%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Multiset_Ordering_NPC/0003_Multiset_Ordering_NP_Hard/prob_00266_013080__14210254_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1840 ( 710 unt; 568 typ;   0 def)
%            Number of atoms       : 3143 (1414 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 10464 ( 242   ~;  56   |; 195   &;8800   @)
%                                         (   0 <=>;1171  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Number of types       :  113 ( 112 usr)
%            Number of type conns  : 1361 (1361   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  459 ( 456 usr;  27 con; 0-3 aty)
%            Number of variables   : 3368 ( 250   ^;3007   !; 111   ?;3368   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:30:46.349
%------------------------------------------------------------------------------
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thf(ty_n_t__Int__Oint,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (456)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Int__Oint,type,
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thf(sy_c_Groups__List_Omonoid__list_OF_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_v_vs____,type,
    vs: list_a ).

thf(sy_v_xs,type,
    xs: list_S5253404628753938141on_nat ).

thf(sy_v_ys,type,
    ys: list_S5253404628753938141on_nat ).

% Relevant facts (1266)
thf(fact_0_assms_I2_J,axiom,
    ( ( propos8254374689592995604_cnf_a @ cnf )
    = s2 ) ).

% assms(2)
thf(fact_1_length__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_2_length__map,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( size_size_list_nat @ ( map_a_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_3_length__map,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( size_size_list_a @ ( map_nat_a @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_4_length__map,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( size_size_list_a @ ( map_a_a @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_5_length__map,axiom,
    ! [F: nat > product_prod_a_o,Xs: list_nat] :
      ( ( size_s1727999245659094451od_a_o @ ( map_na1257269399097651883od_a_o @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_6_length__map,axiom,
    ! [F: a > product_prod_a_o,Xs: list_a] :
      ( ( size_s1727999245659094451od_a_o @ ( map_a_1137562357930033543od_a_o @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_7_length__map,axiom,
    ! [F: product_prod_a_o > nat,Xs: list_P3196869577824546119od_a_o] :
      ( ( size_size_list_nat @ ( map_Pr3957037749107795149_o_nat @ F @ Xs ) )
      = ( size_s1727999245659094451od_a_o @ Xs ) ) ).

% length_map
thf(fact_8_length__map,axiom,
    ! [F: product_prod_a_o > a,Xs: list_P3196869577824546119od_a_o] :
      ( ( size_size_list_a @ ( map_Pr8253109945893590849_a_o_a @ F @ Xs ) )
      = ( size_s1727999245659094451od_a_o @ Xs ) ) ).

% length_map
thf(fact_9_length__map,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,Xs: list_nat] :
      ( ( size_s3379185846251495609od_a_o @ ( map_na764632570197019441od_a_o @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_10_length__map,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,Xs: list_a] :
      ( ( size_s3379185846251495609od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_11_lcnf,axiom,
    ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ cnf ) @ s2 ).

% lcnf
thf(fact_12_calculation,axiom,
    ( ( size_s7490768528479217773on_nat @ s )
    = ( groups4561878855575611511st_nat @ ( map_li381573783031109843_o_nat @ size_s1727999245659094451od_a_o @ cnf ) ) ) ).

% calculation
thf(fact_13_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_14_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_15_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_16_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_17_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_18_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_a_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_19_map__eq__imp__length__eq,axiom,
    ! [F: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
      ( ( ( map_a_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_20_map__eq__imp__length__eq,axiom,
    ! [F: product_prod_a_o > nat,Xs: list_P3196869577824546119od_a_o,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Pr3957037749107795149_o_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_21_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: product_prod_a_o > nat,Ys: list_P3196869577824546119od_a_o] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Pr3957037749107795149_o_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s1727999245659094451od_a_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_22_map__eq__imp__length__eq,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o,G: nat > nat,Ys: list_nat] :
      ( ( ( map_li381573783031109843_o_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_23_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: list_P3196869577824546119od_a_o > nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_li381573783031109843_o_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_24_map__eq__imp__length__eq,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_25_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: sum_su4907927656612468301on_nat > nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Su4809418691747827683at_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_26_map__eq__imp__length__eq,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,G: a > sum_su4907927656612468301on_nat,Ys: list_a] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_27_vs__def,axiom,
    ( vs
    = ( multis4741186478440321083_cnf_a @ cnf ) ) ).

% vs_def
thf(fact_28_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_29_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_30_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_31_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_32_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_33_lvs,axiom,
    ord_less_eq_nat @ ( size_size_list_a @ vs ) @ s2 ).

% lvs
thf(fact_34_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_35_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_36_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_37_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_38_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_39_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_40_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_41_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_42_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_43_ord__eq__le__subst,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_44_ord__eq__le__subst,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_45_ord__eq__le__subst,axiom,
    ! [A: int,F: int > int,B: int,C: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_46_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_47_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_48_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_49_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_50_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_51_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_52_order__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_53_order__subst2,axiom,
    ! [A: int,B: int,F: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_54_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_55_order__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_56_order__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_57_order__subst1,axiom,
    ! [A: int,F: int > int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_58_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_59_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_60_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_61_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_62_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_63_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_64_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_65_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_66_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_67_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_68_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_69_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: int,B3: int] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_70_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_71_order__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_eq_int @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_72_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_73_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_74_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_75_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_76_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_77_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_78_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_79_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_80_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [X3: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_81_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
    = ( ^ [X3: int,Y4: int] :
          ( ( ord_less_eq_int @ X3 @ Y4 )
          & ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_82_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_83_le__cases3,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z2 ) )
       => ( ( ( ord_less_eq_int @ X @ Z2 )
           => ~ ( ord_less_eq_int @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_int @ Z2 @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z2 )
               => ~ ( ord_less_eq_int @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_int @ Z2 @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_84_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_85_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_86_size__neq__size__imp__neq,axiom,
    ! [X: list_P3196869577824546119od_a_o,Y: list_P3196869577824546119od_a_o] :
      ( ( ( size_s1727999245659094451od_a_o @ X )
       != ( size_s1727999245659094451od_a_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_87_size__neq__size__imp__neq,axiom,
    ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ X )
       != ( size_s5686234912572693873on_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_88_size__neq__size__imp__neq,axiom,
    ! [X: list_l8336264072933094989od_a_o,Y: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ X )
       != ( size_s3379185846251495609od_a_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_89_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_90_size__neq__size__imp__neq,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( ( size_size_list_a @ X )
       != ( size_size_list_a @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_91_size__neq__size__imp__neq,axiom,
    ! [X: list_l8999537460041914605on_nat,Y: list_l8999537460041914605on_nat] :
      ( ( ( size_s650155532264187265on_nat @ X )
       != ( size_s650155532264187265on_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_92_size__neq__size__imp__neq,axiom,
    ! [X: list_l1726306699111145479on_nat,Y: list_l1726306699111145479on_nat] :
      ( ( ( size_s129639936549468787on_nat @ X )
       != ( size_s129639936549468787on_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_93_neq__if__length__neq,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o] :
      ( ( ( size_s1727999245659094451od_a_o @ Xs )
       != ( size_s1727999245659094451od_a_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_94_neq__if__length__neq,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
       != ( size_s5686234912572693873on_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_95_neq__if__length__neq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
       != ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_96_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_97_neq__if__length__neq,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
       != ( size_size_list_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_98_neq__if__length__neq,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
       != ( size_s650155532264187265on_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_99_neq__if__length__neq,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( ( size_s129639936549468787on_nat @ Xs )
       != ( size_s129639936549468787on_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_100_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_P3196869577824546119od_a_o] :
      ( ( size_s1727999245659094451od_a_o @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_101_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_S5253404628753938141on_nat] :
      ( ( size_s5686234912572693873on_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_102_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_103_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_104_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_a] :
      ( ( size_size_list_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_105_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_l8999537460041914605on_nat] :
      ( ( size_s650155532264187265on_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_106_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_l1726306699111145479on_nat] :
      ( ( size_s129639936549468787on_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_107_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B ) )
       => ? [X2: nat] :
            ( ( P @ X2 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_108_size__cnf__def,axiom,
    ( propos8254374689592995604_cnf_a
    = ( ^ [Cnf: list_l8336264072933094989od_a_o] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li381573783031109843_o_nat @ size_s1727999245659094451od_a_o @ Cnf ) ) @ ( size_s3379185846251495609od_a_o @ Cnf ) ) ) ) ).

% size_cnf_def
thf(fact_109_map__eq__map__tailrec,axiom,
    map_na8715451351012060651on_nat = map_ta4085101254008040225on_nat ).

% map_eq_map_tailrec
thf(fact_110_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_111_map__eq__map__tailrec,axiom,
    map_a_2711072540680412167on_nat = map_ta9187454088855476029on_nat ).

% map_eq_map_tailrec
thf(fact_112_map__eq__map__tailrec,axiom,
    map_a_7431629901395152583on_nat = map_ta1296617008540906257on_nat ).

% map_eq_map_tailrec
thf(fact_113_map__eq__map__tailrec,axiom,
    map_a_3887198048842487703on_nat = map_ta5901319884867207117on_nat ).

% map_eq_map_tailrec
thf(fact_114_map__eq__map__tailrec,axiom,
    map_nat_a = map_tailrec_nat_a ).

% map_eq_map_tailrec
thf(fact_115_map__eq__map__tailrec,axiom,
    map_a_1137562357930033543od_a_o = map_ta3598503324509809361od_a_o ).

% map_eq_map_tailrec
thf(fact_116_map__eq__map__tailrec,axiom,
    map_a_5058921920454310925od_a_o = map_ta7247006886706624599od_a_o ).

% map_eq_map_tailrec
thf(fact_117_map__eq__map__tailrec,axiom,
    map_a_nat = map_tailrec_a_nat ).

