoid_3764
    = coe
        MAlonzo.Code.Algebra.Bundles.C_Monoid'46'constructor_16157 mulInt
        (1 :: Integer) d_'42''45'1'45'isMonoid_3750
  -- Data.Nat.Properties.*-1-commutativeMonoid
  d_'42''45'1'45'commutativeMonoid_3766 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeMonoid_962
  d_'42''45'1'45'commutativeMonoid_3766
    = coe
        MAlonzo.Code.Algebra.Bundles.C_CommutativeMonoid'46'constructor_17931
        mulInt (1 :: Integer) d_'42''45'1'45'isCommutativeMonoid_3752
  -- Data.Nat.Properties.+-*-semiring
  d_'43''45''42''45'semiring_3768 ::
    MAlonzo.Code.Algebra.Bundles.T_Semiring_2280
  d_'43''45''42''45'semiring_3768
    = coe
        MAlonzo.Code.Algebra.Bundles.C_Semiring'46'constructor_41765 addInt
        mulInt (0 :: Integer) (1 :: Integer)
        d_'43''45''42''45'isSemiring_3754
  -- Data.Nat.Properties.+-*-commutativeSemiring
  d_'43''45''42''45'commutativeSemiring_3770 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeSemiring_2446
  d_'43''45''42''45'commutativeSemiring_3770
    = coe
        MAlonzo.Code.Algebra.Bundles.C_CommutativeSemiring'46'constructor_44731
        addInt mulInt (0 :: Integer) (1 :: Integer)
        d_'43''45''42''45'isCommutativeSemiring_3756
  -- Data.Nat.Properties.*-cancelʳ-≡
  d_'42''45'cancel'691''45''8801'_3780 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'cancel'691''45''8801'_3780 = erased
  -- Data.Nat.Properties.*-cancelˡ-≡
  d_'42''45'cancel'737''45''8801'_3802 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'cancel'737''45''8801'_3802 = erased
  -- Data.Nat.Properties.m*n≡0⇒m≡0∨n≡0
  d_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_3822 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_3822 v0 ~v1 ~v2
    = du_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_3822 v0
  du_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_3822 ::
    Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  du_m'42'n'8801'0'8658'm'8801'0'8744'n'8801'0_3822 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 erased
        _ -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased
  -- Data.Nat.Properties.m*n≢0
  d_m'42'n'8802'0_3840 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  d_m'42'n'8802'0_3840 ~v0 ~v1 ~v2 ~v3 = du_m'42'n'8802'0_3840
  du_m'42'n'8802'0_3840 :: MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  du_m'42'n'8802'0_3840
    = coe
        MAlonzo.Code.Data.Nat.Base.C_NonZero'46'constructor_3575
        (coe MAlonzo.Code.Agda.Builtin.Unit.C_tt_8)
  -- Data.Nat.Properties.m*n≢0⇒m≢0
  d_m'42'n'8802'0'8658'm'8802'0_3850 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  d_m'42'n'8802'0'8658'm'8802'0_3850 ~v0 ~v1 ~v2
    = du_m'42'n'8802'0'8658'm'8802'0_3850
  du_m'42'n'8802'0'8658'm'8802'0_3850 ::
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  du_m'42'n'8802'0'8658'm'8802'0_3850
    = coe
        MAlonzo.Code.Data.Nat.Base.C_NonZero'46'constructor_3575
        (coe MAlonzo.Code.Agda.Builtin.Unit.C_tt_8)
  -- Data.Nat.Properties.m*n≢0⇒n≢0
  d_m'42'n'8802'0'8658'n'8802'0_3856 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  d_m'42'n'8802'0'8658'n'8802'0_3856 ~v0 ~v1 ~v2
    = du_m'42'n'8802'0'8658'n'8802'0_3856
  du_m'42'n'8802'0'8658'n'8802'0_3856 ::
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  du_m'42'n'8802'0'8658'n'8802'0_3856
    = coe du_m'42'n'8802'0'8658'm'8802'0_3850
  -- Data.Nat.Properties.m*n≡0⇒m≡0
  d_m'42'n'8801'0'8658'm'8801'0_3872 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'42'n'8801'0'8658'm'8801'0_3872 = erased
  -- Data.Nat.Properties.m*n≡1⇒m≡1
  d_m'42'n'8801'1'8658'm'8801'1_3880 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'42'n'8801'1'8658'm'8801'1_3880 = erased
  -- Data.Nat.Properties.m*n≡1⇒n≡1
  d_m'42'n'8801'1'8658'n'8801'1_3894 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'42'n'8801'1'8658'n'8801'1_3894 = erased
  -- Data.Nat.Properties.[m*n]*[o*p]≡[m*o]*[n*p]
  d_'91'm'42'n'93''42''91'o'42'p'93''8801''91'm'42'o'93''42''91'n'42'p'93'_3910 ::
    Integer ->
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'91'm'42'n'93''42''91'o'42'p'93''8801''91'm'42'o'93''42''91'n'42'p'93'_3910
    = erased
  -- Data.Nat.Properties.m≢0∧n>1⇒m*n>1
  d_m'8802'0'8743'n'62'1'8658'm'42'n'62'1_3998 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T_NonTrivial_152 ->
    MAlonzo.Code.Data.Nat.Base.T_NonTrivial_152
  d_m'8802'0'8743'n'62'1'8658'm'42'n'62'1_3998 ~v0 ~v1 ~v2 ~v3
    = du_m'8802'0'8743'n'62'1'8658'm'42'n'62'1_3998
  du_m'8802'0'8743'n'62'1'8658'm'42'n'62'1_3998 ::
    MAlonzo.Code.Data.Nat.Base.T_NonTrivial_152
  du_m'8802'0'8743'n'62'1'8658'm'42'n'62'1_3998
    = coe
        MAlonzo.Code.Data.Nat.Base.C_NonTrivial'46'constructor_5661
        (coe MAlonzo.Code.Agda.Builtin.Unit.C_tt_8)
  -- Data.Nat.Properties.n≢0∧m>1⇒m*n>1
  d_n'8802'0'8743'm'62'1'8658'm'42'n'62'1_4012 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T_NonTrivial_152 ->
    MAlonzo.Code.Data.Nat.Base.T_NonTrivial_152
  d_n'8802'0'8743'm'62'1'8658'm'42'n'62'1_4012 ~v0 ~v1 ~v2 ~v3
    = du_n'8802'0'8743'm'62'1'8658'm'42'n'62'1_4012
  du_n'8802'0'8743'm'62'1'8658'm'42'n'62'1_4012 ::
    MAlonzo.Code.Data.Nat.Base.T_NonTrivial_152
  du_n'8802'0'8743'm'62'1'8658'm'42'n'62'1_4012
    = coe du_m'8802'0'8743'n'62'1'8658'm'42'n'62'1_3998
  -- Data.Nat.Properties.*-cancelʳ-≤
  d_'42''45'cancel'691''45''8804'_4030 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'cancel'691''45''8804'_4030 v0 ~v1 ~v2 ~v3 ~v4
    = du_'42''45'cancel'691''45''8804'_4030 v0
  du_'42''45'cancel'691''45''8804'_4030 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'cancel'691''45''8804'_4030 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (coe du_'42''45'cancel'691''45''8804'_4030 (coe v1)))
  -- Data.Nat.Properties.*-cancelˡ-≤
  d_'42''45'cancel'737''45''8804'_4044 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'cancel'737''45''8804'_4044 v0 ~v1 ~v2 ~v3
    = du_'42''45'cancel'737''45''8804'_4044 v0
  du_'42''45'cancel'737''45''8804'_4044 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'cancel'737''45''8804'_4044 v0 v1
    = coe du_'42''45'cancel'691''45''8804'_4030 (coe v0)
  -- Data.Nat.Properties.*-mono-≤
  d_'42''45'mono'45''8804'_4060 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'mono'45''8804'_4060 ~v0 v1 ~v2 v3 v4 v5
    = du_'42''45'mono'45''8804'_4060 v1 v3 v4 v5
  du_'42''45'mono'45''8804'_4060 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'mono'45''8804'_4060 v0 v1 v2 v3
    = case coe v2 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
          -> let v7 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  du_'43''45'mono'45''8804'_3530 (coe mulInt (coe v7) (coe v1))
                  (coe v3)
                  (coe
                     du_'42''45'mono'45''8804'_4060 (coe v7) (coe v1) (coe v6)
                     (coe v3)))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.*-monoˡ-≤
  d_'42''45'mono'737''45''8804'_4070 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'mono'737''45''8804'_4070 v0 ~v1 v2 v3
    = du_'42''45'mono'737''45''8804'_4070 v0 v2 v3
  du_'42''45'mono'737''45''8804'_4070 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'mono'737''45''8804'_4070 v0 v1 v2
    = coe
        du_'42''45'mono'45''8804'_4060 (coe v1) (coe v0) (coe v2)
        (coe d_'8804''45'refl_2776 (coe v0))
  -- Data.Nat.Properties.*-monoʳ-≤
  d_'42''45'mono'691''45''8804'_4080 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'mono'691''45''8804'_4080 v0 ~v1 v2 v3
    = du_'42''45'mono'691''45''8804'_4080 v0 v2 v3
  du_'42''45'mono'691''45''8804'_4080 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'mono'691''45''8804'_4080 v0 v1 v2
    = coe
        du_'42''45'mono'45''8804'_4060 (coe v0) (coe v1)
        (coe d_'8804''45'refl_2776 (coe v0)) (coe v2)
  -- Data.Nat.Properties.*-mono-<
  d_'42''45'mono'45''60'_4086 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'mono'45''60'_4086 ~v0 v1 ~v2 v3 v4 v5
    = du_'42''45'mono'45''60'_4086 v1 v3 v4 v5
  du_'42''45'mono'45''60'_4086 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'mono'45''60'_4086 v0 v1 v2 v3
    = case coe v2 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
          -> case coe v6 of
               MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                 -> coe seq (coe v3) (coe du_0'60'1'43'n_3074)
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9
                 -> case coe v3 of
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v12
                        -> coe
                             du_'43''45'mono'45''60'_3580
                             (mulInt (coe subInt (coe v0) (coe (1 :: Integer))) (coe v1))
                             (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v12)
                             (coe
                                du_'42''45'mono'45''60'_4086
                                (coe subInt (coe v0) (coe (1 :: Integer))) (coe v1)
                                (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9)
                                (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v12))
                      _ -> MAlonzo.RTE.mazUnreachableError
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.*-monoˡ-<
  d_'42''45'mono'737''45''60'_4100 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'mono'737''45''60'_4100 v0 ~v1 v2 v3 v4
    = du_'42''45'mono'737''45''60'_4100 v0 v2 v3 v4
  du_'42''45'mono'737''45''60'_4100 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'mono'737''45''60'_4100 v0 v1 v2 v3
    = case coe v3 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
          -> case coe v6 of
               MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26 -> coe du_0'60'1'43'n_3074
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9
                 -> let v10 = subInt (coe v1) (coe (1 :: Integer)) in
                    coe
                      (coe
                         du_'43''45'mono'45''8804''45''60'_3570
                         (coe
                            MAlonzo.Code.Function.Base.du_'8739'_'10217''45'__298
                            (\ v11 v12 -> v12) (\ v11 -> mulInt (coe v11) (coe v0)) v10
                            (subInt (coe v2) (coe (1 :: Integer))))
                         (coe d_'8804''45'refl_2776 (coe v0))
                         (coe
                            du_'42''45'mono'737''45''60'_4100 (coe v0) (coe v10)
                            (coe subInt (coe v2) (coe (1 :: Integer)))
                            (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9)))
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.*-monoʳ-<
  d_'42''45'mono'691''45''60'_4114 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'mono'691''45''60'_4114 v0 ~v1 ~v2 v3 v4
    = du_'42''45'mono'691''45''60'_4114 v0 v3 v4
  du_'42''45'mono'691''45''60'_4114 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'mono'691''45''60'_4114 v0 v1 v2
    = case coe v0 of
        1 -> case coe v2 of
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
                 -> coe
                      du_'43''45'mono'45''8804'_3530 (coe (0 :: Integer))
                      (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5)
                      (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> case coe v2 of
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
                 -> coe
                      du_'43''45'mono'45''8804'_3530
                      (coe mulInt (coe subInt (coe v0) (coe (1 :: Integer))) (coe v1))
                      (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5)
                      (coe
                         du_'60''8658''8804'_2854
                         (coe
                            du_'42''45'mono'691''45''60'_4114
                            (coe subInt (coe v0) (coe (1 :: Integer))) (coe v1)
                            (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5)))
               _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.m≤m*n
  d_m'8804'm'42'n_4128 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'm'42'n_4128 v0 v1 ~v2 = du_m'8804'm'42'n_4128 v0 v1
  du_m'8804'm'42'n_4128 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'm'42'n_4128 v0 v1
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__46
        (coe
           MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_start_96
           (coe d_'8804''45'isPreorder_2810)
           (\ v2 v3 v4 -> coe du_'60''8658''8804'_2854 v4))
        v0 (mulInt (coe v0) (coe v1))
        (coe
           MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
           (\ v2 v3 v4 v5 v6 -> v6) v0 (mulInt (coe v0) (coe (1 :: Integer)))
           (mulInt (coe v0) (coe v1))
           (coe
              MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8804'_308
              (coe
                 MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8804''45'go_138
                 (coe d_'8804''45'isPreorder_2810)
                 (\ v2 v3 v4 v5 v6 -> coe du_'8804''45''60''45'trans_2986 v5 v6))
              (mulInt (coe v0) (coe (1 :: Integer))) (mulInt (coe v0) (coe v1))
              (mulInt (coe v0) (coe v1))
              (coe
                 MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du__'8718'_492
                 (coe
                    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
                    (coe d_'8804''45'isPreorder_2810))
                 (coe mulInt (coe v0) (coe v1)))
              (coe
                 du_'42''45'mono'691''45''8804'_4080 (coe v0) (coe v1)
                 (coe du_0'60'1'43'n_3074)))
           erased)
  -- Data.Nat.Properties.m≤n*m
  d_m'8804'n'42'm_4140 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'42'm_4140 v0 v1 ~v2 = du_m'8804'n'42'm_4140 v0 v1
  du_m'8804'n'42'm_4140 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'n'42'm_4140 v0 v1
    = coe
        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__46
        (coe
           MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_start_96
           (coe d_'8804''45'isPreorder_2810)
           (\ v2 v3 v4 -> coe du_'60''8658''8804'_2854 v4))
        v0 (mulInt (coe v1) (coe v0))
        (coe
           MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8804'_308
           (coe
              MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8804''45'go_138
              (coe d_'8804''45'isPreorder_2810)
              (\ v2 v3 v4 v5 v6 -> coe du_'8804''45''60''45'trans_2986 v5 v6))
           v0 (mulInt (coe v0) (coe v1)) (mulInt (coe v1) (coe v0))
           (coe
              MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
              (\ v2 v3 v4 v5 v6 -> v6) (mulInt (coe v0) (coe v1))
              (mulInt (coe v1) (coe v0)) (mulInt (coe v1) (coe v0))
              (coe
                 MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du__'8718'_492
                 (coe
                    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
                    (coe d_'8804''45'isPreorder_2810))
                 (coe mulInt (coe v1) (coe v0)))
              erased)
           (coe du_m'8804'm'42'n_4128 (coe v0) (coe v1)))
  -- Data.Nat.Properties.m<m*n
  d_m'60'm'42'n_4152 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'm'42'n_4152 v0 v1 ~v2 v3 = du_m'60'm'42'n_4152 v0 v1 v3
  du_m'60'm'42'n_4152 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'm'42'n_4152 v0 v1 v2
    = let v3 = subInt (coe v0) (coe (1 :: Integer)) in
      coe
        (case coe v2 of
           MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
             -> coe
                  seq (coe v6)
                  (coe
                     MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__126
                     (coe
                        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_strictRelation_202)
                     (coe v0) (coe mulInt (coe v0) (coe v1))
                     (coe
                        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''60'_312
                        (coe
                           MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'60''45'go_152
                           (\ v7 v8 v9 v10 v11 -> coe du_'60''45'trans_2980 v8 v10 v11)
                           (coe
                              MAlonzo.Code.Relation.Binary.PropositionalEquality.Core.du_resp'8322'_160)
                           (\ v7 v8 v9 v10 v11 ->
                              coe du_'60''45''8804''45'trans_2992 v10 v11))
                        v0 (addInt (coe v1) (coe v3)) (mulInt (coe v0) (coe v1))
                        (coe
                           MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8804'_308
                           (coe
                              MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8804''45'go_138
                              (coe d_'8804''45'isPreorder_2810)
                              (\ v7 v8 v9 v10 v11 ->
                                 coe du_'8804''45''60''45'trans_2986 v10 v11))
                           (addInt (coe v1) (coe v3))
                           (addInt (coe v1) (coe mulInt (coe v3) (coe v1)))
                           (mulInt (coe v0) (coe v1))
                           (coe
                              MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du__'8718'_492
                              (coe
                                 MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
                                 (coe d_'8804''45'isPreorder_2810))
                              (coe mulInt (coe v0) (coe v1)))
                           (coe
                              du_'43''45'mono'691''45''8804'_3554 (coe v1)
                              (coe mulInt (coe v3) (coe v1))
                              (coe du_m'8804'm'42'n_4128 (coe v3) (coe v1))))
                        (coe
                           MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                           (coe
                              MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                              (coe du_m'8804'n'43'm_3494 (coe v3))))))
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.m<n⇒m<n*o
  d_m'60'n'8658'm'60'n'42'o_4166 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'm'60'n'42'o_4166 ~v0 v1 v2 ~v3 v4
    = du_m'60'n'8658'm'60'n'42'o_4166 v1 v2 v4
  du_m'60'n'8658'm'60'n'42'o_4166 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'8658'm'60'n'42'o_4166 v0 v1 v2
    = coe
        du_'60''45''8804''45'trans_2992 (coe v2)
        (coe du_m'8804'm'42'n_4128 (coe v0) (coe v1))
  -- Data.Nat.Properties.m<n⇒m<o*n
  d_m'60'n'8658'm'60'o'42'n_4182 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'm'60'o'42'n_4182 v0 v1 v2 ~v3 v4
    = du_m'60'n'8658'm'60'o'42'n_4182 v0 v1 v2 v4
  du_m'60'n'8658'm'60'o'42'n_4182 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'8658'm'60'o'42'n_4182 v0 v1 v2 v3
    = let v4
            = coe
                MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_strictRelation_202 in
      coe
        (coe
           MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__126
           (coe v4) (coe v0) (coe mulInt (coe v2) (coe v1))
           (coe
              MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''60'_312
              (coe
                 MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'60''45'go_152
                 (\ v5 v6 v7 v8 v9 -> coe du_'60''45'trans_2980 v6 v8 v9)
                 (coe
                    MAlonzo.Code.Relation.Binary.PropositionalEquality.Core.du_resp'8322'_160)
                 (\ v5 v6 v7 v8 v9 -> coe du_'60''45''8804''45'trans_2992 v8 v9))
              v0 (mulInt (coe v1) (coe v2)) (mulInt (coe v2) (coe v1))
              (coe
                 MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
                 (\ v5 v6 v7 v8 v9 -> v9) (mulInt (coe v1) (coe v2))
                 (mulInt (coe v2) (coe v1)) (mulInt (coe v2) (coe v1))
                 (coe
                    MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du__'8718'_492
                    (coe
                       MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
                       (coe d_'8804''45'isPreorder_2810))
                    (coe mulInt (coe v2) (coe v1)))