% map_eq_map_tailrec
thf(fact_118_map__eq__map__tailrec,axiom,
    map_a_a = map_tailrec_a_a ).

% map_eq_map_tailrec
thf(fact_119_Greatest__equality,axiom,
    ! [P: int > $o,X: int] :
      ( ( P @ X )
     => ( ! [Y2: int] :
            ( ( P @ Y2 )
           => ( ord_less_eq_int @ Y2 @ X ) )
       => ( ( order_Greatest_int @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_120_Greatest__equality,axiom,
    ! [P: nat > $o,X: nat] :
      ( ( P @ X )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ X ) )
       => ( ( order_Greatest_nat @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_121_GreatestI2__order,axiom,
    ! [P: int > $o,X: int,Q: int > $o] :
      ( ( P @ X )
     => ( ! [Y2: int] :
            ( ( P @ Y2 )
           => ( ord_less_eq_int @ Y2 @ X ) )
       => ( ! [X2: int] :
              ( ( P @ X2 )
             => ( ! [Y5: int] :
                    ( ( P @ Y5 )
                   => ( ord_less_eq_int @ Y5 @ X2 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_122_GreatestI2__order,axiom,
    ! [P: nat > $o,X: nat,Q: nat > $o] :
      ( ( P @ X )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ X ) )
       => ( ! [X2: nat] :
              ( ( P @ X2 )
             => ( ! [Y5: nat] :
                    ( ( P @ Y5 )
                   => ( ord_less_eq_nat @ Y5 @ X2 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_123_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M2: nat] :
      ( ( P @ X )
     => ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( ord_less_eq_nat @ X2 @ M2 ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X4: nat] :
                    ( ( P @ X4 )
                   => ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_124__092_060open_062length_Axs_A_092_060le_062_Alength_Avs_A_L_Alength_Avs_092_060close_062,axiom,
    ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ xs ) @ ( plus_plus_nat @ ( size_size_list_a @ vs ) @ ( size_size_list_a @ vs ) ) ).

% \<open>length xs \<le> length vs + length vs\<close>
thf(fact_125_le__rel__bool__arg__iff,axiom,
    ( ord_less_eq_o_nat
    = ( ^ [X5: $o > nat,Y6: $o > nat] :
          ( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_126_le__rel__bool__arg__iff,axiom,
    ( ord_less_eq_o_int
    = ( ^ [X5: $o > int,Y6: $o > int] :
          ( ( ord_less_eq_int @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_less_eq_int @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_127_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_128_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_129_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_130_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_131_count__list__map__ge,axiom,
    ! [Xs: list_nat,X: nat,F: nat > list_P9127870862658512897on_nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( count_6856440285005804925on_nat @ ( map_na8715451351012060651on_nat @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_132_count__list__map__ge,axiom,
    ! [Xs: list_nat,X: nat,F: nat > nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( count_list_nat @ ( map_nat_nat @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_133_count__list__map__ge,axiom,
    ! [Xs: list_a,X: a,F: a > sum_su4907927656612468301on_nat] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_3605398483369874897on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_134_count__list__map__ge,axiom,
    ! [Xs: list_a,X: a,F: a > list_P9127870862658512897on_nat] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_6856440285005804925on_nat @ ( map_a_7431629901395152583on_nat @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_135_count__list__map__ge,axiom,
    ! [Xs: list_a,X: a,F: a > list_S5253404628753938141on_nat] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_3480697274282583137on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_136_count__list__map__ge,axiom,
    ! [Xs: list_nat,X: nat,F: nat > a] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( count_list_a @ ( map_nat_a @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_137_count__list__map__ge,axiom,
    ! [Xs: list_a,X: a,F: a > product_prod_a_o] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_843742752111076157od_a_o @ ( map_a_1137562357930033543od_a_o @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_138_count__list__map__ge,axiom,
    ! [Xs: list_a,X: a,F: a > list_P3196869577824546119od_a_o] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_5344270436093135555od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_139_count__list__map__ge,axiom,
    ! [Xs: list_a,X: a,F: a > nat] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_list_nat @ ( map_a_nat @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_140_count__list__map__ge,axiom,
    ! [Xs: list_a,X: a,F: a > a] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_list_a @ ( map_a_a @ F @ Xs ) @ ( F @ X ) ) ) ).

% count_list_map_ge
thf(fact_141_count__le__length,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,X: product_prod_a_o] : ( ord_less_eq_nat @ ( count_843742752111076157od_a_o @ Xs @ X ) @ ( size_s1727999245659094451od_a_o @ Xs ) ) ).

% count_le_length
thf(fact_142_count__le__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] : ( ord_less_eq_nat @ ( count_3605398483369874897on_nat @ Xs @ X ) @ ( size_s5686234912572693873on_nat @ Xs ) ) ).

% count_le_length
thf(fact_143_count__le__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] : ( ord_less_eq_nat @ ( count_5344270436093135555od_a_o @ Xs @ X ) @ ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% count_le_length
thf(fact_144_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).

% count_le_length
thf(fact_145_count__le__length,axiom,
    ! [Xs: list_a,X: a] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( size_size_list_a @ Xs ) ) ).

% count_le_length
thf(fact_146_count__le__length,axiom,
    ! [Xs: list_l8999537460041914605on_nat,X: list_S5253404628753938141on_nat] : ( ord_less_eq_nat @ ( count_3480697274282583137on_nat @ Xs @ X ) @ ( size_s650155532264187265on_nat @ Xs ) ) ).

% count_le_length
thf(fact_147_count__le__length,axiom,
    ! [Xs: list_l1726306699111145479on_nat,X: list_P9127870862658512897on_nat] : ( ord_less_eq_nat @ ( count_6856440285005804925on_nat @ Xs @ X ) @ ( size_s129639936549468787on_nat @ Xs ) ) ).

% count_le_length
thf(fact_148_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_149_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_150_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_151_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_152_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_153_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_154_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_155_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_156_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_157_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_158_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_159_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
        ? [K2: nat] :
          ( N3
          = ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_160_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_161_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_162_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_163_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_164_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_165_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_166_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_167_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_168_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_169_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_170_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_171_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_172_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_173_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_174_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_175_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_176_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_177_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_178_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_179_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_180_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_181_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_182_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_183_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_184_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_185_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_186_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_187_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_188_group__cancel_Oadd1,axiom,
    ! [A4: nat,K: nat,A: nat,B: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A4 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_189_group__cancel_Oadd1,axiom,
    ! [A4: int,K: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A4 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_190_group__cancel_Oadd2,axiom,
    ! [B4: nat,K: nat,B: nat,A: nat] :
      ( ( B4
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B4 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_191_group__cancel_Oadd2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B4 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_192_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_193_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_194_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_195_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_196_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_197_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_198_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_199_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.left_commute
thf(fact_200_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_201_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_202_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_203_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_204_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_205_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_206_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_207_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_208_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
        ? [C3: nat] :
          ( B2
          = ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_209_gen__length__def,axiom,
    ( gen_le449265310187613268od_a_o
    = ( ^ [N3: nat,Xs3: list_P3196869577824546119od_a_o] : ( plus_plus_nat @ N3 @ ( size_s1727999245659094451od_a_o @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_210_gen__length__def,axiom,
    ( gen_le4617574567093687674on_nat
    = ( ^ [N3: nat,Xs3: list_S5253404628753938141on_nat] : ( plus_plus_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_211_gen__length__def,axiom,
    ( gen_le5767993353563561050od_a_o
    = ( ^ [N3: nat,Xs3: list_l8336264072933094989od_a_o] : ( plus_plus_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_212_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N3: nat,Xs3: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_213_gen__length__def,axiom,
    ( gen_length_a
    = ( ^ [N3: nat,Xs3: list_a] : ( plus_plus_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_214_gen__length__def,axiom,
    ( gen_le9217917933129907082on_nat
    = ( ^ [N3: nat,Xs3: list_l8999537460041914605on_nat] : ( plus_plus_nat @ N3 @ ( size_s650155532264187265on_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_215_gen__length__def,axiom,
    ( gen_le496107124138664724on_nat
    = ( ^ [N3: nat,Xs3: list_l1726306699111145479on_nat] : ( plus_plus_nat @ N3 @ ( size_s129639936549468787on_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_216_length__splice,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o] :
      ( ( size_s1727999245659094451od_a_o @ ( splice3762031825553873818od_a_o @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s1727999245659094451od_a_o @ Xs ) @ ( size_s1727999245659094451od_a_o @ Ys ) ) ) ).

% length_splice
thf(fact_217_length__splice,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( splice6264165949443683764on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% length_splice
thf(fact_218_length__splice,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( splice1611073882655813792od_a_o @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% length_splice
thf(fact_219_length__splice,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( splice_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_splice
thf(fact_220_length__splice,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( size_size_list_a @ ( splice_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).

% length_splice
thf(fact_221_length__splice,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( size_s650155532264187265on_nat @ ( splice3260574510159476420on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s650155532264187265on_nat @ Xs ) @ ( size_s650155532264187265on_nat @ Ys ) ) ) ).

% length_splice
thf(fact_222_length__splice,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( size_s129639936549468787on_nat @ ( splice1185315791880949594on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s129639936549468787on_nat @ Xs ) @ ( size_s129639936549468787on_nat @ Ys ) ) ) ).

% length_splice
thf(fact_223_size__list__conv__sum__list,axiom,
    ( size_l3983519830990705097od_a_o
    = ( ^ [F2: product_prod_a_o > nat,Xs3: list_P3196869577824546119od_a_o] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_Pr3957037749107795149_o_nat @ F2 @ Xs3 ) ) @ ( size_s1727999245659094451od_a_o @ Xs3 ) ) ) ) ).

% size_list_conv_sum_list
thf(fact_224_size__list__conv__sum__list,axiom,
    ( size_l7723201268554483269on_nat
    = ( ^ [F2: sum_su4907927656612468301on_nat > nat,Xs3: list_S5253404628753938141on_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_Su4809418691747827683at_nat @ F2 @ Xs3 ) ) @ ( size_s5686234912572693873on_nat @ Xs3 ) ) ) ) ).

% size_list_conv_sum_list
thf(fact_225_size__list__conv__sum__list,axiom,
    ( size_l4174101860454368079od_a_o
    = ( ^ [F2: list_P3196869577824546119od_a_o > nat,Xs3: list_l8336264072933094989od_a_o] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li381573783031109843_o_nat @ F2 @ Xs3 ) ) @ ( size_s3379185846251495609od_a_o @ Xs3 ) ) ) ) ).

% size_list_conv_sum_list
thf(fact_226_size__list__conv__sum__list,axiom,
    ( size_list_nat
    = ( ^ [F2: nat > nat,Xs3: list_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F2 @ Xs3 ) ) @ ( size_size_list_nat @ Xs3 ) ) ) ) ).

% size_list_conv_sum_list
thf(fact_227_size__list__conv__sum__list,axiom,
    ( size_list_a
    = ( ^ [F2: a > nat,Xs3: list_a] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_a_nat @ F2 @ Xs3 ) ) @ ( size_size_list_a @ Xs3 ) ) ) ) ).

% size_list_conv_sum_list
thf(fact_228_size__list__conv__sum__list,axiom,
    ( size_l7481798006778809045on_nat
    = ( ^ [F2: list_S5253404628753938141on_nat > nat,Xs3: list_l8999537460041914605on_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li2142278717278591475at_nat @ F2 @ Xs3 ) ) @ ( size_s650155532264187265on_nat @ Xs3 ) ) ) ) ).

% size_list_conv_sum_list
thf(fact_229_size__list__conv__sum__list,axiom,
    ( size_l1756674476620258825on_nat
    = ( ^ [F2: list_P9127870862658512897on_nat > nat,Xs3: list_l1726306699111145479on_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li8334353970798598029at_nat @ F2 @ Xs3 ) ) @ ( size_s129639936549468787on_nat @ Xs3 ) ) ) ) ).

% size_list_conv_sum_list
thf(fact_230_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_231_length__remdups__leq,axiom,
    ! [Xs: list_P3196869577824546119od_a_o] : ( ord_less_eq_nat @ ( size_s1727999245659094451od_a_o @ ( remdup8097019607593547996od_a_o @ Xs ) ) @ ( size_s1727999245659094451od_a_o @ Xs ) ) ).

% length_remdups_leq
thf(fact_232_length__remdups__leq,axiom,
    ! [Xs: list_S5253404628753938141on_nat] : ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ ( remdup2859832054282949362on_nat @ Xs ) ) @ ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_remdups_leq
thf(fact_233_length__remdups__leq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] : ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ ( remdup7454993672696595682od_a_o @ Xs ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_remdups_leq
thf(fact_234_length__remdups__leq,axiom,
    ! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_remdups_leq
thf(fact_235_length__remdups__leq,axiom,
    ! [Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( remdups_a @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).

% length_remdups_leq
thf(fact_236_length__remdups__leq,axiom,
    ! [Xs: list_l8999537460041914605on_nat] : ( ord_less_eq_nat @ ( size_s650155532264187265on_nat @ ( remdup2030676518929194242on_nat @ Xs ) ) @ ( size_s650155532264187265on_nat @ Xs ) ) ).

% length_remdups_leq
thf(fact_237_length__remdups__leq,axiom,
    ! [Xs: list_l1726306699111145479on_nat] : ( ord_less_eq_nat @ ( size_s129639936549468787on_nat @ ( remdup4059826874966911900on_nat @ Xs ) ) @ ( size_s129639936549468787on_nat @ Xs ) ) ).

% length_remdups_leq
thf(fact_238_count__list__append,axiom,
    ! [Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( count_list_nat @ ( append_nat @ Xs @ Ys ) @ X )
      = ( plus_plus_nat @ ( count_list_nat @ Xs @ X ) @ ( count_list_nat @ Ys @ X ) ) ) ).

% count_list_append
thf(fact_239_count__list__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
      ( ( count_3605398483369874897on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ X )
      = ( plus_plus_nat @ ( count_3605398483369874897on_nat @ Xs @ X ) @ ( count_3605398483369874897on_nat @ Ys @ X ) ) ) ).

% count_list_append
thf(fact_240_count__list__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat] :
      ( ( count_9157224493928226551on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ X )
      = ( plus_plus_nat @ ( count_9157224493928226551on_nat @ Xs @ X ) @ ( count_9157224493928226551on_nat @ Ys @ X ) ) ) ).

% count_list_append
thf(fact_241_length__removeAll__less__eq,axiom,
    ! [X: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] : ( ord_less_eq_nat @ ( size_s1727999245659094451od_a_o @ ( remove6062766054978212813od_a_o @ X @ Xs ) ) @ ( size_s1727999245659094451od_a_o @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_242_length__removeAll__less__eq,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] : ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ ( remove433775707702860097on_nat @ X @ Xs ) ) @ ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_243_length__removeAll__less__eq,axiom,
    ! [X: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] : ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ ( remove3990864149796893523od_a_o @ X @ Xs ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_244_length__removeAll__less__eq,axiom,
    ! [X: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_245_length__removeAll__less__eq,axiom,
    ! [X: a,Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( removeAll_a @ X @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_246_length__removeAll__less__eq,axiom,
    ! [X: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat] : ( ord_less_eq_nat @ ( size_s650155532264187265on_nat @ ( remove4496858567263678417on_nat @ X @ Xs ) ) @ ( size_s650155532264187265on_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_247_length__removeAll__less__eq,axiom,
    ! [X: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat] : ( ord_less_eq_nat @ ( size_s129639936549468787on_nat @ ( remove7054922355537174029on_nat @ X @ Xs ) ) @ ( size_s129639936549468787on_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_248_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C )
      = ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_249_append_Oassoc,axiom,
    ! [A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,C: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ ( append3523812819975775986on_nat @ A @ B ) @ C )
      = ( append3523812819975775986on_nat @ A @ ( append3523812819975775986on_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_250_append_Oassoc,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,C: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( append2865821678206098582on_nat @ A @ B ) @ C )
      = ( append2865821678206098582on_nat @ A @ ( append2865821678206098582on_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_251_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_252_append__assoc,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ Zs )
      = ( append3523812819975775986on_nat @ Xs @ ( append3523812819975775986on_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_253_append__assoc,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ Zs )
      = ( append2865821678206098582on_nat @ Xs @ ( append2865821678206098582on_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_254_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_255_append__same__eq,axiom,
    ! [Ys: list_S5253404628753938141on_nat,Xs: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys @ Xs )
        = ( append3523812819975775986on_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_256_append__same__eq,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Ys @ Xs )
        = ( append2865821678206098582on_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_257_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_258_same__append__eq,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys )
        = ( append3523812819975775986on_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_259_same__append__eq,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys )
        = ( append2865821678206098582on_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_260_append__eq__append__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Us: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
      ( ( ( ( size_s7490768528479217773on_nat @ Xs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
        | ( ( size_s7490768528479217773on_nat @ Us )
          = ( size_s7490768528479217773on_nat @ Vs ) ) )
     => ( ( ( append2865821678206098582on_nat @ Xs @ Us )
          = ( append2865821678206098582on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_261_append__eq__append__conv,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o,Us: list_P3196869577824546119od_a_o,Vs: list_P3196869577824546119od_a_o] :
      ( ( ( ( size_s1727999245659094451od_a_o @ Xs )
          = ( size_s1727999245659094451od_a_o @ Ys ) )
        | ( ( size_s1727999245659094451od_a_o @ Us )
          = ( size_s1727999245659094451od_a_o @ Vs ) ) )
     => ( ( ( append2462165029785658588od_a_o @ Xs @ Us )
          = ( append2462165029785658588od_a_o @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_262_append__eq__append__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        | ( ( size_s5686234912572693873on_nat @ Us )
          = ( size_s5686234912572693873on_nat @ Vs ) ) )
     => ( ( ( append3523812819975775986on_nat @ Xs @ Us )
          = ( append3523812819975775986on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_263_append__eq__append__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Us: list_l8336264072933094989od_a_o,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        | ( ( size_s3379185846251495609od_a_o @ Us )
          = ( size_s3379185846251495609od_a_o @ Vs ) ) )
     => ( ( ( append2159120467191820514od_a_o @ Xs @ Us )
          = ( append2159120467191820514od_a_o @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_264_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_265_append__eq__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
      ( ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        | ( ( size_size_list_a @ Us )
          = ( size_size_list_a @ Vs ) ) )
     => ( ( ( append_a @ Xs @ Us )
          = ( append_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_266_append__eq__append__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,Us: list_l8999537460041914605on_nat,Vs: list_l8999537460041914605on_nat] :
      ( ( ( ( size_s650155532264187265on_nat @ Xs )
          = ( size_s650155532264187265on_nat @ Ys ) )
        | ( ( size_s650155532264187265on_nat @ Us )
          = ( size_s650155532264187265on_nat @ Vs ) ) )
     => ( ( ( append6174841615030841602on_nat @ Xs @ Us )
          = ( append6174841615030841602on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_267_append__eq__append__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Us: list_l1726306699111145479on_nat,Vs: list_l1726306699111145479on_nat] :
      ( ( ( ( size_s129639936549468787on_nat @ Xs )
          = ( size_s129639936549468787on_nat @ Ys ) )
        | ( ( size_s129639936549468787on_nat @ Us )
          = ( size_s129639936549468787on_nat @ Vs ) ) )
     => ( ( ( append3839213209832495004on_nat @ Xs @ Us )
          = ( append3839213209832495004on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_268_map__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_269_map__append,axiom,
    ! [F: nat > a,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_a @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_a @ ( map_nat_a @ F @ Xs ) @ ( map_nat_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_270_map__append,axiom,
    ! [F: a > nat,Xs: list_a,Ys: list_a] :
      ( ( map_a_nat @ F @ ( append_a @ Xs @ Ys ) )
      = ( append_nat @ ( map_a_nat @ F @ Xs ) @ ( map_a_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_271_map__append,axiom,
    ! [F: a > a,Xs: list_a,Ys: list_a] :
      ( ( map_a_a @ F @ ( append_a @ Xs @ Ys ) )
      = ( append_a @ ( map_a_a @ F @ Xs ) @ ( map_a_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_272_map__append,axiom,
    ! [F: a > product_prod_a_o,Xs: list_a,Ys: list_a] :
      ( ( map_a_1137562357930033543od_a_o @ F @ ( append_a @ Xs @ Ys ) )
      = ( append2462165029785658588od_a_o @ ( map_a_1137562357930033543od_a_o @ F @ Xs ) @ ( map_a_1137562357930033543od_a_o @ F @ Ys ) ) ) ).