                 erased)
              (coe du_m'60'n'8658'm'60'n'42'o_4166 (coe v1) (coe v2) (coe v3))))
  -- Data.Nat.Properties.*-cancelʳ-<
  d_'42''45'cancel'691''45''60'_4192 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'cancel'691''45''60'_4192 v0 v1 v2 ~v3
    = du_'42''45'cancel'691''45''60'_4192 v0 v1 v2
  du_'42''45'cancel'691''45''60'_4192 ::
    Integer ->
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'42''45'cancel'691''45''60'_4192 v0 v1 v2
    = let v3
            = let v3 = subInt (coe v1) (coe (1 :: Integer)) in
              coe
                (let v4 = subInt (coe v2) (coe (1 :: Integer)) in
                 coe
                   (coe
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                      (coe
                         du_'42''45'cancel'691''45''60'_4192 (coe v0) (coe v3)
                         (coe v4)))) in
      coe
        (coe
           seq (coe v0)
           (case coe v1 of
              0 -> case coe v2 of
                     _ | coe geqInt (coe v2) (coe (1 :: Integer)) ->
                         coe du_0'60'1'43'n_3074
                     _ -> coe v3
              _ -> let v4 = subInt (coe v1) (coe (1 :: Integer)) in
                   coe
                     (case coe v2 of
                        _ | coe geqInt (coe v2) (coe (1 :: Integer)) ->
                            let v5 = subInt (coe v2) (coe (1 :: Integer)) in
                            coe
                              (coe
                                 MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                                 (coe
                                    du_'42''45'cancel'691''45''60'_4192 (coe v0) (coe v4) (coe v5)))
                        _ -> coe v3)))
  -- Data.Nat.Properties.*-cancelˡ-<
  d_'42''45'cancel'737''45''60'_4208 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'42''45'cancel'737''45''60'_4208 v0 v1 v2 v3
    = coe
        du_'42''45'cancel'691''45''60'_4192 (coe v0) (coe v1) (coe v2)
  -- Data.Nat.Properties.*-cancel-<
  d_'42''45'cancel'45''60'_4224 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'cancel'45''60'_4224
    = coe
        MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
        (coe d_'42''45'cancel'737''45''60'_4208)
        (\ v0 v1 v2 v3 -> coe du_'42''45'cancel'691''45''60'_4192 v0 v1 v2)
  -- Data.Nat.Properties.^-identityʳ
  d_'94''45'identity'691'_4226 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'94''45'identity'691'_4226 = erased
  -- Data.Nat.Properties.^-zeroˡ
  d_'94''45'zero'737'_4230 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'94''45'zero'737'_4230 = erased
  -- Data.Nat.Properties.^-distribˡ-+-*
  d_'94''45'distrib'737''45''43''45''42'_4240 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'94''45'distrib'737''45''43''45''42'_4240 = erased
  -- Data.Nat.Properties.^-semigroup-morphism
  d_'94''45'semigroup'45'morphism_4258 ::
    Integer -> MAlonzo.Code.Algebra.Morphism.T_IsSemigroupMorphism_148
  d_'94''45'semigroup'45'morphism_4258 ~v0
    = du_'94''45'semigroup'45'morphism_4258
  du_'94''45'semigroup'45'morphism_4258 ::
    MAlonzo.Code.Algebra.Morphism.T_IsSemigroupMorphism_148
  du_'94''45'semigroup'45'morphism_4258
    = coe
        MAlonzo.Code.Algebra.Morphism.C_IsSemigroupMorphism'46'constructor_1081
        erased erased
  -- Data.Nat.Properties.^-monoid-morphism
  d_'94''45'monoid'45'morphism_4266 ::
    Integer -> MAlonzo.Code.Algebra.Morphism.T_IsMonoidMorphism_306
  d_'94''45'monoid'45'morphism_4266 ~v0
    = du_'94''45'monoid'45'morphism_4266
  du_'94''45'monoid'45'morphism_4266 ::
    MAlonzo.Code.Algebra.Morphism.T_IsMonoidMorphism_306
  du_'94''45'monoid'45'morphism_4266
    = coe
        MAlonzo.Code.Algebra.Morphism.C_IsMonoidMorphism'46'constructor_2139
        (coe du_'94''45'semigroup'45'morphism_4258) erased
  -- Data.Nat.Properties.^-*-assoc
  d_'94''45''42''45'assoc_4274 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'94''45''42''45'assoc_4274 = erased
  -- Data.Nat.Properties.m^n≡0⇒m≡0
  d_m'94'n'8801'0'8658'm'8801'0_4296 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'94'n'8801'0'8658'm'8801'0_4296 = erased
  -- Data.Nat.Properties.m^n≡1⇒n≡0∨m≡1
  d_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_4308 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_4308 ~v0 v1 ~v2
    = du_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_4308 v1
  du_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_4308 ::
    Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  du_m'94'n'8801'1'8658'n'8801'0'8744'm'8801'1_4308 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 erased
        _ -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased
  -- Data.Nat.Properties.m^n≢0
  d_m'94'n'8802'0_4324 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  d_m'94'n'8802'0_4324 v0 v1 ~v2 = du_m'94'n'8802'0_4324 v0 v1
  du_m'94'n'8802'0_4324 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  du_m'94'n'8802'0_4324 v0 v1
    = coe
        MAlonzo.Code.Data.Nat.Base.du_'8802''45'nonZero_126
        (coe MAlonzo.Code.Data.Nat.Base.d__'94'__272 (coe v0) (coe v1))
  -- Data.Nat.Properties.m^n>0
  d_m'94'n'62'0_4336 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'94'n'62'0_4336 ~v0 ~v1 ~v2 = du_m'94'n'62'0_4336
  du_m'94'n'62'0_4336 :: MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'94'n'62'0_4336
    = coe MAlonzo.Code.Data.Nat.Base.du_'62''45'nonZero'8315''185'_146
  -- Data.Nat.Properties.^-monoˡ-≤
  d_'94''45'mono'737''45''8804'_4346 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'94''45'mono'737''45''8804'_4346 v0 ~v1 v2 v3
    = du_'94''45'mono'737''45''8804'_4346 v0 v2 v3
  du_'94''45'mono'737''45''8804'_4346 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'94''45'mono'737''45''8804'_4346 v0 v1 v2
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
               (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
        _ -> let v3 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  du_'42''45'mono'45''8804'_4060 (coe v1)
                  (coe MAlonzo.Code.Data.Nat.Base.d__'94'__272 (coe v1) (coe v3))
                  (coe v2)
                  (coe
                     du_'94''45'mono'737''45''8804'_4346 (coe v3) (coe v1) (coe v2)))
  -- Data.Nat.Properties.^-monoʳ-≤
  d_'94''45'mono'691''45''8804'_4360 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'94''45'mono'691''45''8804'_4360 v0 ~v1 v2 v3 v4
    = du_'94''45'mono'691''45''8804'_4360 v0 v2 v3 v4
  du_'94''45'mono'691''45''8804'_4360 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'94''45'mono'691''45''8804'_4360 v0 v1 v2 v3
    = case coe v3 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe
               du_n'8802'0'8658'n'62'0_3090
               (coe MAlonzo.Code.Data.Nat.Base.d__'94'__272 (coe v0) (coe v2))
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
          -> let v7 = subInt (coe v1) (coe (1 :: Integer)) in
             coe
               (let v8 = subInt (coe v2) (coe (1 :: Integer)) in
                coe
                  (coe
                     du_'42''45'mono'691''45''8804'_4080 (coe v0)
                     (coe
                        MAlonzo.Code.Function.Base.du_'8739'_'10217''45'__298
                        (\ v9 v10 -> v10)
                        (MAlonzo.Code.Data.Nat.Base.d__'94'__272 (coe v0)) v7 v8)
                     (coe
                        du_'94''45'mono'691''45''8804'_4360 (coe v0) (coe v7) (coe v8)
                        (coe v6))))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.^-monoˡ-<
  d_'94''45'mono'737''45''60'_4376 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'94''45'mono'737''45''60'_4376 v0 ~v1 v2 v3 v4
    = du_'94''45'mono'737''45''60'_4376 v0 v2 v3 v4
  du_'94''45'mono'737''45''60'_4376 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'94''45'mono'737''45''60'_4376 v0 v1 v2 v3
    = case coe v0 of
        1 -> coe
               du_'42''45'mono'737''45''60'_4100 (coe (1 :: Integer)) (coe v1)
               (coe v2) (coe v3)
        _ -> coe
               du_'42''45'mono'45''60'_4086 (coe v2)
               (coe
                  MAlonzo.Code.Function.Base.du_'8739'_'10217''45'__298
                  (\ v4 v5 -> v5)
                  (\ v4 ->
                     MAlonzo.Code.Data.Nat.Base.d__'94'__272
                       (coe v4) (coe subInt (coe v0) (coe (1 :: Integer))))
                  v1 v2)
               (coe v3)
               (coe
                  du_'94''45'mono'737''45''60'_4376
                  (coe subInt (coe v0) (coe (1 :: Integer))) (coe v1) (coe v2)
                  (coe v3))
  -- Data.Nat.Properties.^-monoʳ-<
  d_'94''45'mono'691''45''60'_4388 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'94''45'mono'691''45''60'_4388 v0 v1 v2 v3 v4
    = case coe v2 of
        0 -> let v5 = subInt (coe v3) (coe (1 :: Integer)) in
             coe
               (case coe v4 of
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v8
                    -> coe
                         seq (coe v8)
                         (coe
                            du_'42''45'mono'45''8804'_4060 (coe v0)
                            (coe MAlonzo.Code.Data.Nat.Base.d__'94'__272 (coe v0) (coe v5))
                            (coe v1) (coe du_m'94'n'62'0_4336))
                  _ -> MAlonzo.RTE.mazUnreachableError)
        _ -> let v5 = subInt (coe v2) (coe (1 :: Integer)) in
             coe
               (let v6 = subInt (coe v3) (coe (1 :: Integer)) in
                coe
                  (case coe v4 of
                     MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9
                       -> coe
                            du_'42''45'mono'691''45''60'_4114 (coe v0)
                            (coe
                               MAlonzo.Code.Function.Base.du_'8739'_'10217''45'__298
                               (\ v10 v11 -> v11)
                               (MAlonzo.Code.Data.Nat.Base.d__'94'__272 (coe v0)) v5 v6)
                            (coe
                               d_'94''45'mono'691''45''60'_4388 (coe v0) (coe v1) (coe v5)
                               (coe v6) (coe v9))
                     _ -> MAlonzo.RTE.mazUnreachableError))
  -- Data.Nat.Properties.m≤n⇒m⊔n≡n
  d_m'8804'n'8658'm'8852'n'8801'n_4406 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8804'n'8658'm'8852'n'8801'n_4406 = erased
  -- Data.Nat.Properties.m≥n⇒m⊔n≡m
  d_m'8805'n'8658'm'8852'n'8801'm_4412 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8805'n'8658'm'8852'n'8801'm_4412 = erased
  -- Data.Nat.Properties.m≤n⇒m⊓n≡m
  d_m'8804'n'8658'm'8851'n'8801'm_4422 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8804'n'8658'm'8851'n'8801'm_4422 = erased
  -- Data.Nat.Properties.m≥n⇒m⊓n≡n
  d_m'8805'n'8658'm'8851'n'8801'n_4428 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8805'n'8658'm'8851'n'8801'n_4428 = erased
  -- Data.Nat.Properties.⊓-operator
  d_'8851''45'operator_4438 ::
    MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.T_MinOperator_98
  d_'8851''45'operator_4438
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.C_MinOperator'46'constructor_1121
        (coe MAlonzo.Code.Data.Nat.Base.d__'8851'__232) erased erased
  -- Data.Nat.Properties.⊔-operator
  d_'8852''45'operator_4440 ::
    MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.T_MaxOperator_128
  d_'8852''45'operator_4440
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.C_MaxOperator'46'constructor_1665
        (coe MAlonzo.Code.Data.Nat.Base.d__'8852'__204) erased erased
  -- Data.Nat.Properties.⊔≡⊔′
  d_'8852''8801''8852''8242'_4446 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8852''8801''8852''8242'_4446 = erased
  -- Data.Nat.Properties.⊓≡⊓′
  d_'8851''8801''8851''8242'_4472 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''8801''8851''8242'_4472 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤x
  d_x'8851'y'8804'x_4504 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8804'x_4504
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'x_2808
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.x≤y⇒x⊓z≤y
  d_x'8804'y'8658'x'8851'z'8804'y_4506 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8804'y'8658'x'8851'z'8804'y_4506
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'x'8851'z'8804'y_3160
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.x≤y⇒z⊓x≤y
  d_x'8804'y'8658'z'8851'x'8804'y_4508 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8804'y'8658'z'8851'x'8804'y_4508
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'z'8851'x'8804'y_3172
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.x≤y⇒x⊓z≤y
  d_x'8804'y'8658'x'8851'z'8804'y_4510 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8804'y'8658'x'8851'z'8804'y_4510
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'x'8851'z'8804'y_3160
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.x≤y⇒z⊓x≤y
  d_x'8804'y'8658'z'8851'x'8804'y_4512 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8804'y'8658'z'8851'x'8804'y_4512
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'z'8851'x'8804'y_3172
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.x≤y⊓z⇒x≤y
  d_x'8804'y'8851'z'8658'x'8804'y_4514 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8804'y'8851'z'8658'x'8804'y_4514
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'y_3184
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.x≤y⊓z⇒x≤z
  d_x'8804'y'8851'z'8658'x'8804'z_4516 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8804'y'8851'z'8658'x'8804'z_4516
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'z_3198
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤y
  d_x'8851'y'8804'y_4518 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8804'y_4518
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_2834
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≈x⇒x≤y
  d_x'8851'y'8776'x'8658'x'8804'y_4520 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8776'x'8658'x'8804'y_4520
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'x'8658'x'8804'y_3068
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≈y⇒y≤x
  d_x'8851'y'8776'y'8658'y'8804'x_4522 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8776'y'8658'y'8804'x_4522
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'y'8658'y'8804'x_3100
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤x
  d_x'8851'y'8804'x_4524 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8804'x_4524
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'x_2808
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤x⊔y
  d_x'8851'y'8804'x'8852'y_4526 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8804'x'8852'y_4526
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_x'8851'y'8804'x'8852'y_3318
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≤y
  d_x'8851'y'8804'y_4528 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8804'y_4528
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_2834
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≈x⇒x≤y
  d_x'8851'y'8776'x'8658'x'8804'y_4530 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8776'x'8658'x'8804'y_4530
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'x'8658'x'8804'y_3068
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.x⊓y≈y⇒y≤x
  d_x'8851'y'8776'y'8658'y'8804'x_4532 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8851'y'8776'y'8658'y'8804'x_4532
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'y'8658'y'8804'x_3100
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.x≤y⊓z⇒x≤y
  d_x'8804'y'8851'z'8658'x'8804'y_4534 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8804'y'8851'z'8658'x'8804'y_4534
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'y_3184
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.x≤y⊓z⇒x≤z
  d_x'8804'y'8851'z'8658'x'8804'z_4536 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_x'8804'y'8851'z'8658'x'8804'z_4536
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'z_3198
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-absorbs-⊔
  d_'8851''45'absorbs'45''8852'_4538 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'absorbs'45''8852'_4538 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-assoc
  d_'8851''45'assoc_4540 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'assoc_4540 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-band
  d_'8851''45'band_4542 :: MAlonzo.Code.Algebra.Bundles.T_Band_596
  d_'8851''45'band_4542
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'band_3052
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-comm
  d_'8851''45'comm_4544 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'comm_4544 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-commutativeSemigroup
  d_'8851''45'commutativeSemigroup_4546 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeSemigroup_662
  d_'8851''45'commutativeSemigroup_4546
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'commutativeSemigroup_3054
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-distrib-⊔
  d_'8851''45'distrib'45''8852'_4554 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'8851''45'distrib'45''8852'_4554
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45'distrib'45''8852'_3138
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-distribʳ-⊔
  d_'8851''45'distrib'691''45''8852'_4556 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'distrib'691''45''8852'_4556 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-distribˡ-⊔
  d_'8851''45'distrib'737''45''8852'_4558 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'distrib'737''45''8852'_4558 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-glb
  d_'8851''45'glb_4560 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'glb_4560
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_3278
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-idem
  d_'8851''45'idem_4562 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'idem_4562 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isBand
  d_'8851''45'isBand_4570 ::
    MAlonzo.Code.Algebra.Structures.T_IsBand_508
  d_'8851''45'isBand_4570
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isBand_3034
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isCommutativeSemigroup
  d_'8851''45'isCommutativeSemigroup_4572 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
  d_'8851''45'isCommutativeSemigroup_4572
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isCommutativeSemigroup_3036
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isMagma
  d_'8851''45'isMagma_4574 ::
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176
  d_'8851''45'isMagma_4574
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isMagma_3030
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isSelectiveMagma
  d_'8851''45'isSelectiveMagma_4578 ::
    MAlonzo.Code.Algebra.Structures.T_IsSelectiveMagma_436
  d_'8851''45'isSelectiveMagma_4578
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSelectiveMagma_3038
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isSemigroup
  d_'8851''45'isSemigroup_4580 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_'8851''45'isSemigroup_4580
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemigroup_3032
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-magma
  d_'8851''45'magma_4582 :: MAlonzo.Code.Algebra.Bundles.T_Magma_68
  d_'8851''45'magma_4582
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'magma_3048
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-mono-≤
  d_'8851''45'mono'45''8804'_4584 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'mono'45''8804'_4584
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'45''8804'_3206
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-monoʳ-≤
  d_'8851''45'mono'691''45''8804'_4588 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'mono'691''45''8804'_4588
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'691''45''8804'_3266
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-monoˡ-≤
  d_'8851''45'mono'737''45''8804'_4590 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'mono'737''45''8804'_4590
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'737''45''8804'_3256
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-sel
  d_'8851''45'sel_4594 ::
    Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_'8851''45'sel_4594
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'sel_2988
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-selectiveMagma
  d_'8851''45'selectiveMagma_4596 ::
    MAlonzo.Code.Algebra.Bundles.T_SelectiveMagma_122
  d_'8851''45'selectiveMagma_4596
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'selectiveMagma_3056
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-semigroup
  d_'8851''45'semigroup_4598 ::
    MAlonzo.Code.Algebra.Bundles.T_Semigroup_536
  d_'8851''45'semigroup_4598
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'semigroup_3050
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-triangulate
  d_'8851''45'triangulate_4600 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'triangulate_4600 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-⊔-absorptive
  d_'8851''45''8852''45'absorptive_4608 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'8851''45''8852''45'absorptive_4608
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'absorptive_3218
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-properties.⊔-absorbs-⊓
  d_'8852''45'absorbs'45''8851'_4610 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8852''45'absorbs'45''8851'_4610 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-assoc
  d_'8851''45'assoc_4612 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'assoc_4612 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-band
  d_'8851''45'band_4614 :: MAlonzo.Code.Algebra.Bundles.T_Band_596
  d_'8851''45'band_4614
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'band_3052
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-comm
  d_'8851''45'comm_4616 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'comm_4616 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-commutativeSemigroup
  d_'8851''45'commutativeSemigroup_4618 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeSemigroup_662
  d_'8851''45'commutativeSemigroup_4618
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'commutativeSemigroup_3054
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊔-distrib-⊓
  d_'8852''45'distrib'45''8851'_4626 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'8852''45'distrib'45''8851'_4626
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8852''45'distrib'45''8851'_3170
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-properties.⊔-distribʳ-⊓
  d_'8852''45'distrib'691''45''8851'_4628 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8852''45'distrib'691''45''8851'_4628 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊔-distribˡ-⊓
  d_'8852''45'distrib'737''45''8851'_4630 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8852''45'distrib'737''45''8851'_4630 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-idem
  d_'8851''45'idem_4632 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'idem_4632 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isBand
  d_'8851''45'isBand_4640 ::
    MAlonzo.Code.Algebra.Structures.T_IsBand_508
  d_'8851''45'isBand_4640
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isBand_3034
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isCommutativeSemigroup
  d_'8851''45'isCommutativeSemigroup_4642 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemigroup_548
  d_'8851''45'isCommutativeSemigroup_4642
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isCommutativeSemigroup_3036
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isMagma
  d_'8851''45'isMagma_4644 ::
    MAlonzo.Code.Algebra.Structures.T_IsMagma_176
  d_'8851''45'isMagma_4644
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isMagma_3030
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isSelectiveMagma
  d_'8851''45'isSelectiveMagma_4648 ::
    MAlonzo.Code.Algebra.Structures.T_IsSelectiveMagma_436
  d_'8851''45'isSelectiveMagma_4648
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSelectiveMagma_3038
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-isSemigroup
  d_'8851''45'isSemigroup_4650 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemigroup_472
  d_'8851''45'isSemigroup_4650
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemigroup_3032
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-glb
  d_'8851''45'glb_4652 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'glb_4652
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_3278
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-magma
  d_'8851''45'magma_4654 :: MAlonzo.Code.Algebra.Bundles.T_Magma_68
  d_'8851''45'magma_4654
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'magma_3048
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-mono-≤
  d_'8851''45'mono'45''8804'_4656 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'mono'45''8804'_4656
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'45''8804'_3206
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-monoʳ-≤
  d_'8851''45'mono'691''45''8804'_4660 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'mono'691''45''8804'_4660
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'691''45''8804'_3266
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-monoˡ-≤
  d_'8851''45'mono'737''45''8804'_4662 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'mono'737''45''8804'_4662
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'737''45''8804'_3256
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-sel
  d_'8851''45'sel_4664 ::
    Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_'8851''45'sel_4664
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'sel_2988
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-selectiveMagma
  d_'8851''45'selectiveMagma_4666 ::
    MAlonzo.Code.Algebra.Bundles.T_SelectiveMagma_122
  d_'8851''45'selectiveMagma_4666
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'selectiveMagma_3056
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-semigroup
  d_'8851''45'semigroup_4668 ::
    MAlonzo.Code.Algebra.Bundles.T_Semigroup_536
  d_'8851''45'semigroup_4668
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'semigroup_3050
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-properties.⊓-triangulate
  d_'8851''45'triangulate_4670 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'triangulate_4670 = erased
  -- Data.Nat.Properties.⊓-⊔-properties.⊔-⊓-absorptive
  d_'8852''45''8851''45'absorptive_4678 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'8852''45''8851''45'absorptive_4678
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'absorptive_3216
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊓-isSemilattice
  d_'8851''45'isSemilattice_4682 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_590
  d_'8851''45'isSemilattice_4682
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinOp.du_'8851''45'isSemilattice_602
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊓-semilattice
  d_'8851''45'semilattice_4684 ::
    MAlonzo.Code.Algebra.Lattice.Bundles.T_Semilattice_10
  d_'8851''45'semilattice_4684
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinOp.du_'8851''45'semilattice_604
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊓-⊔-distributiveLattice
  d_'8851''45''8852''45'distributiveLattice_4686 ::
    MAlonzo.Code.Algebra.Lattice.Bundles.T_DistributiveLattice_584
  d_'8851''45''8852''45'distributiveLattice_4686
    = coe
        MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'distributiveLattice_808
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊓-⊔-isDistributiveLattice
  d_'8851''45''8852''45'isDistributiveLattice_4688 ::
    MAlonzo.Code.Algebra.Lattice.Structures.T_IsDistributiveLattice_3036
  d_'8851''45''8852''45'isDistributiveLattice_4688
    = coe
        MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'isDistributiveLattice_798
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊓-⊔-isLattice
  d_'8851''45''8852''45'isLattice_4690 ::
    MAlonzo.Code.Algebra.Lattice.Structures.T_IsLattice_2962
  d_'8851''45''8852''45'isLattice_4690
    = coe
        MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'isLattice_796
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊓-⊔-lattice
  d_'8851''45''8852''45'lattice_4692 ::
    MAlonzo.Code.Algebra.Lattice.Bundles.T_Lattice_500
  d_'8851''45''8852''45'lattice_4692
    = coe
        MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.du_'8851''45''8852''45'lattice_804
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊓-isSemilattice
  d_'8851''45'isSemilattice_4694 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeBand_590
  d_'8851''45'isSemilattice_4694
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinOp.du_'8851''45'isSemilattice_602
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊓-semilattice
  d_'8851''45'semilattice_4696 ::
    MAlonzo.Code.Algebra.Lattice.Bundles.T_Semilattice_10
  d_'8851''45'semilattice_4696
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinOp.du_'8851''45'semilattice_604
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊔-⊓-distributiveLattice
  d_'8852''45''8851''45'distributiveLattice_4698 ::
    MAlonzo.Code.Algebra.Lattice.Bundles.T_DistributiveLattice_584
  d_'8852''45''8851''45'distributiveLattice_4698
    = coe
        MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'distributiveLattice_806
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊔-⊓-isDistributiveLattice
  d_'8852''45''8851''45'isDistributiveLattice_4700 ::
    MAlonzo.Code.Algebra.Lattice.Structures.T_IsDistributiveLattice_3036
  d_'8852''45''8851''45'isDistributiveLattice_4700
    = coe
        MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'isDistributiveLattice_800
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊔-⊓-isLattice
  d_'8852''45''8851''45'isLattice_4702 ::
    MAlonzo.Code.Algebra.Lattice.Structures.T_IsLattice_2962
  d_'8852''45''8851''45'isLattice_4702
    = coe
        MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'isLattice_794
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊓-⊔-latticeProperties.⊔-⊓-lattice
  d_'8852''45''8851''45'lattice_4704 ::
    MAlonzo.Code.Algebra.Lattice.Bundles.T_Lattice_500
  d_'8852''45''8851''45'lattice_4704
    = coe
        MAlonzo.Code.Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.du_'8852''45''8851''45'lattice_802
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440)
  -- Data.Nat.Properties.⊔-identityˡ
  d_'8852''45'identity'737'_4706 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8852''45'identity'737'_4706 = erased
  -- Data.Nat.Properties.⊔-identityʳ
  d_'8852''45'identity'691'_4708 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8852''45'identity'691'_4708 = erased
  -- Data.Nat.Properties.⊔-identity
  d_'8852''45'identity_4712 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'8852''45'identity_4712
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.⊔-0-isMonoid
  d_'8852''45'0'45'isMonoid_4714 ::
    MAlonzo.Code.Algebra.Structures.T_IsMonoid_686
  d_'8852''45'0'45'isMonoid_4714
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsMonoid'46'constructor_15873
        (let v0 = d_'8804''45'totalPreorder_2822 in
         coe
           (let v1 = d_'8852''45'operator_4440 in
            coe
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'isSemigroup_3032
                 (coe
                    MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                    (coe v0))
                 (coe
                    MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                    (coe v1)))))
        (coe d_'8852''45'identity_4712)
  -- Data.Nat.Properties.⊔-0-isCommutativeMonoid
  d_'8852''45'0'45'isCommutativeMonoid_4716 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeMonoid_736
  d_'8852''45'0'45'isCommutativeMonoid_4716
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeMonoid'46'constructor_17695
        (coe d_'8852''45'0'45'isMonoid_4714)
        (let v0 = d_'8804''45'totalPreorder_2822 in
         coe
           (let v1 = d_'8852''45'operator_4440 in
            coe
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'comm_2856
                 (coe
                    MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                    (coe v0))
                 (coe
                    MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                    (coe v1)))))
  -- Data.Nat.Properties.⊔-0-monoid
  d_'8852''45'0'45'monoid_4718 ::
    MAlonzo.Code.Algebra.Bundles.T_Monoid_882
  d_'8852''45'0'45'monoid_4718
    = coe
        MAlonzo.Code.Algebra.Bundles.C_Monoid'46'constructor_16157
        MAlonzo.Code.Data.Nat.Base.d__'8852'__204 (0 :: Integer)
        d_'8852''45'0'45'isMonoid_4714
  -- Data.Nat.Properties.⊔-0-commutativeMonoid
  d_'8852''45'0'45'commutativeMonoid_4720 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeMonoid_962
  d_'8852''45'0'45'commutativeMonoid_4720
    = coe
        MAlonzo.Code.Algebra.Bundles.C_CommutativeMonoid'46'constructor_17931
        MAlonzo.Code.Data.Nat.Base.d__'8852'__204 (0 :: Integer)
        d_'8852''45'0'45'isCommutativeMonoid_4716
  -- Data.Nat.Properties.mono-≤-distrib-⊔
  d_mono'45''8804''45'distrib'45''8852'_4728 ::
    (Integer -> Integer) ->
    (Integer ->
     Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22) ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_mono'45''8804''45'distrib'45''8852'_4728 = erased
  -- Data.Nat.Properties.mono-≤-distrib-⊓
  d_mono'45''8804''45'distrib'45''8851'_4738 ::
    (Integer -> Integer) ->
    (Integer ->
     Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22) ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_mono'45''8804''45'distrib'45''8851'_4738 = erased
  -- Data.Nat.Properties.antimono-≤-distrib-⊓
  d_antimono'45''8804''45'distrib'45''8851'_4748 ::
    (Integer -> Integer) ->
    (Integer ->
     Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22) ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_antimono'45''8804''45'distrib'45''8851'_4748 = erased
  -- Data.Nat.Properties.antimono-≤-distrib-⊔
  d_antimono'45''8804''45'distrib'45''8852'_4758 ::
    (Integer -> Integer) ->
    (Integer ->
     Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22) ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_antimono'45''8804''45'distrib'45''8852'_4758 = erased
  -- Data.Nat.Properties.m<n⇒m<n⊔o
  d_m'60'n'8658'm'60'n'8852'o_4764 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'm'60'n'8852'o_4764 v0 v1
    = let v2 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v3 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'x'8851'z'8804'y_3160
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v2))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v3))
              (coe v1) (coe addInt (coe (1 :: Integer)) (coe v0))))
  -- Data.Nat.Properties.m<n⇒m<o⊔n
  d_m'60'n'8658'm'60'o'8852'n_4768 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'm'60'o'8852'n_4768 v0 v1
    = let v2 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v3 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8658'z'8851'x'8804'y_3172
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v2))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v3))
              (coe v1) (coe addInt (coe (1 :: Integer)) (coe v0))))
  -- Data.Nat.Properties.m⊔n<o⇒m<o
  d_m'8852'n'60'o'8658'm'60'o_4776 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8852'n'60'o'8658'm'60'o_4776 v0 v1 ~v2 v3
    = du_m'8852'n'60'o'8658'm'60'o_4776 v0 v1 v3
  du_m'8852'n'60'o'8658'm'60'o_4776 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8852'n'60'o'8658'm'60'o_4776 v0 v1 v2
    = coe
        du_'8804''45''60''45'trans_2986
        (let v3 = d_'8804''45'totalPreorder_2822 in
         coe
           (let v4 = d_'8852''45'operator_4440 in
            coe
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'x_2808
                 (coe
                    MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                    (coe v3))
                 (coe
                    MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                    (coe v4))
                 (coe v0) (coe v1))))
        (coe v2)
  -- Data.Nat.Properties.m⊔n<o⇒n<o
  d_m'8852'n'60'o'8658'n'60'o_4790 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8852'n'60'o'8658'n'60'o_4790 v0 v1 ~v2 v3
    = du_m'8852'n'60'o'8658'n'60'o_4790 v0 v1 v3
  du_m'8852'n'60'o'8658'n'60'o_4790 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8852'n'60'o'8658'n'60'o_4790 v0 v1 v2
    = coe
        du_'8804''45''60''45'trans_2986
        (let v3 = d_'8804''45'totalPreorder_2822 in
         coe
           (let v4 = d_'8852''45'operator_4440 in
            coe
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_2834
                 (coe
                    MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                    (coe v3))
                 (coe
                    MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                    (coe v4))
                 (coe v0) (coe v1))))
        (coe v2)
  -- Data.Nat.Properties.⊔-mono-<
  d_'8852''45'mono'45''60'_4798 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8852''45'mono'45''60'_4798 v0 v1 v2 v3
    = let v4 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v5 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'45''8804'_3206
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v4))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v5))
              (coe v1) (coe addInt (coe (1 :: Integer)) (coe v0)) (coe v3)
              (coe addInt (coe (1 :: Integer)) (coe v2))))
  -- Data.Nat.Properties.⊔-pres-<m
  d_'8852''45'pres'45''60'm_4800 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8852''45'pres'45''60'm_4800 v0 v1 v2 v3 v4
    = coe d_'8852''45'mono'45''60'_4798 v0 v1 v2 v1 v3 v4
  -- Data.Nat.Properties.+-distribˡ-⊔
  d_'43''45'distrib'737''45''8852'_4810 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'distrib'737''45''8852'_4810 = erased
  -- Data.Nat.Properties.+-distribʳ-⊔
  d_'43''45'distrib'691''45''8852'_4822 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'distrib'691''45''8852'_4822 = erased
  -- Data.Nat.Properties.+-distrib-⊔
  d_'43''45'distrib'45''8852'_4824 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'43''45'distrib'45''8852'_4824
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.m⊔n≤m+n
  d_m'8852'n'8804'm'43'n_4830 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8852'n'8804'm'43'n_4830 v0 v1
    = let v2
            = coe
                MAlonzo.Code.Data.Sum.Base.du_'91'_'44'_'93'_52
                (coe
                   (\ v2 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 erased))
                (coe
                   (\ v2 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased))
                (coe d_'8804''45'total_2790 (coe v1) (coe v0)) in
      coe
        (case coe v2 of
           MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 v3
             -> coe du_m'8804'm'43'n_3482 (coe v0)
           MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 v3
             -> coe du_m'8804'n'43'm_3494 (coe v1)
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.*-distribˡ-⊔
  d_'42''45'distrib'737''45''8852'_4860 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'737''45''8852'_4860 = erased
  -- Data.Nat.Properties.*-distribʳ-⊔
  d_'42''45'distrib'691''45''8852'_4882 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'691''45''8852'_4882 = erased
  -- Data.Nat.Properties.*-distrib-⊔
  d_'42''45'distrib'45''8852'_4884 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'distrib'45''8852'_4884
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.⊓-zeroˡ
  d_'8851''45'zero'737'_4886 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'zero'737'_4886 = erased
  -- Data.Nat.Properties.⊓-zeroʳ
  d_'8851''45'zero'691'_4888 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'zero'691'_4888 = erased
  -- Data.Nat.Properties.⊓-zero
  d_'8851''45'zero_4892 :: MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'8851''45'zero_4892
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.⊔-⊓-isSemiringWithoutOne
  d_'8852''45''8851''45'isSemiringWithoutOne_4894 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiringWithoutOne_1298
  d_'8852''45''8851''45'isSemiringWithoutOne_4894