% map_append
thf(fact_273_map__append,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,Xs: list_a,Ys: list_a] :
      ( ( map_a_5058921920454310925od_a_o @ F @ ( append_a @ Xs @ Ys ) )
      = ( append2159120467191820514od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ Xs ) @ ( map_a_5058921920454310925od_a_o @ F @ Ys ) ) ) ).

% map_append
thf(fact_274_map__append,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_na5706685718985795555on_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( map_na5706685718985795555on_nat @ F @ Xs ) @ ( map_na5706685718985795555on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_275_map__append,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( map_Su4809418691747827683at_nat @ F @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_Su4809418691747827683at_nat @ F @ Xs ) @ ( map_Su4809418691747827683at_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_276_map__append,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,Ys: list_a] :
      ( ( map_a_2711072540680412167on_nat @ F @ ( append_a @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) @ ( map_a_2711072540680412167on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_277_map__append,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,Ys: list_a] :
      ( ( map_a_3887198048842487703on_nat @ F @ ( append_a @ Xs @ Ys ) )
      = ( append6174841615030841602on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) @ ( map_a_3887198048842487703on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_278_length__remdups__eq,axiom,
    ! [Xs: list_P3196869577824546119od_a_o] :
      ( ( ( size_s1727999245659094451od_a_o @ ( remdup8097019607593547996od_a_o @ Xs ) )
        = ( size_s1727999245659094451od_a_o @ Xs ) )
      = ( ( remdup8097019607593547996od_a_o @ Xs )
        = Xs ) ) ).

% length_remdups_eq
thf(fact_279_length__remdups__eq,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ ( remdup2859832054282949362on_nat @ Xs ) )
        = ( size_s5686234912572693873on_nat @ Xs ) )
      = ( ( remdup2859832054282949362on_nat @ Xs )
        = Xs ) ) ).

% length_remdups_eq
thf(fact_280_length__remdups__eq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ ( remdup7454993672696595682od_a_o @ Xs ) )
        = ( size_s3379185846251495609od_a_o @ Xs ) )
      = ( ( remdup7454993672696595682od_a_o @ Xs )
        = Xs ) ) ).

% length_remdups_eq
thf(fact_281_length__remdups__eq,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ ( remdups_nat @ Xs ) )
        = ( size_size_list_nat @ Xs ) )
      = ( ( remdups_nat @ Xs )
        = Xs ) ) ).

% length_remdups_eq
thf(fact_282_length__remdups__eq,axiom,
    ! [Xs: list_a] :
      ( ( ( size_size_list_a @ ( remdups_a @ Xs ) )
        = ( size_size_list_a @ Xs ) )
      = ( ( remdups_a @ Xs )
        = Xs ) ) ).

% length_remdups_eq
thf(fact_283_length__remdups__eq,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( ( size_s650155532264187265on_nat @ ( remdup2030676518929194242on_nat @ Xs ) )
        = ( size_s650155532264187265on_nat @ Xs ) )
      = ( ( remdup2030676518929194242on_nat @ Xs )
        = Xs ) ) ).

% length_remdups_eq
thf(fact_284_length__remdups__eq,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( ( size_s129639936549468787on_nat @ ( remdup4059826874966911900on_nat @ Xs ) )
        = ( size_s129639936549468787on_nat @ Xs ) )
      = ( ( remdup4059826874966911900on_nat @ Xs )
        = Xs ) ) ).

% length_remdups_eq
thf(fact_285_removeAll__append,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( removeAll_nat @ X @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( removeAll_nat @ X @ Xs ) @ ( removeAll_nat @ X @ Ys ) ) ) ).

% removeAll_append
thf(fact_286_removeAll__append,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( remove433775707702860097on_nat @ X @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( remove433775707702860097on_nat @ X @ Xs ) @ ( remove433775707702860097on_nat @ X @ Ys ) ) ) ).

% removeAll_append
thf(fact_287_removeAll__append,axiom,
    ! [X: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( remove6084986979817164679on_nat @ X @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( remove6084986979817164679on_nat @ X @ Xs ) @ ( remove6084986979817164679on_nat @ X @ Ys ) ) ) ).

% removeAll_append
thf(fact_288_length__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( size_s7490768528479217773on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ ( size_s7490768528479217773on_nat @ Ys ) ) ) ).

% length_append
thf(fact_289_length__append,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o] :
      ( ( size_s1727999245659094451od_a_o @ ( append2462165029785658588od_a_o @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s1727999245659094451od_a_o @ Xs ) @ ( size_s1727999245659094451od_a_o @ Ys ) ) ) ).

% length_append
thf(fact_290_length__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% length_append
thf(fact_291_length__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% length_append
thf(fact_292_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_293_length__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).

% length_append
thf(fact_294_length__append,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( size_s650155532264187265on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s650155532264187265on_nat @ Xs ) @ ( size_s650155532264187265on_nat @ Ys ) ) ) ).

% length_append
thf(fact_295_length__append,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( size_s129639936549468787on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s129639936549468787on_nat @ Xs ) @ ( size_s129639936549468787on_nat @ Ys ) ) ) ).

% length_append
thf(fact_296_sum__list__append,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( groups4559388385066561235st_int @ ( append_int @ Xs @ Ys ) )
      = ( plus_plus_int @ ( groups4559388385066561235st_int @ Xs ) @ ( groups4559388385066561235st_int @ Ys ) ) ) ).

% sum_list_append
thf(fact_297_sum__list__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( groups4561878855575611511st_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).

% sum_list_append
thf(fact_298_size__list__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( size_list_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ F @ Ys ) ) ) ).

% size_list_append
thf(fact_299_size__list__append,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( size_l7723201268554483269on_nat @ F @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_l7723201268554483269on_nat @ F @ Xs ) @ ( size_l7723201268554483269on_nat @ F @ Ys ) ) ) ).

% size_list_append
thf(fact_300_size__list__append,axiom,
    ! [F: produc1313295348606329723on_nat > nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( size_l6838359727932374403on_nat @ F @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_l6838359727932374403on_nat @ F @ Xs ) @ ( size_l6838359727932374403on_nat @ F @ Ys ) ) ) ).

% size_list_append
thf(fact_301_remdups__append2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( remdup2859832054282949362on_nat @ ( append3523812819975775986on_nat @ Xs @ ( remdup2859832054282949362on_nat @ Ys ) ) )
      = ( remdup2859832054282949362on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) ) ) ).

% remdups_append2
thf(fact_302_remdups__append2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( remdup7182839291886678166on_nat @ ( append2865821678206098582on_nat @ Xs @ ( remdup7182839291886678166on_nat @ Ys ) ) )
      = ( remdup7182839291886678166on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) ) ) ).

% remdups_append2
thf(fact_303_remdups__append2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( remdups_nat @ ( append_nat @ Xs @ ( remdups_nat @ Ys ) ) )
      = ( remdups_nat @ ( append_nat @ Xs @ Ys ) ) ) ).

% remdups_append2
thf(fact_304_remdups__remdups,axiom,
    ! [Xs: list_nat] :
      ( ( remdups_nat @ ( remdups_nat @ Xs ) )
      = ( remdups_nat @ Xs ) ) ).

% remdups_remdups
thf(fact_305_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_306_append__eq__appendI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Xs1: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Us: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append3523812819975775986on_nat @ Xs1 @ Us ) )
       => ( ( append3523812819975775986on_nat @ Xs @ Ys )
          = ( append3523812819975775986on_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_307_append__eq__appendI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Xs1: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Us: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append2865821678206098582on_nat @ Xs1 @ Us ) )
       => ( ( append2865821678206098582on_nat @ Xs @ Ys )
          = ( append2865821678206098582on_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_308_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_309_append__eq__append__conv2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,Ts: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys )
        = ( append3523812819975775986on_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_S5253404628753938141on_nat] :
            ( ( ( Xs
                = ( append3523812819975775986on_nat @ Zs @ Us2 ) )
              & ( ( append3523812819975775986on_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append3523812819975775986on_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append3523812819975775986on_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_310_append__eq__append__conv2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,Ts: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys )
        = ( append2865821678206098582on_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_P9127870862658512897on_nat] :
            ( ( ( Xs
                = ( append2865821678206098582on_nat @ Zs @ Us2 ) )
              & ( ( append2865821678206098582on_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append2865821678206098582on_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append2865821678206098582on_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_311_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_nat_nat @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_312_append__eq__map__conv,axiom,
    ! [Ys: list_a,Zs: list_a,F: nat > a,Xs: list_nat] :
      ( ( ( append_a @ Ys @ Zs )
        = ( map_nat_a @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_a @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_a @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_313_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: a > nat,Xs: list_a] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_a_nat @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_314_append__eq__map__conv,axiom,
    ! [Ys: list_a,Zs: list_a,F: a > a,Xs: list_a] :
      ( ( ( append_a @ Ys @ Zs )
        = ( map_a_a @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_a @ F @ Us2 ) )
            & ( Zs
              = ( map_a_a @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_315_append__eq__map__conv,axiom,
    ! [Ys: list_P3196869577824546119od_a_o,Zs: list_P3196869577824546119od_a_o,F: a > product_prod_a_o,Xs: list_a] :
      ( ( ( append2462165029785658588od_a_o @ Ys @ Zs )
        = ( map_a_1137562357930033543od_a_o @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_1137562357930033543od_a_o @ F @ Us2 ) )
            & ( Zs
              = ( map_a_1137562357930033543od_a_o @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_316_append__eq__map__conv,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o,F: a > list_P3196869577824546119od_a_o,Xs: list_a] :
      ( ( ( append2159120467191820514od_a_o @ Ys @ Zs )
        = ( map_a_5058921920454310925od_a_o @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_5058921920454310925od_a_o @ F @ Us2 ) )
            & ( Zs
              = ( map_a_5058921920454310925od_a_o @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_317_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_Su4809418691747827683at_nat @ F @ Xs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Su4809418691747827683at_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Su4809418691747827683at_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_318_append__eq__map__conv,axiom,
    ! [Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,F: nat > sum_su4907927656612468301on_nat,Xs: list_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys @ Zs )
        = ( map_na5706685718985795555on_nat @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_na5706685718985795555on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_na5706685718985795555on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_319_append__eq__map__conv,axiom,
    ! [Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( ( append3523812819975775986on_nat @ Ys @ Zs )
        = ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_2711072540680412167on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_2711072540680412167on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_320_append__eq__map__conv,axiom,
    ! [Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat,F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( ( append6174841615030841602on_nat @ Ys @ Zs )
        = ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_3887198048842487703on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_3887198048842487703on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_321_map__eq__append__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_322_map__eq__append__conv,axiom,
    ! [F: nat > a,Xs: list_nat,Ys: list_a,Zs: list_a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( append_a @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_a @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_a @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_323_map__eq__append__conv,axiom,
    ! [F: a > nat,Xs: list_a,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_a_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_324_map__eq__append__conv,axiom,
    ! [F: a > a,Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = ( append_a @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_a @ F @ Us2 ) )
            & ( Zs
              = ( map_a_a @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_325_map__eq__append__conv,axiom,
    ! [F: a > product_prod_a_o,Xs: list_a,Ys: list_P3196869577824546119od_a_o,Zs: list_P3196869577824546119od_a_o] :
      ( ( ( map_a_1137562357930033543od_a_o @ F @ Xs )
        = ( append2462165029785658588od_a_o @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_1137562357930033543od_a_o @ F @ Us2 ) )
            & ( Zs
              = ( map_a_1137562357930033543od_a_o @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_326_map__eq__append__conv,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,Xs: list_a,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( ( map_a_5058921920454310925od_a_o @ F @ Xs )
        = ( append2159120467191820514od_a_o @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_5058921920454310925od_a_o @ F @ Us2 ) )
            & ( Zs
              = ( map_a_5058921920454310925od_a_o @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_327_map__eq__append__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Su4809418691747827683at_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Su4809418691747827683at_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_328_map__eq__append__conv,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_na5706685718985795555on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_na5706685718985795555on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_329_map__eq__append__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_2711072540680412167on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_2711072540680412167on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_330_map__eq__append__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( append6174841615030841602on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_3887198048842487703on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_3887198048842487703on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_331_remdups__map__remdups,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( remdup4059826874966911900on_nat @ ( map_na8715451351012060651on_nat @ F @ ( remdups_nat @ Xs ) ) )
      = ( remdup4059826874966911900on_nat @ ( map_na8715451351012060651on_nat @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_332_remdups__map__remdups,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( remdups_nat @ ( map_nat_nat @ F @ ( remdups_nat @ Xs ) ) )
      = ( remdups_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_333_remdups__map__remdups,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( remdup2859832054282949362on_nat @ ( map_a_2711072540680412167on_nat @ F @ ( remdups_a @ Xs ) ) )
      = ( remdup2859832054282949362on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_334_remdups__map__remdups,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( remdup4059826874966911900on_nat @ ( map_a_7431629901395152583on_nat @ F @ ( remdups_a @ Xs ) ) )
      = ( remdup4059826874966911900on_nat @ ( map_a_7431629901395152583on_nat @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_335_remdups__map__remdups,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( remdup2030676518929194242on_nat @ ( map_a_3887198048842487703on_nat @ F @ ( remdups_a @ Xs ) ) )
      = ( remdup2030676518929194242on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_336_remdups__map__remdups,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( remdups_a @ ( map_nat_a @ F @ ( remdups_nat @ Xs ) ) )
      = ( remdups_a @ ( map_nat_a @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_337_remdups__map__remdups,axiom,
    ! [F: a > product_prod_a_o,Xs: list_a] :
      ( ( remdup8097019607593547996od_a_o @ ( map_a_1137562357930033543od_a_o @ F @ ( remdups_a @ Xs ) ) )
      = ( remdup8097019607593547996od_a_o @ ( map_a_1137562357930033543od_a_o @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_338_remdups__map__remdups,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,Xs: list_a] :
      ( ( remdup7454993672696595682od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ ( remdups_a @ Xs ) ) )
      = ( remdup7454993672696595682od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_339_remdups__map__remdups,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( remdups_nat @ ( map_a_nat @ F @ ( remdups_a @ Xs ) ) )
      = ( remdups_nat @ ( map_a_nat @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_340_remdups__map__remdups,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( remdups_a @ ( map_a_a @ F @ ( remdups_a @ Xs ) ) )
      = ( remdups_a @ ( map_a_a @ F @ Xs ) ) ) ).

% remdups_map_remdups
thf(fact_341_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( enumer158069492004818656on_nat @ N @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append7842736379566049733on_nat @ ( enumer158069492004818656on_nat @ N @ Xs ) @ ( enumer158069492004818656on_nat @ ( plus_plus_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_342_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o] :
      ( ( enumer6454736120857440038od_a_o @ N @ ( append2462165029785658588od_a_o @ Xs @ Ys ) )
      = ( append953226819290480395od_a_o @ ( enumer6454736120857440038od_a_o @ N @ Xs ) @ ( enumer6454736120857440038od_a_o @ ( plus_plus_nat @ N @ ( size_s1727999245659094451od_a_o @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_343_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( enumer7657777930897539368on_nat @ N @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( append6099216408989362713on_nat @ ( enumer7657777930897539368on_nat @ N @ Xs ) @ ( enumer7657777930897539368on_nat @ ( plus_plus_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_344_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( enumer7178362783773462060od_a_o @ N @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
      = ( append5252407214551111185od_a_o @ ( enumer7178362783773462060od_a_o @ N @ Xs ) @ ( enumer7178362783773462060od_a_o @ ( plus_plus_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_345_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
      = ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_346_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_a,Ys: list_a] :
      ( ( enumerate_a @ N @ ( append_a @ Xs @ Ys ) )
      = ( append1694031006427026248_nat_a @ ( enumerate_a @ N @ Xs ) @ ( enumerate_a @ ( plus_plus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_347_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( enumer9047912371274807352on_nat @ N @ ( append6174841615030841602on_nat @ Xs @ Ys ) )
      = ( append5882401969570823721on_nat @ ( enumer9047912371274807352on_nat @ N @ Xs ) @ ( enumer9047912371274807352on_nat @ ( plus_plus_nat @ N @ ( size_s650155532264187265on_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_348_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( enumer6979154339162643686on_nat @ N @ ( append3839213209832495004on_nat @ Xs @ Ys ) )
      = ( append3332032915773263819on_nat @ ( enumer6979154339162643686on_nat @ N @ Xs ) @ ( enumer6979154339162643686on_nat @ ( plus_plus_nat @ N @ ( size_s129639936549468787on_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_349_length__concat,axiom,
    ! [Xss: list_l1726306699111145479on_nat] :
      ( ( size_s7490768528479217773on_nat @ ( concat937493773106420796on_nat @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_li8334353970798598029at_nat @ size_s7490768528479217773on_nat @ Xss ) ) ) ).

% length_concat
thf(fact_350_length__concat,axiom,
    ! [Xss: list_l8336264072933094989od_a_o] :
      ( ( size_s1727999245659094451od_a_o @ ( concat7947350394041715842od_a_o @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_li381573783031109843_o_nat @ size_s1727999245659094451od_a_o @ Xss ) ) ) ).

% length_concat
thf(fact_351_length__concat,axiom,
    ! [Xss: list_l8999537460041914605on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( concat6161509665823917772on_nat @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_li2142278717278591475at_nat @ size_s5686234912572693873on_nat @ Xss ) ) ) ).

% length_concat
thf(fact_352_length__concat,axiom,
    ! [Xss: list_l7431997615882138323od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( concat3529464442532842888od_a_o @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_li999134049962033241_o_nat @ size_s3379185846251495609od_a_o @ Xss ) ) ) ).

% length_concat
thf(fact_353_length__concat,axiom,
    ! [Xss: list_list_nat] :
      ( ( size_size_list_nat @ ( concat_nat @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xss ) ) ) ).