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsSemiringWithoutOne'46'constructor_37629
        (coe d_'8852''45'0'45'isCommutativeMonoid_4716) erased
        (coe
           MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'assoc_2944
           (coe d_'8804''45'totalPreorder_2822)
           (coe d_'8851''45'operator_4438))
        (coe
           MAlonzo.Code.Algebra.Construct.NaturalChoice.MinMaxOp.du_'8851''45'distrib'45''8852'_3138
           (coe d_'8804''45'totalPreorder_2822)
           (coe d_'8851''45'operator_4438) (coe d_'8852''45'operator_4440))
        (coe d_'8851''45'zero_4892)
  -- Data.Nat.Properties.⊔-⊓-isCommutativeSemiringWithoutOne
  d_'8852''45''8851''45'isCommutativeSemiringWithoutOne_4896 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiringWithoutOne_1382
  d_'8852''45''8851''45'isCommutativeSemiringWithoutOne_4896
    = coe
        MAlonzo.Code.Algebra.Structures.C_IsCommutativeSemiringWithoutOne'46'constructor_41457
        (coe d_'8852''45''8851''45'isSemiringWithoutOne_4894)
        (coe
           MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'comm_2856
           (coe d_'8804''45'totalPreorder_2822)
           (coe d_'8851''45'operator_4438))
  -- Data.Nat.Properties.⊔-⊓-commutativeSemiringWithoutOne
  d_'8852''45''8851''45'commutativeSemiringWithoutOne_4898 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeSemiringWithoutOne_2002
  d_'8852''45''8851''45'commutativeSemiringWithoutOne_4898
    = coe
        MAlonzo.Code.Algebra.Bundles.C_CommutativeSemiringWithoutOne'46'constructor_36869
        MAlonzo.Code.Data.Nat.Base.d__'8852'__204
        MAlonzo.Code.Data.Nat.Base.d__'8851'__232 (0 :: Integer)
        d_'8852''45''8851''45'isCommutativeSemiringWithoutOne_4896
  -- Data.Nat.Properties.m<n⇒m⊓o<n
  d_m'60'n'8658'm'8851'o'60'n_4902 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'm'8851'o'60'n_4902 v0 ~v1 v2 v3
    = du_m'60'n'8658'm'8851'o'60'n_4902 v0 v2 v3
  du_m'60'n'8658'm'8851'o'60'n_4902 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'8658'm'8851'o'60'n_4902 v0 v1 v2
    = coe
        du_'8804''45''60''45'trans_2986
        (coe
           MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'x_2808
           (coe d_'8804''45'totalPreorder_2822)
           (coe d_'8851''45'operator_4438) (coe v0) (coe v1))
        (coe v2)
  -- Data.Nat.Properties.m<n⇒o⊓m<n
  d_m'60'n'8658'o'8851'm'60'n_4910 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'o'8851'm'60'n_4910 v0 ~v1 v2 v3
    = du_m'60'n'8658'o'8851'm'60'n_4910 v0 v2 v3
  du_m'60'n'8658'o'8851'm'60'n_4910 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'8658'o'8851'm'60'n_4910 v0 v1 v2
    = coe
        du_'8804''45''60''45'trans_2986
        (coe
           MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_2834
           (coe d_'8804''45'totalPreorder_2822)
           (coe d_'8851''45'operator_4438) (coe v1) (coe v0))
        (coe v2)
  -- Data.Nat.Properties.m<n⊓o⇒m<n
  d_m'60'n'8851'o'8658'm'60'n_4920 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8851'o'8658'm'60'n_4920 v0
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'y_3184
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438)
        (coe addInt (coe (1 :: Integer)) (coe v0))
  -- Data.Nat.Properties.m<n⊓o⇒m<o
  d_m'60'n'8851'o'8658'm'60'o_4926 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8851'o'8658'm'60'o_4926 v0
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8804'y'8851'z'8658'x'8804'z_3198
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438)
        (coe addInt (coe (1 :: Integer)) (coe v0))
  -- Data.Nat.Properties.⊓-mono-<
  d_'8851''45'mono'45''60'_4928 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'mono'45''60'_4928 v0 v1 v2 v3
    = coe
        MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'mono'45''8804'_3206
        (coe d_'8804''45'totalPreorder_2822)
        (coe d_'8851''45'operator_4438)
        (coe addInt (coe (1 :: Integer)) (coe v0)) (coe v1)
        (coe addInt (coe (1 :: Integer)) (coe v2)) (coe v3)
  -- Data.Nat.Properties.⊓-pres-m<
  d_'8851''45'pres'45'm'60'_4930 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'pres'45'm'60'_4930 v0 v1 v2 v3 v4
    = coe d_'8851''45'mono'45''60'_4928 v0 v1 v0 v2 v3 v4
  -- Data.Nat.Properties.+-distribˡ-⊓
  d_'43''45'distrib'737''45''8851'_4940 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'distrib'737''45''8851'_4940 = erased
  -- Data.Nat.Properties.+-distribʳ-⊓
  d_'43''45'distrib'691''45''8851'_4952 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45'distrib'691''45''8851'_4952 = erased
  -- Data.Nat.Properties.+-distrib-⊓
  d_'43''45'distrib'45''8851'_4954 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'43''45'distrib'45''8851'_4954
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.m⊓n≤m+n
  d_m'8851'n'8804'm'43'n_4960 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8851'n'8804'm'43'n_4960 v0 v1
    = let v2
            = coe
                MAlonzo.Code.Data.Sum.Base.du_'91'_'44'_'93'_52
                (coe
                   (\ v2 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 erased))
                (coe
                   (\ v2 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased))
                (coe d_'8804''45'total_2790 (coe v0) (coe v1)) in
      coe
        (case coe v2 of
           MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 v3
             -> coe du_m'8804'm'43'n_3482 (coe v0)
           MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 v3
             -> coe du_m'8804'n'43'm_3494 (coe v1)
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.*-distribˡ-⊓
  d_'42''45'distrib'737''45''8851'_4990 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'737''45''8851'_4990 = erased
  -- Data.Nat.Properties.*-distribʳ-⊓
  d_'42''45'distrib'691''45''8851'_5012 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'691''45''8851'_5012 = erased
  -- Data.Nat.Properties.*-distrib-⊓
  d_'42''45'distrib'45''8851'_5014 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'distrib'45''8851'_5014
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.0∸n≡0
  d_0'8760'n'8801'0_5016 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_0'8760'n'8801'0_5016 = erased
  -- Data.Nat.Properties.n∸n≡0
  d_n'8760'n'8801'0_5020 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_n'8760'n'8801'0_5020 = erased
  -- Data.Nat.Properties.pred[m∸n]≡m∸[1+n]
  d_pred'91'm'8760'n'93''8801'm'8760''91'1'43'n'93'_5028 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_pred'91'm'8760'n'93''8801'm'8760''91'1'43'n'93'_5028 = erased
  -- Data.Nat.Properties.m∸n≤m
  d_m'8760'n'8804'm_5042 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8760'n'8804'm_5042 v0 v1
    = case coe v1 of
        0 -> coe
               d_'8804''45'refl_2776
               (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v0 (0 :: Integer))
        _ -> let v2 = subInt (coe v1) (coe (1 :: Integer)) in
             coe
               (case coe v0 of
                  0 -> coe
                         d_'8804''45'refl_2776
                         (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 (0 :: Integer) v1)
                  _ -> let v3 = subInt (coe v0) (coe (1 :: Integer)) in
                       coe
                         (coe
                            du_'8804''45'trans_2784
                            (coe d_m'8760'n'8804'm_5042 (coe v3) (coe v2))
                            (coe d_n'8804'1'43'n_2844 (coe v3))))
  -- Data.Nat.Properties.m≮m∸n
  d_m'8814'm'8760'n_5056 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'8814'm'8760'n_5056 = erased
  -- Data.Nat.Properties.1+m≢m∸n
  d_1'43'm'8802'm'8760'n_5068 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_1'43'm'8802'm'8760'n_5068 = erased
  -- Data.Nat.Properties.∸-mono
  d_'8760''45'mono_5076 ::
    Integer ->
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8760''45'mono_5076 v0 v1 v2 v3 v4 v5
    = let v6
            = seq
                (coe v5)
                (coe
                   du_'8804''45'trans_2784
                   (coe d_m'8760'n'8804'm_5042 (coe v0) (coe v2)) (coe v4)) in
      coe
        (case coe v4 of
           MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
             -> case coe v5 of
                  MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                    -> coe
                         du_'8804''45'trans_2784
                         (coe d_m'8760'n'8804'm_5042 (coe v0) (coe v2))
                         (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v10
                    -> case coe v2 of
                         _ | coe geqInt (coe v2) (coe (1 :: Integer)) ->
                             case coe v3 of
                               _ | coe geqInt (coe v3) (coe (1 :: Integer)) ->
                                   coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                               _ -> coe v6
                         _ -> coe v6
                  _ -> MAlonzo.RTE.mazUnreachableError
           MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9
             -> let v10
                      = seq
                          (coe v5)
                          (coe
                             du_'8804''45'trans_2784
                             (coe d_m'8760'n'8804'm_5042 (coe v0) (coe v2))
                             (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9)) in
                coe
                  (case coe v0 of
                     _ | coe geqInt (coe v0) (coe (1 :: Integer)) ->
                         let v11 = subInt (coe v0) (coe (1 :: Integer)) in
                         coe
                           (let v12
                                  = seq
                                      (coe v5)
                                      (coe
                                         du_'8804''45'trans_2784
                                         (coe d_m'8760'n'8804'm_5042 (coe v0) (coe v2))
                                         (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9)) in
                            coe
                              (case coe v1 of
                                 _ | coe geqInt (coe v1) (coe (1 :: Integer)) ->
                                     let v13 = subInt (coe v1) (coe (1 :: Integer)) in
                                     coe
                                       (case coe v5 of
                                          MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                                            -> coe
                                                 du_'8804''45'trans_2784
                                                 (coe d_m'8760'n'8804'm_5042 (coe v0) (coe v2))
                                                 (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9)
                                          MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v16
                                            -> case coe v2 of
                                                 _ | coe geqInt (coe v2) (coe (1 :: Integer)) ->
                                                     let v17 = subInt (coe v2) (coe (1 :: Integer)) in
                                                     coe
                                                       (case coe v3 of
                                                          _ | coe
                                                                geqInt (coe v3)
                                                                (coe (1 :: Integer)) ->
                                                              let v18
                                                                    = subInt
                                                                        (coe v3)
                                                                        (coe (1 :: Integer)) in
                                                              coe
                                                                (coe
                                                                   d_'8760''45'mono_5076 (coe v11)
                                                                   (coe v13) (coe v17) (coe v18)
                                                                   (coe v9) (coe v16))
                                                          _ -> coe v12)
                                                 _ -> coe v12
                                          _ -> MAlonzo.RTE.mazUnreachableError)
                                 _ -> coe v12))
                     _ -> coe v10)
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.∸-monoˡ-≤
  d_'8760''45'mono'737''45''8804'_5090 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8760''45'mono'737''45''8804'_5090 v0 v1 v2 v3
    = coe
        d_'8760''45'mono_5076 (coe v0) (coe v1) (coe v2) (coe v2) (coe v3)
        (coe d_'8804''45'refl_2776 (coe v2))
  -- Data.Nat.Properties.∸-monoʳ-≤
  d_'8760''45'mono'691''45''8804'_5098 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8760''45'mono'691''45''8804'_5098 v0 v1 v2 v3
    = coe
        d_'8760''45'mono_5076 (coe v2) (coe v2) (coe v1) (coe v0)
        (coe d_'8804''45'refl_2776 (coe v2)) (coe v3)
  -- Data.Nat.Properties.∸-monoˡ-<
  d_'8760''45'mono'737''45''60'_5108 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8760''45'mono'737''45''60'_5108 ~v0 v1 ~v2 v3 v4
    = du_'8760''45'mono'737''45''60'_5108 v1 v3 v4
  du_'8760''45'mono'737''45''60'_5108 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8760''45'mono'737''45''60'_5108 v0 v1 v2
    = case coe v0 of
        0 -> coe v1
        _ -> let v3 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
                    -> case coe v2 of
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9
                           -> coe
                                du_'8760''45'mono'737''45''60'_5108 (coe v3) (coe v6) (coe v9)
                         _ -> MAlonzo.RTE.mazUnreachableError
                  _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.∸-monoʳ-<
  d_'8760''45'mono'691''45''60'_5134 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8760''45'mono'691''45''60'_5134 v0 v1 v2 v3 v4
    = let v5 = subInt (coe v1) (coe (1 :: Integer)) in
      coe
        (case coe v2 of
           0 -> coe
                  seq (coe v3)
                  (coe
                     seq (coe v4)
                     (coe
                        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                        (d_m'8760'n'8804'm_5042
                           (coe subInt (coe v0) (coe (1 :: Integer))) (coe v5))))
           _ -> let v6 = subInt (coe v2) (coe (1 :: Integer)) in
                coe
                  (case coe v3 of
                     MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v9
                       -> case coe v4 of
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v12
                              -> let v13 = subInt (coe v0) (coe (1 :: Integer)) in
                                 coe
                                   (coe
                                      d_'8760''45'mono'691''45''60'_5134 (coe v13) (coe v5) (coe v6)
                                      (coe v9) (coe v12))
                            _ -> MAlonzo.RTE.mazUnreachableError
                     _ -> MAlonzo.RTE.mazUnreachableError))
  -- Data.Nat.Properties.∸-cancelʳ-≤
  d_'8760''45'cancel'691''45''8804'_5156 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8760''45'cancel'691''45''8804'_5156 ~v0 v1 ~v2 v3 ~v4
    = du_'8760''45'cancel'691''45''8804'_5156 v1 v3
  du_'8760''45'cancel'691''45''8804'_5156 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8760''45'cancel'691''45''8804'_5156 v0 v1
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> case coe v0 of
               0 -> coe
                      MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
                      erased
               _ -> let v5 = subInt (coe v0) (coe (1 :: Integer)) in
                    coe
                      (coe
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                         (coe du_'8760''45'cancel'691''45''8804'_5156 (coe v5) (coe v4)))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.∸-cancelʳ-<
  d_'8760''45'cancel'691''45''60'_5176 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8760''45'cancel'691''45''60'_5176 v0 v1 ~v2 ~v3
    = du_'8760''45'cancel'691''45''60'_5176 v0 v1
  du_'8760''45'cancel'691''45''60'_5176 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8760''45'cancel'691''45''60'_5176 v0 v1
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
               erased
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe du_0'60'1'43'n_3074
                  _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                            (coe du_'8760''45'cancel'691''45''60'_5176 (coe v2) (coe v3))))
  -- Data.Nat.Properties.∸-cancelˡ-≡
  d_'8760''45'cancel'737''45''8801'_5196 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8760''45'cancel'737''45''8801'_5196 = erased
  -- Data.Nat.Properties.∸-cancelʳ-≡
  d_'8760''45'cancel'691''45''8801'_5212 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8760''45'cancel'691''45''8801'_5212 = erased
  -- Data.Nat.Properties.m∸n≡0⇒m≤n
  d_m'8760'n'8801'0'8658'm'8804'n_5222 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8760'n'8801'0'8658'm'8804'n_5222 v0 ~v1 ~v2
    = du_m'8760'n'8801'0'8658'm'8804'n_5222 v0
  du_m'8760'n'8801'0'8658'm'8804'n_5222 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8760'n'8801'0'8658'm'8804'n_5222 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (coe du_m'8760'n'8801'0'8658'm'8804'n_5222 (coe v1)))
  -- Data.Nat.Properties.m≤n⇒m∸n≡0
  d_m'8804'n'8658'm'8760'n'8801'0_5230 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8804'n'8658'm'8760'n'8801'0_5230 = erased
  -- Data.Nat.Properties.m<n⇒0<n∸m
  d_m'60'n'8658'0'60'n'8760'm_5236 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'8658'0'60'n'8760'm_5236 v0 ~v1 v2
    = du_m'60'n'8658'0'60'n'8760'm_5236 v0 v2
  du_m'60'n'8658'0'60'n'8760'm_5236 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'8658'0'60'n'8760'm_5236 v0 v1
    = case coe v0 of
        0 -> coe du_0'60'1'43'n_3074
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
                    -> coe du_m'60'n'8658'0'60'n'8760'm_5236 (coe v2) (coe v5)
                  _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.m∸n≢0⇒n<m
  d_m'8760'n'8802'0'8658'n'60'm_5246 ::
    Integer ->
    Integer ->
    (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8760'n'8802'0'8658'n'60'm_5246 v0 v1 ~v2
    = du_m'8760'n'8802'0'8658'n'60'm_5246 v0 v1
  du_m'8760'n'8802'0'8658'n'60'm_5246 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8760'n'8802'0'8658'n'60'm_5246 v0 v1
    = let v2
            = coe
                MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
                (\ v2 ->
                   coe
                     du_'8804''7495''8658''8804'_2746
                     (coe addInt (coe (1 :: Integer)) (coe v1)))
                (coe du_'8804''8658''8804''7495'_2758)
                (coe
                   MAlonzo.Code.Relation.Nullary.Decidable.Core.d_T'63'_66
                   (coe
                      MAlonzo.Code.Data.Nat.Base.d__'8804''7495'__14
                      (coe addInt (coe (1 :: Integer)) (coe v1)) (coe v0))) in
      coe
        (case coe v2 of
           MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v3 v4
             -> if coe v3
                  then case coe v4 of
                         MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v5 -> coe v5
                         _ -> MAlonzo.RTE.mazUnreachableError
                  else coe
                         seq (coe v4)
                         (coe
                            MAlonzo.Code.Relation.Nullary.Negation.Core.du_contradiction_44
                            erased)
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.m>n⇒m∸n≢0
  d_m'62'n'8658'm'8760'n'8802'0_5274 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_m'62'n'8658'm'8760'n'8802'0_5274 = erased
  -- Data.Nat.Properties.m≤n⇒n∸m≤n
  d_m'8804'n'8658'n'8760'm'8804'n_5280 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'8658'n'8760'm'8804'n_5280 ~v0 v1 v2
    = du_m'8804'n'8658'n'8760'm'8804'n_5280 v1 v2
  du_m'8804'n'8658'n'8760'm'8804'n_5280 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'n'8658'n'8760'm'8804'n_5280 v0 v1
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe
               d_'8804''45'refl_2776
               (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v0 (0 :: Integer))
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> let v5 = subInt (coe v0) (coe (1 :: Integer)) in
             coe (coe du_m'8804'n'8658'n'8760'm'8804'n_5280 (coe v5) (coe v4))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.+-∸-comm
  d_'43''45''8760''45'comm_5290 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45''8760''45'comm_5290 = erased
  -- Data.Nat.Properties.∸-+-assoc
  d_'8760''45''43''45'assoc_5308 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8760''45''43''45'assoc_5308 = erased
  -- Data.Nat.Properties.+-∸-assoc
  d_'43''45''8760''45'assoc_5332 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'43''45''8760''45'assoc_5332 = erased
  -- Data.Nat.Properties.m≤n+o⇒m∸n≤o
  d_m'8804'n'43'o'8658'm'8760'n'8804'o_5354 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'43'o'8658'm'8760'n'8804'o_5354 v0 v1 ~v2 v3
    = du_m'8804'n'43'o'8658'm'8760'n'8804'o_5354 v0 v1 v3
  du_m'8804'n'43'o'8658'm'8760'n'8804'o_5354 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'n'43'o'8658'm'8760'n'8804'o_5354 v0 v1 v2
    = case coe v1 of
        0 -> coe v2
        _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
             coe
               (case coe v0 of
                  0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                  _ -> let v4 = subInt (coe v0) (coe (1 :: Integer)) in
                       coe
                         (coe
                            du_m'8804'n'43'o'8658'm'8760'n'8804'o_5354 (coe v4) (coe v3)
                            (coe
                               MAlonzo.Code.Data.Nat.Base.du_s'8804's'8315''185'_62 (coe v2))))
  -- Data.Nat.Properties.m<n+o⇒m∸n<o
  d_m'60'n'43'o'8658'm'8760'n'60'o_5374 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'60'n'43'o'8658'm'8760'n'60'o_5374 v0 v1 ~v2 ~v3 v4
    = du_m'60'n'43'o'8658'm'8760'n'60'o_5374 v0 v1 v4
  du_m'60'n'43'o'8658'm'8760'n'60'o_5374 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'60'n'43'o'8658'm'8760'n'60'o_5374 v0 v1 v2
    = case coe v1 of
        0 -> coe v2
        _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
             coe
               (case coe v0 of
                  0 -> coe
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                         (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
                  _ -> let v4 = subInt (coe v0) (coe (1 :: Integer)) in
                       coe
                         (coe
                            du_m'60'n'43'o'8658'm'8760'n'60'o_5374 (coe v4) (coe v3)
                            (coe MAlonzo.Code.Data.Nat.Base.du_s'60's'8315''185'_70 (coe v2))))
  -- Data.Nat.Properties.m+n≤o⇒m≤o∸n
  d_m'43'n'8804'o'8658'm'8804'o'8760'n_5398 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'43'n'8804'o'8658'm'8804'o'8760'n_5398 v0 ~v1 ~v2 v3
    = du_m'43'n'8804'o'8658'm'8804'o'8760'n_5398 v0 v3
  du_m'43'n'8804'o'8658'm'8804'o'8760'n_5398 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'43'n'8804'o'8658'm'8804'o'8760'n_5398 v0 v1
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v5
                    -> coe
                         MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                         (coe du_m'43'n'8804'o'8658'm'8804'o'8760'n_5398 (coe v2) (coe v5))
                  _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.m≤o∸n⇒m+n≤o
  d_m'8804'o'8760'n'8658'm'43'n'8804'o_5418 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'o'8760'n'8658'm'43'n'8804'o_5418 ~v0 ~v1 ~v2 v3 v4
    = du_m'8804'o'8760'n'8658'm'43'n'8804'o_5418 v3 v4
  du_m'8804'o'8760'n'8658'm'43'n'8804'o_5418 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_m'8804'o'8760'n'8658'm'43'n'8804'o_5418 v0 v1
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26 -> coe v1
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> coe
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
               (coe du_m'8804'o'8760'n'8658'm'43'n'8804'o_5418 (coe v4) (coe v1))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.m≤n+m∸n
  d_m'8804'n'43'm'8760'n_5444 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'43'm'8760'n_5444 v0 v1
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe d_'8804''45'refl_2776 (coe v0)
                  _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                            (d_m'8804'n'43'm'8760'n_5444 (coe v2) (coe v3))))
  -- Data.Nat.Properties.m+n∸n≡m
  d_m'43'n'8760'n'8801'm_5458 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'43'n'8760'n'8801'm_5458 = erased
  -- Data.Nat.Properties.m+n∸m≡n
  d_m'43'n'8760'm'8801'n_5470 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'43'n'8760'm'8801'n_5470 = erased
  -- Data.Nat.Properties.m+[n∸m]≡n
  d_m'43''91'n'8760'm'93''8801'n_5478 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'43''91'n'8760'm'93''8801'n_5478 = erased
  -- Data.Nat.Properties.m∸n+n≡m
  d_m'8760'n'43'n'8801'm_5492 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8760'n'43'n'8801'm_5492 = erased
  -- Data.Nat.Properties.m∸[m∸n]≡n
  d_m'8760''91'm'8760'n'93''8801'n_5504 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8760''91'm'8760'n'93''8801'n_5504 = erased
  -- Data.Nat.Properties.[m+n]∸[m+o]≡n∸o
  d_'91'm'43'n'93''8760''91'm'43'o'93''8801'n'8760'o_5520 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'91'm'43'n'93''8760''91'm'43'o'93''8801'n'8760'o_5520 = erased
  -- Data.Nat.Properties.*-distribʳ-∸
  d_'42''45'distrib'691''45''8760'_5532 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'691''45''8760'_5532 = erased
  -- Data.Nat.Properties.*-distribˡ-∸
  d_'42''45'distrib'737''45''8760'_5552 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'737''45''8760'_5552 = erased
  -- Data.Nat.Properties.*-distrib-∸
  d_'42''45'distrib'45''8760'_5554 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'distrib'45''8760'_5554
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.even≢odd
  d_even'8802'odd_5560 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_even'8802'odd_5560 = erased
  -- Data.Nat.Properties.m⊓n+n∸m≡n
  d_m'8851'n'43'n'8760'm'8801'n_5576 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8851'n'43'n'8760'm'8801'n_5576 = erased
  -- Data.Nat.Properties.[m∸n]⊓[n∸m]≡0
  d_'91'm'8760'n'93''8851''91'n'8760'm'93''8801'0_5590 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'91'm'8760'n'93''8851''91'n'8760'm'93''8801'0_5590 = erased
  -- Data.Nat.Properties.∸-distribˡ-⊓-⊔
  d_'8760''45'distrib'737''45''8851''45''8852'_5606 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8760''45'distrib'737''45''8851''45''8852'_5606 = erased
  -- Data.Nat.Properties.∸-distribʳ-⊓
  d_'8760''45'distrib'691''45''8851'_5614 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8760''45'distrib'691''45''8851'_5614 = erased
  -- Data.Nat.Properties.∸-distribˡ-⊔-⊓
  d_'8760''45'distrib'737''45''8852''45''8851'_5628 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8760''45'distrib'737''45''8852''45''8851'_5628 = erased
  -- Data.Nat.Properties.∸-distribʳ-⊔
  d_'8760''45'distrib'691''45''8852'_5636 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8760''45'distrib'691''45''8852'_5636 = erased
  -- Data.Nat.Properties.pred[n]≤n
  d_pred'91'n'93''8804'n_5644 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_pred'91'n'93''8804'n_5644 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe (coe d_n'8804'1'43'n_2844 (coe v1))
  -- Data.Nat.Properties.≤pred⇒≤
  d_'8804'pred'8658''8804'_5648 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804'pred'8658''8804'_5648 ~v0 v1 v2
    = du_'8804'pred'8658''8804'_5648 v1 v2
  du_'8804'pred'8658''8804'_5648 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804'pred'8658''8804'_5648 v0 v1 = coe seq (coe v0) (coe v1)
  -- Data.Nat.Properties.≤⇒pred≤
  d_'8804''8658'pred'8804'_5656 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''8658'pred'8804'_5656 v0 ~v1 v2
    = du_'8804''8658'pred'8804'_5656 v0 v2
  du_'8804''8658'pred'8804'_5656 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''8658'pred'8804'_5656 v0 v1
    = case coe v0 of
        0 -> coe v1
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  du_'8804''45'trans_2784 (coe d_n'8804'1'43'n_2844 (coe v2))
                  (coe v1))
  -- Data.Nat.Properties.<⇒≤pred
  d_'60''8658''8804'pred_5664 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''8658''8804'pred_5664 ~v0 ~v1 v2
    = du_'60''8658''8804'pred_5664 v2
  du_'60''8658''8804'pred_5664 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'60''8658''8804'pred_5664 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3 -> coe v3
        _                                           -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.suc-pred
  d_suc'45'pred_5672 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_suc'45'pred_5672 = erased
  -- Data.Nat.Properties.pred-mono-≤
  d_pred'45'mono'45''8804'_5676 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_pred'45'mono'45''8804'_5676 v0 ~v1 v2
    = du_pred'45'mono'45''8804'_5676 v0 v2
  du_pred'45'mono'45''8804'_5676 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_pred'45'mono'45''8804'_5676 v0 v1
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> coe
               MAlonzo.Code.Data.Nat.Base.du_s'8804's'8315''185'_62 (coe v1)
  -- Data.Nat.Properties.pred-mono-<
  d_pred'45'mono'45''60'_5680 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_pred'45'mono'45''60'_5680 v0 v1 ~v2
    = du_pred'45'mono'45''60'_5680 v0 v1
  du_pred'45'mono'45''60'_5680 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_pred'45'mono'45''60'_5680 v0 v1
    = case coe v0 of
        0 -> coe (\ v2 -> MAlonzo.RTE.mazUnreachableError)
        _ -> case coe v1 of
               0 -> coe (\ v2 -> MAlonzo.RTE.mazUnreachableError)
               _ -> coe MAlonzo.Code.Data.Nat.Base.du_s'60's'8315''185'_70
  -- Data.Nat.Properties.pred-cancel-≤
  d_pred'45'cancel'45''8804'_5682 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_pred'45'cancel'45''8804'_5682 v0 v1 v2
    = case coe v0 of
        0 -> coe
               seq (coe v1)
               (coe
                  MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                  (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26))
        _ -> case coe v1 of
               0 -> coe
                      seq (coe v2)
                      (coe
                         MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
                         (coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased))
               _ -> coe
                      MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                      (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v2)
  -- Data.Nat.Properties.pred-cancel-<
  d_pred'45'cancel'45''60'_5686 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_pred'45'cancel'45''60'_5686 v0 v1
    = case coe v0 of
        0 -> case coe v1 of
               0 -> coe (\ v2 -> MAlonzo.RTE.mazUnreachableError)
               _ -> coe
                      (\ v2 ->
                         coe
                           MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                           (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26))
        _ -> case coe v1 of
               0 -> coe (\ v2 -> MAlonzo.RTE.mazUnreachableError)
               _ -> coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
  -- Data.Nat.Properties.pred-injective
  d_pred'45'injective_5688 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_pred'45'injective_5688 = erased
  -- Data.Nat.Properties.pred-cancel-≡
  d_pred'45'cancel'45''8801'_5694 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_pred'45'cancel'45''8801'_5694 v0 v1
    = case coe v0 of
        0 -> case coe v1 of
               0 -> coe
                      (\ v2 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased)
               _ -> coe
                      (\ v2 ->
                         coe
                           MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
                           (coe
                              MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
                              (coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased)))
        _ -> case coe v1 of
               0 -> coe
                      (\ v2 ->
                         coe
                           MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
                           (coe
                              MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                              (coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased)))
               _ -> coe
                      (\ v2 -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased)
  -- Data.Nat.Properties.m≡n⇒∣m-n∣≡0
  d_m'8801'n'8658''8739'm'45'n'8739''8801'0_5696 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8801'n'8658''8739'm'45'n'8739''8801'0_5696 = erased
  -- Data.Nat.Properties.∣m-n∣≡0⇒m≡n
  d_'8739'm'45'n'8739''8801'0'8658'm'8801'n_5700 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739'm'45'n'8739''8801'0'8658'm'8801'n_5700 = erased
  -- Data.Nat.Properties.m≤n⇒∣n-m∣≡n∸m
  d_m'8804'n'8658''8739'n'45'm'8739''8801'n'8760'm_5710 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8804'n'8658''8739'n'45'm'8739''8801'n'8760'm_5710 = erased
  -- Data.Nat.Properties.m≤n⇒∣m-n∣≡n∸m
  d_m'8804'n'8658''8739'm'45'n'8739''8801'n'8760'm_5716 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8804'n'8658''8739'm'45'n'8739''8801'n'8760'm_5716 = erased
  -- Data.Nat.Properties.∣m-n∣≡m∸n⇒n≤m
  d_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_5722 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_5722 v0 v1 ~v2
    = du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_5722 v0 v1
  du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_5722 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_5722 v0 v1
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                  _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                            (coe
                               du_'8739'm'45'n'8739''8801'm'8760'n'8658'n'8804'm_5722 (coe v2)
                               (coe v3))))
  -- Data.Nat.Properties.∣n-n∣≡0
  d_'8739'n'45'n'8739''8801'0_5738 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739'n'45'n'8739''8801'0_5738 = erased
  -- Data.Nat.Properties.∣m-m+n∣≡n
  d_'8739'm'45'm'43'n'8739''8801'n_5746 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739'm'45'm'43'n'8739''8801'n_5746 = erased
  -- Data.Nat.Properties.∣m+n-m+o∣≡∣n-o∣
  d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'8739'_5760 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'8739'_5760
    = erased
  -- Data.Nat.Properties.m∸n≤∣m-n∣
  d_m'8760'n'8804''8739'm'45'n'8739'_5776 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8760'n'8804''8739'm'45'n'8739'_5776 v0 v1
    = let v2 = d_'8804''45'total_2790 (coe v0) (coe v1) in
      coe
        (case coe v2 of
           MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 v3
             -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
           MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 v3
             -> coe
                  d_'8804''45'refl_2776
                  (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v0 v1)
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.∣m-n∣≤m⊔n
  d_'8739'm'45'n'8739''8804'm'8852'n_5806 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8739'm'45'n'8739''8804'm'8852'n_5806 v0 v1
    = case coe v0 of
        0 -> coe
               d_'8804''45'refl_2776
               (coe
                  MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                  (coe (0 :: Integer)) (coe v1))
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe
                         d_'8804''45'refl_2776
                         (coe
                            MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v0)
                            (coe (0 :: Integer)))
                  _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe d_'8739'm'45'n'8739''8804'm'8852'n_5806 (coe v2) (coe v3)))
  -- Data.Nat.Properties.∣-∣-identityˡ
  d_'8739''45''8739''45'identity'737'_5816 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739''45''8739''45'identity'737'_5816 = erased
  -- Data.Nat.Properties.∣-∣-identityʳ
  d_'8739''45''8739''45'identity'691'_5820 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739''45''8739''45'identity'691'_5820 = erased
  -- Data.Nat.Properties.∣-∣-identity
  d_'8739''45''8739''45'identity_5824 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'8739''45''8739''45'identity_5824
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.∣-∣-comm
  d_'8739''45''8739''45'comm_5826 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739''45''8739''45'comm_5826 = erased
  -- Data.Nat.Properties.∣m-n∣≡[m∸n]∨[n∸m]
  d_'8739'm'45'n'8739''8801''91'm'8760'n'93''8744''91'n'8760'm'93'_5840 ::
    Integer -> Integer -> MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_'8739'm'45'n'8739''8801''91'm'8760'n'93''8744''91'n'8760'm'93'_5840 v0
                                                                        v1
    = let v2 = d_'8804''45'total_2790 (coe v0) (coe v1) in
      coe
        (case coe v2 of
           MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 v3
             -> coe
                  MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42
                  (let v4
                         = coe
                             MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_eqRelation_238 in
                   coe
                     (coe
                        MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__126
                        (coe v4)
                        (coe
                           MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v0)
                           (coe v1))
                        (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v1 v0)
                        (coe
                           MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
                           (\ v5 v6 v7 v8 v9 -> v9)
                           (MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                              (coe v0) (coe v1))
                           (MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                              (coe v1) (coe v0))
                           (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v1 v0)
                           (coe
                              MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
                              (\ v5 v6 v7 v8 v9 -> v9)
                              (MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                 (coe v1) (coe v0))
                              (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v1 v0)
                              (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v1 v0)
                              (coe
                                 MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du__'8718'_492
                                 (coe
                                    MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
                                    (coe d_'8804''45'isPreorder_2810))
                                 (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v1 v0))
                              erased)
                           erased)))
           MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 v3
             -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38 erased
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties.*-distribˡ-∣-∣-aux
  d_'42''45'distrib'737''45''8739''45''8739''45'aux_5868 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'737''45''8739''45''8739''45'aux_5868 = erased
  -- Data.Nat.Properties.*-distribˡ-∣-∣