% length_concat
thf(fact_354_length__concat,axiom,
    ! [Xss: list_list_a] :
      ( ( size_size_list_a @ ( concat_a @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_list_a_nat @ size_size_list_a @ Xss ) ) ) ).

% length_concat
thf(fact_355_length__concat,axiom,
    ! [Xss: list_l8365940747029150845on_nat] :
      ( ( size_s650155532264187265on_nat @ ( concat4774721536428481500on_nat @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_li3435095097222882691at_nat @ size_s650155532264187265on_nat @ Xss ) ) ) ).

% length_concat
thf(fact_356_length__concat,axiom,
    ! [Xss: list_l6719245731233385357on_nat] :
      ( ( size_s129639936549468787on_nat @ ( concat4341727598008994882on_nat @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_li8636112849402572307at_nat @ size_s129639936549468787on_nat @ Xss ) ) ) ).

% length_concat
thf(fact_357_nth__append__length__plus,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,N: nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ N ) )
      = ( nth_Pr3336893728888831618on_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_358_nth__append__length__plus,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o,N: nat] :
      ( ( nth_Product_prod_a_o @ ( append2462165029785658588od_a_o @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s1727999245659094451od_a_o @ Xs ) @ N ) )
      = ( nth_Product_prod_a_o @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_359_nth__append__length__plus,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,N: nat] :
      ( ( nth_Su8471564416988646406on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ N ) )
      = ( nth_Su8471564416988646406on_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_360_nth__append__length__plus,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,N: nat] :
      ( ( nth_li3248255784598043854od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ N ) )
      = ( nth_li3248255784598043854od_a_o @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_361_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_362_nth__append__length__plus,axiom,
    ! [Xs: list_a,Ys: list_a,N: nat] :
      ( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
      = ( nth_a @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_363_nth__append__length__plus,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,N: nat] :
      ( ( nth_li4091121329102240790on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s650155532264187265on_nat @ Xs ) @ N ) )
      = ( nth_li4091121329102240790on_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_364_nth__append__length__plus,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,N: nat] :
      ( ( nth_li1302981970230039432on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s129639936549468787on_nat @ Xs ) @ N ) )
      = ( nth_li1302981970230039432on_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_365_sum__list__filter__le__nat,axiom,
    ! [F: nat > nat,P: nat > $o,Xs: list_nat] : ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs ) ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).

% sum_list_filter_le_nat
thf(fact_366_sum__list__filter__le__nat,axiom,
    ! [F: a > nat,P: a > $o,Xs: list_a] : ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_a_nat @ F @ ( filter_a @ P @ Xs ) ) ) @ ( groups4561878855575611511st_nat @ ( map_a_nat @ F @ Xs ) ) ) ).

% sum_list_filter_le_nat
thf(fact_367_length__code,axiom,
    ( size_s1727999245659094451od_a_o
    = ( gen_le449265310187613268od_a_o @ zero_zero_nat ) ) ).

% length_code
thf(fact_368_length__code,axiom,
    ( size_s5686234912572693873on_nat
    = ( gen_le4617574567093687674on_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_369_length__code,axiom,
    ( size_s3379185846251495609od_a_o
    = ( gen_le5767993353563561050od_a_o @ zero_zero_nat ) ) ).

% length_code
thf(fact_370_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_371_length__code,axiom,
    ( size_size_list_a
    = ( gen_length_a @ zero_zero_nat ) ) ).

% length_code
thf(fact_372_length__code,axiom,
    ( size_s650155532264187265on_nat
    = ( gen_le9217917933129907082on_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_373_length__code,axiom,
    ( size_s129639936549468787on_nat
    = ( gen_le496107124138664724on_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_374_sum__list__abs,axiom,
    ! [Xs: list_int] : ( ord_less_eq_int @ ( abs_abs_int @ ( groups4559388385066561235st_int @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_int_int @ abs_abs_int @ Xs ) ) ) ).

% sum_list_abs
thf(fact_375_sum__mset_Ounion,axiom,
    ! [M2: multiset_nat,N4: multiset_nat] :
      ( ( comm_m762188921832702859et_nat @ ( plus_p6334493942879108393et_nat @ M2 @ N4 ) )
      = ( plus_plus_nat @ ( comm_m762188921832702859et_nat @ M2 ) @ ( comm_m762188921832702859et_nat @ N4 ) ) ) ).

% sum_mset.union
thf(fact_376_sum__mset_Ounion,axiom,
    ! [M2: multiset_int,N4: multiset_int] :
      ( ( comm_m759698451323652583et_int @ ( plus_p2156642923369911685et_int @ M2 @ N4 ) )
      = ( plus_plus_int @ ( comm_m759698451323652583et_int @ M2 ) @ ( comm_m759698451323652583et_int @ N4 ) ) ) ).

% sum_mset.union
thf(fact_377_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_378_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_379_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_380_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_381_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_382_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_383_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_384_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_385_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_386_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_387_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_388_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_389_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_390_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_391_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_392_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_393_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_394_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_395_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_396_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_397_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_398_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_399_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_400_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_401_filter__append,axiom,
    ! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( filter_nat @ P @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( filter_nat @ P @ Xs ) @ ( filter_nat @ P @ Ys ) ) ) ).

% filter_append
thf(fact_402_filter__append,axiom,
    ! [P: sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( filter891921761048319888on_nat @ P @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( filter891921761048319888on_nat @ P @ Xs ) @ ( filter891921761048319888on_nat @ P @ Ys ) ) ) ).

% filter_append
thf(fact_403_filter__append,axiom,
    ! [P: produc1313295348606329723on_nat > $o,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( filter3381104429707572856on_nat @ P @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( filter3381104429707572856on_nat @ P @ Xs ) @ ( filter3381104429707572856on_nat @ P @ Ys ) ) ) ).

% filter_append
thf(fact_404_sum__mset_Oempty,axiom,
    ( ( comm_m762188921832702859et_nat @ zero_z7348594199698428585et_nat )
    = zero_zero_nat ) ).

% sum_mset.empty
thf(fact_405_sum__mset_Oempty,axiom,
    ( ( comm_m759698451323652583et_int @ zero_z3170743180189231877et_int )
    = zero_zero_int ) ).

% sum_mset.empty
thf(fact_406_concat__append,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
      = ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).

% concat_append
thf(fact_407_concat__append,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( concat937493773106420796on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( concat937493773106420796on_nat @ Xs ) @ ( concat937493773106420796on_nat @ Ys ) ) ) ).

% concat_append
thf(fact_408_concat__append,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( concat6161509665823917772on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( concat6161509665823917772on_nat @ Xs ) @ ( concat6161509665823917772on_nat @ Ys ) ) ) ).

% concat_append
thf(fact_409_length__enumerate,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o] :
      ( ( size_s5677016091812162314od_a_o @ ( enumer6454736120857440038od_a_o @ N @ Xs ) )
      = ( size_s1727999245659094451od_a_o @ Xs ) ) ).

% length_enumerate
thf(fact_410_length__enumerate,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( size_s4628643742116541552on_nat @ ( enumer7657777930897539368on_nat @ N @ Xs ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_enumerate
thf(fact_411_length__enumerate,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( size_s8880912519739811088od_a_o @ ( enumer7178362783773462060od_a_o @ N @ Xs ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_enumerate
thf(fact_412_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_413_length__enumerate,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( size_s243904063682394823_nat_a @ ( enumerate_a @ N @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_enumerate
thf(fact_414_length__enumerate,axiom,
    ! [N: nat,Xs: list_l8999537460041914605on_nat] :
      ( ( size_s4773178582899407488on_nat @ ( enumer9047912371274807352on_nat @ N @ Xs ) )
      = ( size_s650155532264187265on_nat @ Xs ) ) ).

% length_enumerate
thf(fact_415_length__enumerate,axiom,
    ! [N: nat,Xs: list_l1726306699111145479on_nat] :
      ( ( size_s1420186467293376970on_nat @ ( enumer6979154339162643686on_nat @ N @ Xs ) )
      = ( size_s129639936549468787on_nat @ Xs ) ) ).

% length_enumerate
thf(fact_416_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_417_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_418_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_419_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_420_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_421_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_422_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_423_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_424_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_425_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_426_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_427_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_428_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_429_filter__concat,axiom,
    ! [P2: produc1313295348606329723on_nat > $o,Xs: list_l1726306699111145479on_nat] :
      ( ( filter3381104429707572856on_nat @ P2 @ ( concat937493773106420796on_nat @ Xs ) )
      = ( concat937493773106420796on_nat @ ( map_li815642509948034692on_nat @ ( filter3381104429707572856on_nat @ P2 ) @ Xs ) ) ) ).

% filter_concat
thf(fact_430_filter__concat,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,Xs: list_l8999537460041914605on_nat] :
      ( ( filter891921761048319888on_nat @ P2 @ ( concat6161509665823917772on_nat @ Xs ) )
      = ( concat6161509665823917772on_nat @ ( map_li4295621887133137792on_nat @ ( filter891921761048319888on_nat @ P2 ) @ Xs ) ) ) ).

% filter_concat
thf(fact_431_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_432_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_433_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_434_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_435_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_436_remdups__filter,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( remdups_nat @ ( filter_nat @ P @ Xs ) )
      = ( filter_nat @ P @ ( remdups_nat @ Xs ) ) ) ).