  d_'42''45'distrib'737''45''8739''45''8739'_5880 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'737''45''8739''45''8739'_5880 = erased
  -- Data.Nat.Properties.*-distribʳ-∣-∣
  d_'42''45'distrib'691''45''8739''45''8739'_5910 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'42''45'distrib'691''45''8739''45''8739'_5910 = erased
  -- Data.Nat.Properties.*-distrib-∣-∣
  d_'42''45'distrib'45''8739''45''8739'_5912 ::
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_'42''45'distrib'45''8739''45''8739'_5912
    = coe MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 erased erased
  -- Data.Nat.Properties.m≤n+∣n-m∣
  d_m'8804'n'43''8739'n'45'm'8739'_5918 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'43''8739'n'45'm'8739'_5918 v0 v1
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe d_'8804''45'refl_2776 (coe v0)
                  _ -> let v3 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (coe
                            MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                            (d_m'8804'n'43''8739'n'45'm'8739'_5918 (coe v2) (coe v3))))
  -- Data.Nat.Properties.m≤n+∣m-n∣
  d_m'8804'n'43''8739'm'45'n'8739'_5932 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804'n'43''8739'm'45'n'8739'_5932 v0 v1
    = coe d_m'8804'n'43''8739'n'45'm'8739'_5918 (coe v0) (coe v1)
  -- Data.Nat.Properties.m≤∣m-n∣+n
  d_m'8804''8739'm'45'n'8739''43'n_5946 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_m'8804''8739'm'45'n'8739''43'n_5946 v0 v1
    = coe d_m'8804'n'43''8739'm'45'n'8739'_5932 (coe v0) (coe v1)
  -- Data.Nat.Properties.∣-∣-triangle
  d_'8739''45''8739''45'triangle_5954 ::
    Integer ->
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8739''45''8739''45'triangle_5954 v0 v1 v2
    = case coe v0 of
        0 -> coe d_m'8804'n'43''8739'n'45'm'8739'_5918 (coe v2) (coe v1)
        _ -> let v3 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v1 of
                  0 -> coe
                         MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__46
                         (coe
                            MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_start_96
                            (coe d_'8804''45'isPreorder_2810)
                            (\ v4 v5 v6 -> coe du_'60''8658''8804'_2854 v6))
                         (MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                            (coe v0) (coe v2))
                         (addInt
                            (coe
                               MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                               (coe (0 :: Integer)) (coe v2))
                            (coe
                               MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v0)
                               (coe (0 :: Integer))))
                         (coe
                            MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8804'_308
                            (coe
                               MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8804''45'go_138
                               (coe d_'8804''45'isPreorder_2810)
                               (\ v4 v5 v6 v7 v8 -> coe du_'8804''45''60''45'trans_2986 v7 v8))
                            (MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                               (coe v0) (coe v2))
                            (MAlonzo.Code.Data.Nat.Base.d__'8852'__204 (coe v0) (coe v2))
                            (addInt
                               (coe
                                  MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                  (coe (0 :: Integer)) (coe v2))
                               (coe
                                  MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v0)
                                  (coe (0 :: Integer))))
                            (coe
                               MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8804'_308
                               (coe
                                  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8804''45'go_138
                                  (coe d_'8804''45'isPreorder_2810)
                                  (\ v4 v5 v6 v7 v8 -> coe du_'8804''45''60''45'trans_2986 v7 v8))
                               (MAlonzo.Code.Data.Nat.Base.d__'8852'__204 (coe v0) (coe v2))
                               (addInt (coe v0) (coe v2))
                               (addInt
                                  (coe
                                     MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                     (coe (0 :: Integer)) (coe v2))
                                  (coe
                                     MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v0)
                                     (coe (0 :: Integer))))
                               (coe
                                  MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
                                  (\ v4 v5 v6 v7 v8 -> v8) (addInt (coe v0) (coe v2))
                                  (addInt
                                     (coe
                                        MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v0)
                                        (coe (0 :: Integer)))
                                     (coe v2))
                                  (addInt
                                     (coe
                                        MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                        (coe (0 :: Integer)) (coe v2))
                                     (coe
                                        MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v0)
                                        (coe (0 :: Integer))))
                                  (coe
                                     MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du__'8718'_492
                                     (coe
                                        MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
                                        (coe d_'8804''45'isPreorder_2810))
                                     (coe
                                        addInt
                                        (coe
                                           MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                           (coe v0) (coe (0 :: Integer)))
                                        (coe v2)))
                                  erased)
                               (d_m'8852'n'8804'm'43'n_4830 (coe v0) (coe v2)))
                            (d_'8739'm'45'n'8739''8804'm'8852'n_5806 (coe v0) (coe v2)))
                  _ -> let v4 = subInt (coe v1) (coe (1 :: Integer)) in
                       coe
                         (case coe v2 of
                            0 -> coe
                                   MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_begin__46
                                   (coe
                                      MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_start_96
                                      (coe d_'8804''45'isPreorder_2810)
                                      (\ v5 v6 v7 -> coe du_'60''8658''8804'_2854 v7))
                                   (MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                      (coe v0) (coe (0 :: Integer)))
                                   (addInt
                                      (coe
                                         MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v0)
                                         (coe v1))
                                      (coe
                                         MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280 (coe v1)
                                         (coe (0 :: Integer))))
                                   (coe
                                      MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
                                      (\ v5 v6 v7 v8 v9 -> v9)
                                      (MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                         (coe v0) (coe (0 :: Integer)))
                                      v0
                                      (addInt
                                         (coe
                                            MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                            (coe v0) (coe v1))
                                         (coe
                                            MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                            (coe v1) (coe (0 :: Integer))))
                                      (coe
                                         MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8804'_308
                                         (coe
                                            MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_'8804''45'go_138
                                            (coe d_'8804''45'isPreorder_2810)
                                            (\ v5 v6 v7 v8 v9 ->
                                               coe du_'8804''45''60''45'trans_2986 v8 v9))
                                         v0
                                         (addInt
                                            (coe
                                               MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                               (coe v0) (coe v1))
                                            (coe v1))
                                         (addInt
                                            (coe
                                               MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                               (coe v0) (coe v1))
                                            (coe
                                               MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                               (coe v1) (coe (0 :: Integer))))
                                         (coe
                                            MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du_step'45''8801''45''10217'_436
                                            (\ v5 v6 v7 v8 v9 -> v9)
                                            (addInt
                                               (coe
                                                  MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                                  (coe v0) (coe v1))
                                               (coe v1))
                                            (addInt
                                               (coe
                                                  MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                                  (coe v0) (coe v1))
                                               (coe
                                                  MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                                  (coe v1) (coe (0 :: Integer))))
                                            (addInt
                                               (coe
                                                  MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                                  (coe v0) (coe v1))
                                               (coe
                                                  MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                                  (coe v1) (coe (0 :: Integer))))
                                            (coe
                                               MAlonzo.Code.Relation.Binary.Reasoning.Syntax.du__'8718'_492
                                               (coe
                                                  MAlonzo.Code.Relation.Binary.Reasoning.Base.Triple.du_stop_166
                                                  (coe d_'8804''45'isPreorder_2810))
                                               (coe
                                                  addInt
                                                  (coe
                                                     MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                                     (coe v0) (coe v1))
                                                  (coe
                                                     MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
                                                     (coe v1) (coe (0 :: Integer)))))
                                            erased)
                                         (d_m'8804''8739'm'45'n'8739''43'n_5946 (coe v0) (coe v1)))
                                      erased)
                            _ -> let v5 = subInt (coe v2) (coe (1 :: Integer)) in
                                 coe
                                   (coe
                                      d_'8739''45''8739''45'triangle_5954 (coe v3) (coe v4)
                                      (coe v5))))
  -- Data.Nat.Properties.∣-∣≡∣-∣′
  d_'8739''45''8739''8801''8739''45''8739''8242'_5986 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739''45''8739''8801''8739''45''8739''8242'_5986 = erased
  -- Data.Nat.Properties.∣-∣-isProtoMetric
  d_'8739''45''8739''45'isProtoMetric_6008 ::
    MAlonzo.Code.Function.Metric.Structures.T_IsProtoMetric_30
  d_'8739''45''8739''45'isProtoMetric_6008
    = coe
        MAlonzo.Code.Function.Metric.Structures.C_IsProtoMetric'46'constructor_2109
        (coe d_'8804''45'isPartialOrder_2814)
        (coe
           MAlonzo.Code.Relation.Binary.PropositionalEquality.Properties.du_isEquivalence_396)
        erased
        (coe (\ v0 v1 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26))
  -- Data.Nat.Properties.∣-∣-isPreMetric
  d_'8739''45''8739''45'isPreMetric_6010 ::
    MAlonzo.Code.Function.Metric.Structures.T_IsPreMetric_102
  d_'8739''45''8739''45'isPreMetric_6010
    = coe
        MAlonzo.Code.Function.Metric.Structures.C_IsPreMetric'46'constructor_6347
        (coe d_'8739''45''8739''45'isProtoMetric_6008) erased
  -- Data.Nat.Properties.∣-∣-isQuasiSemiMetric
  d_'8739''45''8739''45'isQuasiSemiMetric_6012 ::
    MAlonzo.Code.Function.Metric.Structures.T_IsQuasiSemiMetric_174
  d_'8739''45''8739''45'isQuasiSemiMetric_6012
    = coe
        MAlonzo.Code.Function.Metric.Structures.C_IsQuasiSemiMetric'46'constructor_10111
        (coe d_'8739''45''8739''45'isPreMetric_6010) erased
  -- Data.Nat.Properties.∣-∣-isSemiMetric
  d_'8739''45''8739''45'isSemiMetric_6014 ::
    MAlonzo.Code.Function.Metric.Structures.T_IsSemiMetric_250
  d_'8739''45''8739''45'isSemiMetric_6014
    = coe
        MAlonzo.Code.Function.Metric.Structures.C_IsSemiMetric'46'constructor_14005
        (coe d_'8739''45''8739''45'isQuasiSemiMetric_6012) erased
  -- Data.Nat.Properties.∣-∣-isMetric
  d_'8739''45''8739''45'isMetric_6016 ::
    MAlonzo.Code.Function.Metric.Structures.T_IsGeneralMetric_332
  d_'8739''45''8739''45'isMetric_6016
    = coe
        MAlonzo.Code.Function.Metric.Structures.C_IsGeneralMetric'46'constructor_18255
        (coe d_'8739''45''8739''45'isSemiMetric_6014)
        (coe d_'8739''45''8739''45'triangle_5954)
  -- Data.Nat.Properties.∣-∣-quasiSemiMetric
  d_'8739''45''8739''45'quasiSemiMetric_6018 ::
    MAlonzo.Code.Function.Metric.Nat.Bundles.T_QuasiSemiMetric_186
  d_'8739''45''8739''45'quasiSemiMetric_6018
    = coe
        MAlonzo.Code.Function.Metric.Nat.Bundles.C_QuasiSemiMetric'46'constructor_3255
        MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
        d_'8739''45''8739''45'isQuasiSemiMetric_6012
  -- Data.Nat.Properties.∣-∣-semiMetric
  d_'8739''45''8739''45'semiMetric_6020 ::
    MAlonzo.Code.Function.Metric.Nat.Bundles.T_SemiMetric_284
  d_'8739''45''8739''45'semiMetric_6020
    = coe
        MAlonzo.Code.Function.Metric.Nat.Bundles.C_SemiMetric'46'constructor_4991
        MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
        d_'8739''45''8739''45'isSemiMetric_6014
  -- Data.Nat.Properties.∣-∣-preMetric
  d_'8739''45''8739''45'preMetric_6022 ::
    MAlonzo.Code.Function.Metric.Nat.Bundles.T_PreMetric_96
  d_'8739''45''8739''45'preMetric_6022
    = coe
        MAlonzo.Code.Function.Metric.Nat.Bundles.C_PreMetric'46'constructor_1629
        MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
        d_'8739''45''8739''45'isPreMetric_6010
  -- Data.Nat.Properties.∣-∣-metric
  d_'8739''45''8739''45'metric_6024 ::
    MAlonzo.Code.Function.Metric.Nat.Bundles.T_Metric_388
  d_'8739''45''8739''45'metric_6024
    = coe
        MAlonzo.Code.Function.Metric.Nat.Bundles.C_Metric'46'constructor_6797
        MAlonzo.Code.Data.Nat.Base.d_'8739'_'45'_'8739'_280
        d_'8739''45''8739''45'isMetric_6016
  -- Data.Nat.Properties.⌊n/2⌋-mono
  d_'8970'n'47'2'8971''45'mono_6026 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8970'n'47'2'8971''45'mono_6026 ~v0 ~v1 v2
    = du_'8970'n'47'2'8971''45'mono_6026 v2
  du_'8970'n'47'2'8971''45'mono_6026 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8970'n'47'2'8971''45'mono_6026 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v3
          -> case coe v3 of
               MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v6
                 -> coe
                      MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                      (coe du_'8970'n'47'2'8971''45'mono_6026 (coe v6))
               _ -> MAlonzo.RTE.mazUnreachableError
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.⌈n/2⌉-mono
  d_'8968'n'47'2'8969''45'mono_6030 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8968'n'47'2'8969''45'mono_6030 ~v0 ~v1 v2
    = du_'8968'n'47'2'8969''45'mono_6030 v2
  du_'8968'n'47'2'8969''45'mono_6030 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8968'n'47'2'8969''45'mono_6030 v0
    = coe
        du_'8970'n'47'2'8971''45'mono_6026
        (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v0)
  -- Data.Nat.Properties.⌊n/2⌋≤⌈n/2⌉
  d_'8970'n'47'2'8971''8804''8968'n'47'2'8969'_6036 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8970'n'47'2'8971''8804''8968'n'47'2'8969'_6036 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        1 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (2 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (d_'8970'n'47'2'8971''8804''8968'n'47'2'8969'_6036 (coe v1)))
  -- Data.Nat.Properties.⌊n/2⌋+⌈n/2⌉≡n
  d_'8970'n'47'2'8971''43''8968'n'47'2'8969''8801'n_6042 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8970'n'47'2'8971''43''8968'n'47'2'8969''8801'n_6042 = erased
  -- Data.Nat.Properties.⌊n/2⌋≤n
  d_'8970'n'47'2'8971''8804'n_6048 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8970'n'47'2'8971''8804'n_6048 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        1 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (2 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (d_'8970'n'47'2'8971''8804'n_6048 (coe v1)))
  -- Data.Nat.Properties.⌊n/2⌋<n
  d_'8970'n'47'2'8971''60'n_6054 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8970'n'47'2'8971''60'n_6054 v0
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
               (coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26)
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (coe
                     MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                     (d_'8970'n'47'2'8971''8804'n_6048 (coe v1))))
  -- Data.Nat.Properties.n≡⌊n+n/2⌋
  d_n'8801''8970'n'43'n'47'2'8971'_6060 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_n'8801''8970'n'43'n'47'2'8971'_6060 = erased
  -- Data.Nat.Properties.⌈n/2⌉≤n
  d_'8968'n'47'2'8969''8804'n_6068 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8968'n'47'2'8969''8804'n_6068 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
                  (d_'8970'n'47'2'8971''8804'n_6048 (coe v1)))
  -- Data.Nat.Properties.⌈n/2⌉<n
  d_'8968'n'47'2'8969''60'n_6074 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8968'n'47'2'8969''60'n_6074 v0
    = coe
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34
        (d_'8970'n'47'2'8971''60'n_6054 (coe v0))
  -- Data.Nat.Properties.n≡⌈n+n/2⌉
  d_n'8801''8968'n'43'n'47'2'8969'_6080 ::
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_n'8801''8968'n'43'n'47'2'8969'_6080 = erased
  -- Data.Nat.Properties.1≤n!