% remdups_filter
thf(fact_437_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_438_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_439_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_440_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_441_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_442_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_443_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_444_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_445_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_446_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_447_map__concat,axiom,
    ! [F: nat > nat,Xs: list_list_nat] :
      ( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
      = ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_448_map__concat,axiom,
    ! [F: nat > a,Xs: list_list_nat] :
      ( ( map_nat_a @ F @ ( concat_nat @ Xs ) )
      = ( concat_a @ ( map_list_nat_list_a @ ( map_nat_a @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_449_map__concat,axiom,
    ! [F: a > nat,Xs: list_list_a] :
      ( ( map_a_nat @ F @ ( concat_a @ Xs ) )
      = ( concat_nat @ ( map_list_a_list_nat @ ( map_a_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_450_map__concat,axiom,
    ! [F: a > a,Xs: list_list_a] :
      ( ( map_a_a @ F @ ( concat_a @ Xs ) )
      = ( concat_a @ ( map_list_a_list_a @ ( map_a_a @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_451_map__concat,axiom,
    ! [F: a > product_prod_a_o,Xs: list_list_a] :
      ( ( map_a_1137562357930033543od_a_o @ F @ ( concat_a @ Xs ) )
      = ( concat7947350394041715842od_a_o @ ( map_li1480699347688408199od_a_o @ ( map_a_1137562357930033543od_a_o @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_452_map__concat,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,Xs: list_list_a] :
      ( ( map_a_5058921920454310925od_a_o @ F @ ( concat_a @ Xs ) )
      = ( concat3529464442532842888od_a_o @ ( map_li7794534110530448781od_a_o @ ( map_a_5058921920454310925od_a_o @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_453_map__concat,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_list_a] :
      ( ( map_a_2711072540680412167on_nat @ F @ ( concat_a @ Xs ) )
      = ( concat6161509665823917772on_nat @ ( map_li3793676233746214813on_nat @ ( map_a_2711072540680412167on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_454_map__concat,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_list_a] :
      ( ( map_a_3887198048842487703on_nat @ F @ ( concat_a @ Xs ) )
      = ( concat4774721536428481500on_nat @ ( map_li7277942993266323885on_nat @ ( map_a_3887198048842487703on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_455_map__concat,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_l8999537460041914605on_nat] :
      ( ( map_Su241112079427911264on_nat @ F @ ( concat6161509665823917772on_nat @ Xs ) )
      = ( concat6161509665823917772on_nat @ ( map_li4295621887133137792on_nat @ ( map_Su241112079427911264on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_456_map__concat,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_list_nat] :
      ( ( map_na8715451351012060651on_nat @ F @ ( concat_nat @ Xs ) )
      = ( concat4341727598008994882on_nat @ ( map_li3981902705288052833on_nat @ ( map_na8715451351012060651on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_457_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_458_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_459_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_460_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_461_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_462_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_463_length__filter__le,axiom,
    ! [P: product_prod_a_o > $o,Xs: list_P3196869577824546119od_a_o] : ( ord_less_eq_nat @ ( size_s1727999245659094451od_a_o @ ( filter7579345971658946238od_a_o @ P @ Xs ) ) @ ( size_s1727999245659094451od_a_o @ Xs ) ) ).

% length_filter_le
thf(fact_464_length__filter__le,axiom,
    ! [P: sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat] : ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ ( filter891921761048319888on_nat @ P @ Xs ) ) @ ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_filter_le
thf(fact_465_length__filter__le,axiom,
    ! [P: list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o] : ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ ( filter2863170042043607492od_a_o @ P @ Xs ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_filter_le
thf(fact_466_length__filter__le,axiom,
    ! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_filter_le
thf(fact_467_length__filter__le,axiom,
    ! [P: a > $o,Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( filter_a @ P @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).

% length_filter_le
thf(fact_468_length__filter__le,axiom,
    ! [P: list_S5253404628753938141on_nat > $o,Xs: list_l8999537460041914605on_nat] : ( ord_less_eq_nat @ ( size_s650155532264187265on_nat @ ( filter2112716456448801440on_nat @ P @ Xs ) ) @ ( size_s650155532264187265on_nat @ Xs ) ) ).

% length_filter_le
thf(fact_469_length__filter__le,axiom,
    ! [P: list_P9127870862658512897on_nat > $o,Xs: list_l1726306699111145479on_nat] : ( ord_less_eq_nat @ ( size_s129639936549468787on_nat @ ( filter6410360991914645630on_nat @ P @ Xs ) ) @ ( size_s129639936549468787on_nat @ Xs ) ) ).

% length_filter_le
thf(fact_470_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_471_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_472_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_473_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_474_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_475_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_476_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_477_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_478_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_479_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_480_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_481_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_482_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_483_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_484_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_485_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_486_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_487_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_488_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_489_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_490_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P @ A3 @ B3 )
          = ( P @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ A3 @ B3 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_491_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_492_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_493_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_494_nth__drop,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( ( nth_Product_prod_a_o @ ( drop_P2811767759619103239od_a_o @ N @ Xs ) @ I )
        = ( nth_Product_prod_a_o @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_495_nth__drop,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( nth_Su8471564416988646406on_nat @ ( drop_S4725550819094782855on_nat @ N @ Xs ) @ I )
        = ( nth_Su8471564416988646406on_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_496_nth__drop,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( drop_l2897732520037520525od_a_o @ N @ Xs ) @ I )
        = ( nth_li3248255784598043854od_a_o @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_497_nth__drop,axiom,
    ! [N: nat,Xs: list_nat,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
        = ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_498_nth__drop,axiom,
    ! [N: nat,Xs: list_a,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_a @ ( drop_a @ N @ Xs ) @ I )
        = ( nth_a @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_499_nth__drop,axiom,
    ! [N: nat,Xs: list_l8999537460041914605on_nat,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s650155532264187265on_nat @ Xs ) )
     => ( ( nth_li4091121329102240790on_nat @ ( drop_l3393446327006289687on_nat @ N @ Xs ) @ I )
        = ( nth_li4091121329102240790on_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_500_nth__drop,axiom,
    ! [N: nat,Xs: list_l1726306699111145479on_nat,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s129639936549468787on_nat @ Xs ) )
     => ( ( nth_li1302981970230039432on_nat @ ( drop_l138928210601027399on_nat @ N @ Xs ) @ I )
        = ( nth_li1302981970230039432on_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_501_sum__list__def,axiom,
    ( groups4559388385066561235st_int
    = ( groups_monoid_F_int @ plus_plus_int @ zero_zero_int ) ) ).

% sum_list_def
thf(fact_502_sum__list__def,axiom,
    ( groups4561878855575611511st_nat
    = ( groups_monoid_F_nat @ plus_plus_nat @ zero_zero_nat ) ) ).

% sum_list_def
thf(fact_503_drop0,axiom,
    ( ( drop_nat @ zero_zero_nat )
    = ( ^ [X3: list_nat] : X3 ) ) ).

% drop0
thf(fact_504_drop__drop,axiom,
    ! [N: nat,M: nat,Xs: list_nat] :
      ( ( drop_nat @ N @ ( drop_nat @ M @ Xs ) )
      = ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).

% drop_drop
thf(fact_505_drop__0,axiom,
    ! [Xs: list_nat] :
      ( ( drop_nat @ zero_zero_nat @ Xs )
      = Xs ) ).

% drop_0
thf(fact_506_drop__map,axiom,
    ! [N: nat,F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( drop_l138928210601027399on_nat @ N @ ( map_na8715451351012060651on_nat @ F @ Xs ) )
      = ( map_na8715451351012060651on_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_507_drop__map,axiom,
    ! [N: nat,F: nat > nat,Xs: list_nat] :
      ( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_508_drop__map,axiom,
    ! [N: nat,F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( drop_S4725550819094782855on_nat @ N @ ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( map_a_2711072540680412167on_nat @ F @ ( drop_a @ N @ Xs ) ) ) ).

% drop_map
thf(fact_509_drop__map,axiom,
    ! [N: nat,F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( drop_l138928210601027399on_nat @ N @ ( map_a_7431629901395152583on_nat @ F @ Xs ) )
      = ( map_a_7431629901395152583on_nat @ F @ ( drop_a @ N @ Xs ) ) ) ).

% drop_map
thf(fact_510_drop__map,axiom,
    ! [N: nat,F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( drop_l3393446327006289687on_nat @ N @ ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( map_a_3887198048842487703on_nat @ F @ ( drop_a @ N @ Xs ) ) ) ).

% drop_map
thf(fact_511_drop__map,axiom,
    ! [N: nat,F: nat > a,Xs: list_nat] :
      ( ( drop_a @ N @ ( map_nat_a @ F @ Xs ) )
      = ( map_nat_a @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_512_drop__map,axiom,
    ! [N: nat,F: a > product_prod_a_o,Xs: list_a] :
      ( ( drop_P2811767759619103239od_a_o @ N @ ( map_a_1137562357930033543od_a_o @ F @ Xs ) )
      = ( map_a_1137562357930033543od_a_o @ F @ ( drop_a @ N @ Xs ) ) ) ).

% drop_map
thf(fact_513_drop__map,axiom,
    ! [N: nat,F: a > list_P3196869577824546119od_a_o,Xs: list_a] :
      ( ( drop_l2897732520037520525od_a_o @ N @ ( map_a_5058921920454310925od_a_o @ F @ Xs ) )
      = ( map_a_5058921920454310925od_a_o @ F @ ( drop_a @ N @ Xs ) ) ) ).

% drop_map
thf(fact_514_drop__map,axiom,
    ! [N: nat,F: a > nat,Xs: list_a] :
      ( ( drop_nat @ N @ ( map_a_nat @ F @ Xs ) )
      = ( map_a_nat @ F @ ( drop_a @ N @ Xs ) ) ) ).

% drop_map
thf(fact_515_drop__map,axiom,
    ! [N: nat,F: a > a,Xs: list_a] :
      ( ( drop_a @ N @ ( map_a_a @ F @ Xs ) )
      = ( map_a_a @ F @ ( drop_a @ N @ Xs ) ) ) ).