  d_1'8804'n'33'_6088 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_1'8804'n'33'_6088 v0
    = case coe v0 of
        0 -> coe d_'8804''45'refl_2776 (coe (1 :: Integer))
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  du_'42''45'mono'45''8804'_4060 (coe v0)
                  (coe MAlonzo.Code.Data.Nat.Base.d__'33'_332 (coe v1))
                  (coe du_m'8804'm'43'n_3482 (coe (1 :: Integer)))
                  (coe d_1'8804'n'33'_6088 (coe v1)))
  -- Data.Nat.Properties._!≢0
  d__'33''8802'0_6094 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  d__'33''8802'0_6094 v0
    = coe
        MAlonzo.Code.Data.Nat.Base.du_'62''45'nonZero_136
        (coe d_1'8804'n'33'_6088 (coe v0))
  -- Data.Nat.Properties._!*_!≢0
  d__'33''42'_'33''8802'0_6102 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  d__'33''42'_'33''8802'0_6102 ~v0 ~v1
    = du__'33''42'_'33''8802'0_6102
  du__'33''42'_'33''8802'0_6102 ::
    MAlonzo.Code.Data.Nat.Base.T_NonZero_112
  du__'33''42'_'33''8802'0_6102 = coe du_m'42'n'8802'0_3840
  -- Data.Nat.Properties.≤′-trans
  d_'8804''8242''45'trans_6108 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_'8804''8242''45'trans_6108 ~v0 ~v1 ~v2 v3 v4
    = du_'8804''8242''45'trans_6108 v3 v4
  du_'8804''8242''45'trans_6108 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  du_'8804''8242''45'trans_6108 v0 v1
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342 -> coe v0
        MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348 v3
          -> coe
               MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348
               (coe du_'8804''8242''45'trans_6108 (coe v0) (coe v3))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.z≤′n
  d_z'8804''8242'n_6116 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_z'8804''8242'n_6116 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348
                  (d_z'8804''8242'n_6116 (coe v1)))
  -- Data.Nat.Properties.s≤′s
  d_s'8804''8242's_6120 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_s'8804''8242's_6120 ~v0 ~v1 v2 = du_s'8804''8242's_6120 v2
  du_s'8804''8242's_6120 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  du_s'8804''8242's_6120 v0 = coe v0
  -- Data.Nat.Properties.≤′⇒≤
  d_'8804''8242''8658''8804'_6124 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''8242''8658''8804'_6124 v0 ~v1 v2
    = du_'8804''8242''8658''8804'_6124 v0 v2
  du_'8804''8242''8658''8804'_6124 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''8242''8658''8804'_6124 v0 v1
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
          -> coe d_'8804''45'refl_2776 (coe v0)
        MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348 v3
          -> coe du_'8804''8242''8658''8804'_6124 (coe v0) (coe v3)
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.≤⇒≤′
  d_'8804''8658''8804''8242'_6128 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_'8804''8658''8804''8242'_6128 ~v0 v1 v2
    = du_'8804''8658''8804''8242'_6128 v1 v2
  du_'8804''8658''8804''8242'_6128 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  du_'8804''8658''8804''8242'_6128 v0 v1
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
          -> coe d_z'8804''8242'n_6116 (coe v0)
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> let v5 = subInt (coe v0) (coe (1 :: Integer)) in
             coe (coe du_'8804''8658''8804''8242'_6128 (coe v5) (coe v4))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.≤′-step-injective
  d_'8804''8242''45'step'45'injective_6136 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8804''8242''45'step'45'injective_6136 = erased
  -- Data.Nat.Properties.z<′s
  d_z'60''8242's_6138 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_z'60''8242's_6138 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348
                  (d_z'60''8242's_6138 (coe v1)))
  -- Data.Nat.Properties.s<′s
  d_s'60''8242's_6142 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_s'60''8242's_6142 ~v0 ~v1 v2 = du_s'60''8242's_6142 v2
  du_s'60''8242's_6142 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  du_s'60''8242's_6142 v0 = coe v0
  -- Data.Nat.Properties.<⇒<′
  d_'60''8658''60''8242'_6146 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_'60''8658''60''8242'_6146 ~v0 v1 v2
    = du_'60''8658''60''8242'_6146 v1 v2
  du_'60''8658''60''8242'_6146 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  du_'60''8658''60''8242'_6146 v0 v1
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v4
          -> let v5 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (case coe v4 of
                  MAlonzo.Code.Data.Nat.Base.C_z'8804'n_26
                    -> coe d_z'60''8242's_6138 (coe v5)
                  MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v8
                    -> coe
                         du_'60''8658''60''8242'_6146
                         (coe subInt (coe v0) (coe (1 :: Integer)))
                         (coe MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v8)
                  _ -> MAlonzo.RTE.mazUnreachableError)
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.<′⇒<
  d_'60''8242''8658''60'_6150 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''8242''8658''60'_6150 v0 ~v1 v2
    = du_'60''8242''8658''60'_6150 v0 v2
  du_'60''8242''8658''60'_6150 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'60''8242''8658''60'_6150 v0 v1
    = case coe v1 of
        MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
          -> coe d_n'60'1'43'n_3078 (coe v0)
        MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348 v3
          -> coe
               du_m'60'n'8658'm'60'1'43'n_3062
               (coe du_'60''8242''8658''60'_6150 (coe v0) (coe v3))
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.m<1+n⇒m<n∨m≡n′
  d_m'60'1'43'n'8658'm'60'n'8744'm'8801'n'8242'_6154 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Sum.Base.T__'8846'__30
  d_m'60'1'43'n'8658'm'60'n'8744'm'8801'n'8242'_6154 v0 v1 v2
    = let v3
            = coe
                du_'60''8658''60''8242'_6146
                (coe addInt (coe (1 :: Integer)) (coe v1)) (coe v2) in
      coe
        (case coe v3 of
           MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
             -> coe MAlonzo.Code.Data.Sum.Base.C_inj'8322'_42 erased
           MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348 v5
             -> coe
                  MAlonzo.Code.Data.Sum.Base.C_inj'8321'_38
                  (coe du_'60''8242''8658''60'_6150 (coe v0) (coe v5))
           _ -> MAlonzo.RTE.mazUnreachableError)
  -- Data.Nat.Properties._≤′?_
  d__'8804''8242''63'__6168 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8804''8242''63'__6168 v0 v1
    = coe
        MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
        (coe du_'8804''8658''8804''8242'_6128 (coe v1))
        (coe du_'8804''8242''8658''8804'_6124 (coe v0))
        (coe d__'8804''63'__2802 (coe v0) (coe v1))
  -- Data.Nat.Properties._<′?_
  d__'60''8242''63'__6174 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'60''8242''63'__6174 v0 v1
    = coe
        d__'8804''8242''63'__6168
        (coe addInt (coe (1 :: Integer)) (coe v0)) (coe v1)
  -- Data.Nat.Properties._≥′?_
  d__'8805''8242''63'__6180 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8805''8242''63'__6180 v0 v1
    = coe d__'8804''8242''63'__6168 (coe v1) (coe v0)
  -- Data.Nat.Properties._>′?_
  d__'62''8242''63'__6182 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'62''8242''63'__6182 v0 v1
    = coe d__'60''8242''63'__6174 (coe v1) (coe v0)
  -- Data.Nat.Properties.m≤′m+n
  d_m'8804''8242'm'43'n_6188 ::
    Integer ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_m'8804''8242'm'43'n_6188 v0 v1
    = coe
        du_'8804''8658''8804''8242'_6128 (coe addInt (coe v0) (coe v1))
        (coe du_m'8804'm'43'n_3482 (coe v0))
  -- Data.Nat.Properties.n≤′m+n
  d_n'8804''8242'm'43'n_6198 ::
    Integer ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_n'8804''8242'm'43'n_6198 v0 ~v1 = du_n'8804''8242'm'43'n_6198 v0
  du_n'8804''8242'm'43'n_6198 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  du_n'8804''8242'm'43'n_6198 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348
                  (coe du_n'8804''8242'm'43'n_6198 (coe v1)))
  -- Data.Nat.Properties.⌈n/2⌉≤′n
  d_'8968'n'47'2'8969''8804''8242'n_6208 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_'8968'n'47'2'8969''8804''8242'n_6208 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
        1 -> coe MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
        _ -> let v1 = subInt (coe v0) (coe (2 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348
                  (d_'8968'n'47'2'8969''8804''8242'n_6208 (coe v1)))
  -- Data.Nat.Properties.⌊n/2⌋≤′n
  d_'8970'n'47'2'8971''8804''8242'n_6214 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8242'__338
  d_'8970'n'47'2'8971''8804''8242'n_6214 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'refl_342
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_'8804''8242''45'step_348
                  (d_'8968'n'47'2'8969''8804''8242'n_6208 (coe v1)))
  -- Data.Nat.Properties.≤⇒≤″
  d_'8804''8658''8804''8243'_6218 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26
  d_'8804''8658''8804''8243'_6218 v0 v1 ~v2
    = du_'8804''8658''8804''8243'_6218 v0 v1
  du_'8804''8658''8804''8243'_6218 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26
  du_'8804''8658''8804''8243'_6218 v0 v1
    = coe
        MAlonzo.Code.Algebra.Definitions.RawMagma.C__'44'__40
        (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v1 v0) erased
  -- Data.Nat.Properties.<⇒<″
  d_'60''8658''60''8243'_6222 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26
  d_'60''8658''60''8243'_6222 v0 v1 v2
    = coe
        du_'8804''8658''8804''8243'_6218
        (coe addInt (coe (1 :: Integer)) (coe v0)) (coe v1)
  -- Data.Nat.Properties.≤″⇒≤
  d_'8804''8243''8658''8804'_6224 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''8243''8658''8804'_6224 v0 ~v1 v2
    = du_'8804''8243''8658''8804'_6224 v0 v2
  du_'8804''8243''8658''8804'_6224 ::
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''8243''8658''8804'_6224 v0 v1
    = coe seq (coe v1) (coe du_m'8804'm'43'n_3482 (coe v0))
  -- Data.Nat.Properties.≤″-proof
  d_'8804''8243''45'proof_6232 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8804''8243''45'proof_6232 = erased
  -- Data.Nat.Properties.m≤n⇒∃[o]m+o≡n
  d_m'8804'n'8658''8707''91'o'93'm'43'o'8801'n_6238 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  d_m'8804'n'8658''8707''91'o'93'm'43'o'8801'n_6238 v0 v1 ~v2
    = du_m'8804'n'8658''8707''91'o'93'm'43'o'8801'n_6238 v0 v1
  du_m'8804'n'8658''8707''91'o'93'm'43'o'8801'n_6238 ::
    Integer -> Integer -> MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14
  du_m'8804'n'8658''8707''91'o'93'm'43'o'8801'n_6238 v0 v1
    = coe
        MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
        (coe MAlonzo.Code.Agda.Builtin.Nat.d__'45'__22 v1 v0) erased
  -- Data.Nat.Properties.guarded-∸≗∸
  d_guarded'45''8760''8791''8760'_6250 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_guarded'45''8760''8791''8760'_6250 = erased
  -- Data.Nat.Properties.m<ᵇn⇒1+m+[n-1+m]≡n
  d_m'60''7495'n'8658'1'43'm'43''91'n'45'1'43'm'93''8801'n_6258 ::
    Integer ->
    Integer ->
    AgdaAny -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'60''7495'n'8658'1'43'm'43''91'n'45'1'43'm'93''8801'n_6258
    = erased
  -- Data.Nat.Properties.m<ᵇ1+m+n
  d_m'60''7495'1'43'm'43'n_6270 :: Integer -> Integer -> AgdaAny
  d_m'60''7495'1'43'm'43'n_6270 v0 ~v1
    = du_m'60''7495'1'43'm'43'n_6270 v0
  du_m'60''7495'1'43'm'43'n_6270 :: Integer -> AgdaAny
  du_m'60''7495'1'43'm'43'n_6270 v0
    = coe
        du_'60''8658''60''7495'_2728
        (coe
           du_m'8804'm'43'n_3482 (coe addInt (coe (1 :: Integer)) (coe v0)))
  -- Data.Nat.Properties.<ᵇ⇒<″
  d_'60''7495''8658''60''8243'_6274 ::
    Integer ->
    Integer ->
    AgdaAny ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26
  d_'60''7495''8658''60''8243'_6274 v0 v1 ~v2
    = du_'60''7495''8658''60''8243'_6274 v0 v1
  du_'60''7495''8658''60''8243'_6274 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26
  du_'60''7495''8658''60''8243'_6274 v0 v1
    = coe
        d_'60''8658''60''8243'_6222 v0 v1
        (coe du_'60''7495''8658''60'_2716 (coe v0))
  -- Data.Nat.Properties.<″⇒<ᵇ
  d_'60''8243''8658''60''7495'_6284 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    AgdaAny
  d_'60''8243''8658''60''7495'_6284 v0 ~v1 v2
    = du_'60''8243''8658''60''7495'_6284 v0 v2
  du_'60''8243''8658''60''7495'_6284 ::
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    AgdaAny
  du_'60''8243''8658''60''7495'_6284 v0 v1
    = coe
        seq (coe v1)
        (coe
           du_'60''8658''60''7495'_2728
           (coe
              du_m'8804'm'43'n_3482 (coe addInt (coe (1 :: Integer)) (coe v0))))
  -- Data.Nat.Properties._<″?_
  d__'60''8243''63'__6290 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'60''8243''63'__6290 v0 v1
    = coe
        MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
        (\ v2 -> coe du_'60''7495''8658''60''8243'_6274 (coe v0) (coe v1))
        (coe du_'60''8243''8658''60''7495'_6284 (coe v0))
        (coe
           MAlonzo.Code.Relation.Nullary.Decidable.Core.d_T'63'_66
           (coe ltInt (coe v0) (coe v1)))
  -- Data.Nat.Properties._≤″?_
  d__'8804''8243''63'__6296 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8804''8243''63'__6296 v0 v1
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_true_10)
               (coe
                  MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22
                  (coe
                     MAlonzo.Code.Algebra.Definitions.RawMagma.C__'44'__40 (coe v1)
                     erased))
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe (coe d__'60''8243''63'__6290 (coe v2) (coe v1))
  -- Data.Nat.Properties._≥″?_
  d__'8805''8243''63'__6304 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8805''8243''63'__6304 v0 v1
    = coe d__'8804''8243''63'__6296 (coe v1) (coe v0)
  -- Data.Nat.Properties._>″?_
  d__'62''8243''63'__6306 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'62''8243''63'__6306 v0 v1
    = coe d__'60''8243''63'__6290 (coe v1) (coe v0)
  -- Data.Nat.Properties.≤″-irrelevant
  d_'8804''8243''45'irrelevant_6308 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8804''8243''45'irrelevant_6308 = erased
  -- Data.Nat.Properties.<″-irrelevant
  d_'60''8243''45'irrelevant_6322 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'60''8243''45'irrelevant_6322 = erased
  -- Data.Nat.Properties.>″-irrelevant
  d_'62''8243''45'irrelevant_6324 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'62''8243''45'irrelevant_6324 = erased
  -- Data.Nat.Properties.≥″-irrelevant
  d_'8805''8243''45'irrelevant_6326 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8805''8243''45'irrelevant_6326 = erased
  -- Data.Nat.Properties.≤‴⇒≤″
  d_'8804''8244''8658''8804''8243'_6332 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408 ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26
  d_'8804''8244''8658''8804''8243'_6332 ~v0 ~v1 v2
    = du_'8804''8244''8658''8804''8243'_6332 v2
  du_'8804''8244''8658''8804''8243'_6332 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408 ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26
  du_'8804''8244''8658''8804''8243'_6332 v0
    = case coe v0 of
        MAlonzo.Code.Data.Nat.Base.C_'8804''8244''45'refl_412
          -> coe
               MAlonzo.Code.Algebra.Definitions.RawMagma.C__'44'__40
               (coe (0 :: Integer)) erased
        MAlonzo.Code.Data.Nat.Base.C_'8804''8244''45'step_418 v3
          -> coe
               MAlonzo.Code.Algebra.Definitions.RawMagma.C__'44'__40
               (coe
                  addInt (coe (1 :: Integer))
                  (coe
                     MAlonzo.Code.Algebra.Definitions.RawMagma.d_quotient_36
                     (coe du_'8804''8244''8658''8804''8243'_6332 (coe v3))))
               erased
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.m≤‴m+k
  d_m'8804''8244'm'43'k_6346 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  d_m'8804''8244'm'43'k_6346 ~v0 ~v1 v2 ~v3
    = du_m'8804''8244'm'43'k_6346 v2
  du_m'8804''8244'm'43'k_6346 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  du_m'8804''8244'm'43'k_6346 v0
    = case coe v0 of
        0 -> coe MAlonzo.Code.Data.Nat.Base.C_'8804''8244''45'refl_412
        _ -> let v1 = subInt (coe v0) (coe (1 :: Integer)) in
             coe
               (coe
                  MAlonzo.Code.Data.Nat.Base.C_'8804''8244''45'step_418
                  (coe du_m'8804''8244'm'43'k_6346 (coe v1)))
  -- Data.Nat.Properties.≤″⇒≤‴
  d_'8804''8243''8658''8804''8244'_6362 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  d_'8804''8243''8658''8804''8244'_6362 ~v0 ~v1 v2
    = du_'8804''8243''8658''8804''8244'_6362 v2
  du_'8804''8243''8658''8804''8244'_6362 ::
    MAlonzo.Code.Algebra.Definitions.RawMagma.T__'8739''737'__26 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  du_'8804''8243''8658''8804''8244'_6362 v0
    = coe
        du_m'8804''8244'm'43'k_6346
        (coe
           MAlonzo.Code.Algebra.Definitions.RawMagma.d_quotient_36 (coe v0))
  -- Data.Nat.Properties.0≤‴n
  d_0'8804''8244'n_6366 ::
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  d_0'8804''8244'n_6366 v0 = coe du_m'8804''8244'm'43'k_6346 (coe v0)
  -- Data.Nat.Properties.<ᵇ⇒<‴
  d_'60''7495''8658''60''8244'_6368 ::
    Integer ->
    Integer ->
    AgdaAny -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  d_'60''7495''8658''60''8244'_6368 v0 v1 ~v2
    = du_'60''7495''8658''60''8244'_6368 v0 v1
  du_'60''7495''8658''60''8244'_6368 ::
    Integer ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  du_'60''7495''8658''60''8244'_6368 v0 v1
    = coe
        du_'8804''8243''8658''8804''8244'_6362
        (coe du_'60''7495''8658''60''8243'_6274 (coe v0) (coe v1))
  -- Data.Nat.Properties.<‴⇒<ᵇ
  d_'60''8244''8658''60''7495'_6376 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408 -> AgdaAny
  d_'60''8244''8658''60''7495'_6376 v0 ~v1 v2
    = du_'60''8244''8658''60''7495'_6376 v0 v2
  du_'60''8244''8658''60''7495'_6376 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408 -> AgdaAny
  du_'60''8244''8658''60''7495'_6376 v0 v1
    = coe
        du_'60''8243''8658''60''7495'_6284 (coe v0)
        (coe du_'8804''8244''8658''8804''8243'_6332 (coe v1))
  -- Data.Nat.Properties._<‴?_
  d__'60''8244''63'__6380 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'60''8244''63'__6380 v0 v1
    = coe
        MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
        (\ v2 -> coe du_'60''7495''8658''60''8244'_6368 (coe v0) (coe v1))
        (coe du_'60''8244''8658''60''7495'_6376 (coe v0))
        (coe
           MAlonzo.Code.Relation.Nullary.Decidable.Core.d_T'63'_66
           (coe ltInt (coe v0) (coe v1)))
  -- Data.Nat.Properties._≤‴?_
  d__'8804''8244''63'__6386 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8804''8244''63'__6386 v0 v1
    = case coe v0 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_true_10)
               (coe
                  MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22
                  (coe d_0'8804''8244'n_6366 (coe v1)))
        _ -> let v2 = subInt (coe v0) (coe (1 :: Integer)) in
             coe (coe d__'60''8244''63'__6380 (coe v2) (coe v1))
  -- Data.Nat.Properties._≥‴?_
  d__'8805''8244''63'__6394 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'8805''8244''63'__6394 v0 v1
    = coe d__'8804''8244''63'__6386 (coe v1) (coe v0)
  -- Data.Nat.Properties._>‴?_
  d__'62''8244''63'__6396 ::
    Integer ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d__'62''8244''63'__6396 v0 v1
    = coe d__'60''8244''63'__6380 (coe v1) (coe v0)
  -- Data.Nat.Properties.≤⇒≤‴
  d_'8804''8658''8804''8244'_6398 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  d_'8804''8658''8804''8244'_6398 v0 v1 ~v2
    = du_'8804''8658''8804''8244'_6398 v0 v1
  du_'8804''8658''8804''8244'_6398 ::
    Integer ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408
  du_'8804''8658''8804''8244'_6398 v0 v1
    = coe
        du_'8804''8243''8658''8804''8244'_6362
        (coe du_'8804''8658''8804''8243'_6218 (coe v0) (coe v1))
  -- Data.Nat.Properties.≤‴⇒≤
  d_'8804''8244''8658''8804'_6400 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''8244''8658''8804'_6400 v0 ~v1 v2
    = du_'8804''8244''8658''8804'_6400 v0 v2
  du_'8804''8244''8658''8804'_6400 ::
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804''8244'__408 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''8244''8658''8804'_6400 v0 v1
    = coe
        du_'8804''8243''8658''8804'_6224 (coe v0)
        (coe du_'8804''8244''8658''8804''8243'_6332 (coe v1))
  -- Data.Nat.Properties.eq?
  d_eq'63'_6406 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    () ->
    MAlonzo.Code.Function.Bundles.T_Injection_776 ->
    AgdaAny ->
    AgdaAny -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d_eq'63'_6406 ~v0 ~v1 v2 = du_eq'63'_6406 v2
  du_eq'63'_6406 ::
    MAlonzo.Code.Function.Bundles.T_Injection_776 ->
    AgdaAny ->
    AgdaAny -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  du_eq'63'_6406 v0
    = coe
        MAlonzo.Code.Relation.Nullary.Decidable.du_via'45'injection_160
        (coe v0) (coe d__'8799'__2688)
  -- Data.Nat.Properties._.anyUpTo?
  d_anyUpTo'63'_6424 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    (Integer -> ()) ->
    (Integer ->
     MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20) ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d_anyUpTo'63'_6424 ~v0 ~v1 v2 v3 = du_anyUpTo'63'_6424 v2 v3
  du_anyUpTo'63'_6424 ::
    (Integer ->
     MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20) ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  du_anyUpTo'63'_6424 v0 v1
    = case coe v1 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_false_8)
               (coe MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
        _ -> let v2 = subInt (coe v1) (coe (1 :: Integer)) in
             coe
               (let v3 = coe v0 v2 in
                coe
                  (let v4 = coe du_anyUpTo'63'_6424 (coe v0) (coe v2) in
                   coe
                     (let v5
                            = case coe v4 of
                                MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v5 v6
                                  -> coe
                                       seq (coe v5)
                                       (case coe v6 of
                                          MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v7
                                            -> case coe v7 of
                                                 MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 v8 v9
                                                   -> coe
                                                        seq (coe v9)
                                                        (coe
                                                           MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                                           (coe
                                                              MAlonzo.Code.Agda.Builtin.Bool.C_true_10)
                                                           (coe v6))
                                                 _ -> MAlonzo.RTE.mazUnreachableError
                                          _ -> MAlonzo.RTE.mazUnreachableError)
                                _ -> MAlonzo.RTE.mazUnreachableError in
                      coe
                        (case coe v3 of
                           MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v6 v7
                             -> let v8
                                      = case coe v4 of
                                          MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v8 v9
                                            -> case coe v8 of
                                                 MAlonzo.Code.Agda.Builtin.Bool.C_true_10
                                                   -> case coe v9 of
                                                        MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v10
                                                          -> case coe v10 of
                                                               MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 v11 v12
                                                                 -> coe seq (coe v12) (coe v4)
                                                               _ -> MAlonzo.RTE.mazUnreachableError
                                                        _ -> coe v5
                                                 _ -> coe v5
                                          _ -> MAlonzo.RTE.mazUnreachableError in
                                coe
                                  (if coe v6
                                     then let v9
                                                = case coe v4 of
                                                    MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v9 v10
                                                      -> case coe v9 of
                                                           MAlonzo.Code.Agda.Builtin.Bool.C_true_10
                                                             -> case coe v10 of
                                                                  MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v11
                                                                    -> case coe v11 of
                                                                         MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 v12 v13
                                                                           -> coe
                                                                                seq (coe v13) (coe v4)
                                                                         _ -> MAlonzo.RTE.mazUnreachableError
                                                                  _ -> coe v8
                                                           _ -> coe v8
                                                    _ -> MAlonzo.RTE.mazUnreachableError in
                                          coe
                                            (case coe v7 of
                                               MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v10
                                                 -> coe
                                                      MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                                      (coe v6)
                                                      (coe
                                                         MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22
                                                         (coe
                                                            MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
                                                            (coe v2)
                                                            (coe
                                                               MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32
                                                               (coe d_'8804''45'refl_2776 (coe v1))
                                                               (coe v10))))
                                               _ -> coe v9)
                                     else (case coe v4 of
                                             MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v9 v10
                                               -> if coe v9
                                                    then case coe v10 of
                                                           MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v11
                                                             -> case coe v11 of
                                                                  MAlonzo.Code.Agda.Builtin.Sigma.C__'44'__32 v12 v13
                                                                    -> coe seq (coe v13) (coe v4)
                                                                  _ -> MAlonzo.RTE.mazUnreachableError
                                                           _ -> coe v8
                                                    else (case coe v7 of
                                                            MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26
                                                              -> case coe v10 of
                                                                   MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26
                                                                     -> coe
                                                                          MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                                                          (coe v9)
                                                                          (coe
                                                                             MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
                                                                   _ -> coe v8
                                                            _ -> coe v8)
                                             _ -> MAlonzo.RTE.mazUnreachableError))
                           _ -> MAlonzo.RTE.mazUnreachableError))))
  -- Data.Nat.Properties._._.¬Pn<1+v
  d_'172'Pn'60'1'43'v_6458 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    (Integer -> ()) ->
    (Integer ->
     MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20) ->
    Integer ->
    (AgdaAny -> MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    (MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Agda.Builtin.Sigma.T_Σ_14 ->
    MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20
  d_'172'Pn'60'1'43'v_6458 = erased
  -- Data.Nat.Properties._.allUpTo?
  d_allUpTo'63'_6488 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    (Integer -> ()) ->
    (Integer ->
     MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20) ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  d_allUpTo'63'_6488 ~v0 ~v1 v2 v3 = du_allUpTo'63'_6488 v2 v3
  du_allUpTo'63'_6488 ::
    (Integer ->
     MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20) ->
    Integer -> MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20
  du_allUpTo'63'_6488 v0 v1
    = case coe v1 of
        0 -> coe
               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
               (coe MAlonzo.Code.Agda.Builtin.Bool.C_true_10)
               (coe MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 erased)
        _ -> let v2 = subInt (coe v1) (coe (1 :: Integer)) in
             coe
               (let v3 = coe v0 v2 in
                coe
                  (let v4 = coe du_allUpTo'63'_6488 (coe v0) (coe v2) in
                   coe
                     (let v5
                            = case coe v4 of
                                MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v5 v6
                                  -> coe
                                       seq (coe v5)
                                       (coe
                                          seq (coe v6)
                                          (coe
                                             MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                             (coe MAlonzo.Code.Agda.Builtin.Bool.C_false_8)
                                             (coe
                                                MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)))
                                _ -> MAlonzo.RTE.mazUnreachableError in
                      coe
                        (case coe v3 of
                           MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v6 v7
                             -> let v8
                                      = case coe v4 of
                                          MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v8 v9
                                            -> case coe v8 of
                                                 MAlonzo.Code.Agda.Builtin.Bool.C_false_8
                                                   -> case coe v9 of
                                                        MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26
                                                          -> coe
                                                               MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                                               (coe v8)
                                                               (coe
                                                                  MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
                                                        _ -> coe v5
                                                 _ -> coe v5
                                          _ -> MAlonzo.RTE.mazUnreachableError in
                                coe
                                  (if coe v6
                                     then case coe v4 of
                                            MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v9 v10
                                              -> if coe v9
                                                   then case coe v7 of
                                                          MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v11
                                                            -> case coe v10 of
                                                                 MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22 v12
                                                                   -> coe
                                                                        MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                                                        (coe v9)
                                                                        (coe
                                                                           MAlonzo.Code.Relation.Nullary.Reflects.C_of'696'_22
                                                                           (coe
                                                                              du_Pn'60'1'43'v_6520
                                                                              (coe v2) (coe v11)
                                                                              (coe v12)))
                                                                 _ -> coe v8
                                                          _ -> coe v8
                                                   else (case coe v10 of
                                                           MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26
                                                             -> coe
                                                                  MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                                                  (coe v9)
                                                                  (coe
                                                                     MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
                                                           _ -> coe v8)
                                            _ -> MAlonzo.RTE.mazUnreachableError
                                     else (let v9
                                                 = case coe v4 of
                                                     MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v9 v10
                                                       -> case coe v9 of
                                                            MAlonzo.Code.Agda.Builtin.Bool.C_false_8
                                                              -> case coe v10 of
                                                                   MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26
                                                                     -> coe
                                                                          MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                                                          (coe v9)
                                                                          (coe
                                                                             MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
                                                                   _ -> coe v8
                                                            _ -> coe v8
                                                     _ -> MAlonzo.RTE.mazUnreachableError in
                                           coe
                                             (case coe v7 of
                                                MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26
                                                  -> coe
                                                       MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32
                                                       (coe v6)
                                                       (coe
                                                          MAlonzo.Code.Relation.Nullary.Reflects.C_of'8319'_26)
                                                _ -> coe v9)))
                           _ -> MAlonzo.RTE.mazUnreachableError))))
  -- Data.Nat.Properties._._.Pn<1+v
  d_Pn'60'1'43'v_6520 ::
    MAlonzo.Code.Agda.Primitive.T_Level_18 ->
    (Integer -> ()) ->
    (Integer ->
     MAlonzo.Code.Relation.Nullary.Decidable.Core.T_Dec_20) ->
    Integer ->
    AgdaAny ->
    (Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22 -> AgdaAny) ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22 -> AgdaAny
  d_Pn'60'1'43'v_6520 ~v0 ~v1 ~v2 v3 v4 v5 v6 v7
    = du_Pn'60'1'43'v_6520 v3 v4 v5 v6 v7
  du_Pn'60'1'43'v_6520 ::
    Integer ->
    AgdaAny ->
    (Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22 -> AgdaAny) ->
    Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22 -> AgdaAny
  du_Pn'60'1'43'v_6520 v0 v1 v2 v3 v4
    = case coe v4 of
        MAlonzo.Code.Data.Nat.Base.C_s'8804's_34 v7
          -> let v8
                   = coe
                       MAlonzo.Code.Relation.Nullary.Decidable.Core.du_map'8242'_168
                       erased (\ v8 -> coe du_'8801''8658''8801''7495'_2678 (coe v3))
                       (coe
                          MAlonzo.Code.Relation.Nullary.Decidable.Core.d_T'63'_66
                          (coe eqInt (coe v3) (coe v0))) in
             coe
               (case coe v8 of
                  MAlonzo.Code.Relation.Nullary.Decidable.Core.C__because__32 v9 v10
                    -> if coe v9
                         then coe seq (coe v10) (coe v1)
                         else coe
                                seq (coe v10)
                                (coe
                                   v2 v3 (coe du_'8804''8743''8802''8658''60'_2918 (coe v0) (coe v7)))
                  _ -> MAlonzo.RTE.mazUnreachableError)
        _ -> MAlonzo.RTE.mazUnreachableError
  -- Data.Nat.Properties.∀[m≤n⇒m≢o]⇒o<n
  d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'o'60'n_6546 ::
    Integer ->
    Integer ->
    (Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8704''91'm'8804'n'8658'm'8802'o'93''8658'o'60'n_6546 v0 v1 v2
    = coe
        du_'8704''91'm'8804'n'8658'm'8802'o'93''8658'n'60'o_3132 v0 v1
  -- Data.Nat.Properties.∀[m<n⇒m≢o]⇒o≤n
  d_'8704''91'm'60'n'8658'm'8802'o'93''8658'o'8804'n_6554 ::
    Integer ->
    Integer ->
    (Integer ->
     MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
     MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8704''91'm'60'n'8658'm'8802'o'93''8658'o'8804'n_6554 v0 v1 v2
    = coe
        du_'8704''91'm'60'n'8658'm'8802'o'93''8658'n'8804'o_3160 v0 v1
  -- Data.Nat.Properties.*-+-isSemiring
  d_'42''45''43''45'isSemiring_6556 ::
    MAlonzo.Code.Algebra.Structures.T_IsSemiring_1570
  d_'42''45''43''45'isSemiring_6556
    = coe d_'43''45''42''45'isSemiring_3754
  -- Data.Nat.Properties.*-+-isCommutativeSemiring
  d_'42''45''43''45'isCommutativeSemiring_6558 ::
    MAlonzo.Code.Algebra.Structures.T_IsCommutativeSemiring_1678
  d_'42''45''43''45'isCommutativeSemiring_6558
    = coe d_'43''45''42''45'isCommutativeSemiring_3756
  -- Data.Nat.Properties.*-+-semiring
  d_'42''45''43''45'semiring_6560 ::
    MAlonzo.Code.Algebra.Bundles.T_Semiring_2280
  d_'42''45''43''45'semiring_6560
    = coe d_'43''45''42''45'semiring_3768
  -- Data.Nat.Properties.*-+-commutativeSemiring
  d_'42''45''43''45'commutativeSemiring_6562 ::
    MAlonzo.Code.Algebra.Bundles.T_CommutativeSemiring_2446
  d_'42''45''43''45'commutativeSemiring_6562
    = coe d_'43''45''42''45'commutativeSemiring_3770
  -- Data.Nat.Properties.∣m+n-m+o∣≡∣n-o|
  d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'124'_6564 ::
    Integer ->
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8739'm'43'n'45'm'43'o'8739''8801''8739'n'45'o'124'_6564 = erased
  -- Data.Nat.Properties.m≤n⇒n⊔m≡n
  d_m'8804'n'8658'n'8852'm'8801'n_6566 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8804'n'8658'n'8852'm'8801'n_6566 = erased
  -- Data.Nat.Properties.m≤n⇒n⊓m≡m
  d_m'8804'n'8658'n'8851'm'8801'm_6568 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_m'8804'n'8658'n'8851'm'8801'm_6568 = erased
  -- Data.Nat.Properties.n⊔m≡m⇒n≤m
  d_n'8852'm'8801'm'8658'n'8804'm_6570 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_n'8852'm'8801'm'8658'n'8804'm_6570
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'y'8658'y'8804'x_3100
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.n⊔m≡n⇒m≤n
  d_n'8852'm'8801'n'8658'm'8804'n_6572 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_n'8852'm'8801'n'8658'm'8804'n_6572
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8776'x'8658'x'8804'y_3068
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.n≤m⊔n
  d_n'8804'm'8852'n_6574 ::
    Integer -> Integer -> MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_n'8804'm'8852'n_6574
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_x'8851'y'8804'y_2834
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊔-least
  d_'8852''45'least_6576 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8852''45'least_6576
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_3278
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-greatest
  d_'8851''45'greatest_6578 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'greatest_6578
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_3278
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊔-pres-≤m
  d_'8852''45'pres'45''8804'm_6580 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8852''45'pres'45''8804'm_6580
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8852''45'operator_4440 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_3278
              (coe
                 MAlonzo.Code.Relation.Binary.Construct.Flip.EqAndOrd.du_totalPreorder_746
                 (coe v0))
              (coe
                 MAlonzo.Code.Algebra.Construct.NaturalChoice.Base.du_MaxOp'8658'MinOp_174
                 (coe v1))))
  -- Data.Nat.Properties.⊓-pres-m≤
  d_'8851''45'pres'45'm'8804'_6582 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8851''45'pres'45'm'8804'_6582
    = let v0 = d_'8804''45'totalPreorder_2822 in
      coe
        (let v1 = d_'8851''45'operator_4438 in
         coe
           (coe
              MAlonzo.Code.Algebra.Construct.NaturalChoice.MinOp.du_'8851''45'glb_3278
              (coe v0) (coe v1)))
  -- Data.Nat.Properties.⊔-abs-⊓
  d_'8852''45'abs'45''8851'_6584 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8852''45'abs'45''8851'_6584 = erased
  -- Data.Nat.Properties.⊓-abs-⊔
  d_'8851''45'abs'45''8852'_6586 ::
    Integer ->
    Integer -> MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_'8851''45'abs'45''8852'_6586 = erased
  -- Data.Nat.Properties.suc[pred[n]]≡n
  d_suc'91'pred'91'n'93''93''8801'n_6588 ::
    Integer ->
    (MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12 ->
     MAlonzo.Code.Data.Irrelevant.T_Irrelevant_20) ->
    MAlonzo.Code.Agda.Builtin.Equality.T__'8801'__12
  d_suc'91'pred'91'n'93''93''8801'n_6588 = erased
  -- Data.Nat.Properties.≤-step
  d_'8804''45'step_6594 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''45'step_6594 ~v0 ~v1 v2 = du_'8804''45'step_6594 v2
  du_'8804''45'step_6594 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''45'step_6594 v0 = coe v0
  -- Data.Nat.Properties.≤-stepsˡ
  d_'8804''45'steps'737'_6596 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''45'steps'737'_6596 ~v0 ~v1 ~v2 v3
    = du_'8804''45'steps'737'_6596 v3
  du_'8804''45'steps'737'_6596 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''45'steps'737'_6596 v0 = coe v0
  -- Data.Nat.Properties.≤-stepsʳ
  d_'8804''45'steps'691'_6598 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'8804''45'steps'691'_6598 ~v0 ~v1 ~v2 v3
    = du_'8804''45'steps'691'_6598 v3
  du_'8804''45'steps'691'_6598 ::
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  du_'8804''45'steps'691'_6598 v0 = coe v0
  -- Data.Nat.Properties.<-step
  d_'60''45'step_6600 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''45'step_6600 v0 v1 v2
    = coe du_m'60'n'8658'm'60'1'43'n_3062 v2
  -- Data.Nat.Properties.pred-mono
  d_pred'45'mono_6602 ::
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_pred'45'mono_6602 v0 v1 v2
    = coe du_pred'45'mono'45''8804'_5676 v0 v2
  -- Data.Nat.Properties.<-transʳ
  d_'60''45'trans'691'_6604 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''45'trans'691'_6604 v0 v1 v2 v3 v4
    = coe du_'8804''45''60''45'trans_2986 v3 v4
  -- Data.Nat.Properties.<-transˡ
  d_'60''45'trans'737'_6606 ::
    Integer ->
    Integer ->
    Integer ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22 ->
    MAlonzo.Code.Data.Nat.Base.T__'8804'__22
  d_'60''45'trans'737'_6606 v0 v1 v2 v3 v4
    = coe du_'60''45''8804''45'trans_2992 v3 v4
  