% drop_map
thf(fact_516_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_517_maps__def,axiom,
    ( maps_n4421757044315953832on_nat
    = ( ^ [F2: nat > list_P9127870862658512897on_nat,Xs3: list_nat] : ( concat937493773106420796on_nat @ ( map_na8715451351012060651on_nat @ F2 @ Xs3 ) ) ) ) ).

% maps_def
thf(fact_518_maps__def,axiom,
    ( maps_a723151493118966718on_nat
    = ( ^ [F2: a > list_P9127870862658512897on_nat,Xs3: list_a] : ( concat937493773106420796on_nat @ ( map_a_7431629901395152583on_nat @ F2 @ Xs3 ) ) ) ) ).

% maps_def
thf(fact_519_maps__def,axiom,
    ( maps_a7760926417009752522on_nat
    = ( ^ [F2: a > list_S5253404628753938141on_nat,Xs3: list_a] : ( concat6161509665823917772on_nat @ ( map_a_3887198048842487703on_nat @ F2 @ Xs3 ) ) ) ) ).

% maps_def
thf(fact_520_maps__def,axiom,
    ( maps_a4539202661891466244od_a_o
    = ( ^ [F2: a > list_P3196869577824546119od_a_o,Xs3: list_a] : ( concat7947350394041715842od_a_o @ ( map_a_5058921920454310925od_a_o @ F2 @ Xs3 ) ) ) ) ).

% maps_def
thf(fact_521_concat__map__maps,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( concat937493773106420796on_nat @ ( map_na8715451351012060651on_nat @ F @ Xs ) )
      = ( maps_n4421757044315953832on_nat @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_522_concat__map__maps,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( concat937493773106420796on_nat @ ( map_a_7431629901395152583on_nat @ F @ Xs ) )
      = ( maps_a723151493118966718on_nat @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_523_concat__map__maps,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( concat6161509665823917772on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( maps_a7760926417009752522on_nat @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_524_concat__map__maps,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,Xs: list_a] :
      ( ( concat7947350394041715842od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ Xs ) )
      = ( maps_a4539202661891466244od_a_o @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_525_List_Obind__def,axiom,
    ( bind_n8661063611441293282on_nat
    = ( ^ [Xs3: list_nat,F2: nat > list_P9127870862658512897on_nat] : ( concat937493773106420796on_nat @ ( map_na8715451351012060651on_nat @ F2 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_526_List_Obind__def,axiom,
    ( bind_a2320658689571621124on_nat
    = ( ^ [Xs3: list_a,F2: a > list_P9127870862658512897on_nat] : ( concat937493773106420796on_nat @ ( map_a_7431629901395152583on_nat @ F2 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_527_List_Obind__def,axiom,
    ( bind_a8347162265041818628on_nat
    = ( ^ [Xs3: list_a,F2: a > list_S5253404628753938141on_nat] : ( concat6161509665823917772on_nat @ ( map_a_3887198048842487703on_nat @ F2 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_528_List_Obind__def,axiom,
    ( bind_a2614128812081191242od_a_o
    = ( ^ [Xs3: list_a,F2: a > list_P3196869577824546119od_a_o] : ( concat7947350394041715842od_a_o @ ( map_a_5058921920454310925od_a_o @ F2 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_529_sum__mset_Oeq__fold,axiom,
    ( comm_m762188921832702859et_nat
    = ( fold_mset_nat_nat @ plus_plus_nat @ zero_zero_nat ) ) ).

% sum_mset.eq_fold
thf(fact_530_sum__mset_Oeq__fold,axiom,
    ( comm_m759698451323652583et_int
    = ( fold_mset_int_int @ plus_plus_int @ zero_zero_int ) ) ).

% sum_mset.eq_fold
thf(fact_531_elem__le__sum__list,axiom,
    ! [K: nat,Ns: list_nat] :
      ( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
     => ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).

% elem_le_sum_list
thf(fact_532_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_533_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_534_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_535_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_536_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_537_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_538_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_539_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_540_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_541_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_542_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_543_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_544_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_545_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_546_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_547_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_548_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_549_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_550_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_551_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_552_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_553_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_554_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > a] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_a @ ( map_nat_a @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_555_nth__map,axiom,
    ! [N: nat,Xs: list_a,F: a > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_nat @ ( map_a_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_556_nth__map,axiom,
    ! [N: nat,Xs: list_a,F: a > a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_a @ ( map_a_a @ F @ Xs ) @ N )
        = ( F @ ( nth_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_557_nth__map,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o,F: product_prod_a_o > nat] :
      ( ( ord_less_nat @ N @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( ( nth_nat @ ( map_Pr3957037749107795149_o_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_Product_prod_a_o @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_558_nth__map,axiom,
    ! [N: nat,Xs: list_a,F: a > product_prod_a_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_Product_prod_a_o @ ( map_a_1137562357930033543od_a_o @ F @ Xs ) @ N )
        = ( F @ ( nth_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_559_nth__map,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,F: list_P3196869577824546119od_a_o > nat] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_nat @ ( map_li381573783031109843_o_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_560_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( map_na764632570197019441od_a_o @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_561_nth__map,axiom,
    ! [N: nat,Xs: list_a,F: a > list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ Xs ) @ N )
        = ( F @ ( nth_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_562_nth__map,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > nat] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( nth_nat @ ( map_Su4809418691747827683at_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_563_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_564_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_565_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_566_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_567_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_568_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_569_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_570_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_571_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_572_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_573_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_574_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_575_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_576_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_577_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_578_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_579_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_580_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_581_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_582_order__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_583_order__less__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_584_order__less__subst1,axiom,
    ! [A: int,F: int > int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_585_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_586_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_587_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_588_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_589_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > nat,C: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_590_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_591_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_592_ord__eq__less__subst,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_593_ord__eq__less__subst,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_594_ord__eq__less__subst,axiom,
    ! [A: int,F: int > int,B: int,C: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_595_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_596_order__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_597_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_598_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_599_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_600_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_601_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_602_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_603_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_604_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_605_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_606_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M5: nat] :
              ( ( ord_less_nat @ M5 @ N2 )
              & ~ ( P @ M5 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_607_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N2 )
             => ( P @ M5 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_608_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_609_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_610_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_611_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_612_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_613_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_614_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_615_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_616_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_617_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_618_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_619_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_620_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_621_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_622_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_623_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_624_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: int] : ( P @ A3 @ A3 )
       => ( ! [A3: int,B3: int] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_625_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ? [N3: nat] :
          ( ( P4 @ N3 )
          & ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
             => ~ ( P4 @ M4 ) ) ) ) ) ).

% exists_least_iff
thf(fact_626_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_627_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_628_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_629_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_630_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_631_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_632_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_633_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_634_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X2: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X2 )
             => ( P @ Y5 ) )
         => ( P @ X2 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_635_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_636_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_637_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_638_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_639_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_640_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_641_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_642_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_643_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_644_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).

% gt_ex
thf(fact_645_lt__ex,axiom,
    ! [X: int] :
    ? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).

% lt_ex
thf(fact_646_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_647_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_648_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_649_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_650_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_651_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_652_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_653_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_654_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_655_order__less__le__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_656_order__less__le__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_657_order__less__le__subst1,axiom,
    ! [A: int,F: int > int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_658_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_659_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_660_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_661_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_662_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_663_order__le__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_664_order__le__less__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_665_order__le__less__subst1,axiom,
    ! [A: int,F: int > int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_666_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_667_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_668_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_669_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_670_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_671_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_672_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_673_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_674_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_675_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_676_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_677_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_678_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_679_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_680_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y4 )
          & ( X3 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_681_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y4: int] :
          ( ( ord_less_eq_int @ X3 @ Y4 )
          & ( X3 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_682_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y4: nat] :
          ( ( ord_less_nat @ X3 @ Y4 )
          | ( X3 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_683_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y4: int] :
          ( ( ord_less_int @ X3 @ Y4 )
          | ( X3 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_684_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_685_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_686_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_687_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_688_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_689_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_690_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_691_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_692_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_693_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_694_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_695_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_696_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_nat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_697_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_int @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_698_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_699_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_700_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_701_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_702_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_703_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_704_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_705_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_706_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_nat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_707_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_int @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_708_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_709_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_710_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_711_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y4: int] :
          ( ( ord_less_eq_int @ X3 @ Y4 )
          & ~ ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_712_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_713_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_714_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_715_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_716_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_717_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_718_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_719_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_720_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_721_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_722_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
      = ( ord_less_nat @ A5 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_723_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
      = ( ord_less_int @ A5 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_724_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_725_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_726_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_727_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_728_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_729_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_730_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_731_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_732_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_733_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_734_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_735_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_736_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_737_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_738_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_739_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_740_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_741_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_742_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_743_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_744_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_745_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_746_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M5: nat] :
                  ( ( ord_less_nat @ M5 @ N2 )
                  & ~ ( P @ M5 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_747_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_748_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_749_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_750_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_751_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_752_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_753_length__induct,axiom,
    ! [P: list_P3196869577824546119od_a_o > $o,Xs: list_P3196869577824546119od_a_o] :
      ( ! [Xs2: list_P3196869577824546119od_a_o] :
          ( ! [Ys2: list_P3196869577824546119od_a_o] :
              ( ( ord_less_nat @ ( size_s1727999245659094451od_a_o @ Ys2 ) @ ( size_s1727999245659094451od_a_o @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_754_length__induct,axiom,
    ! [P: list_S5253404628753938141on_nat > $o,Xs: list_S5253404628753938141on_nat] :
      ( ! [Xs2: list_S5253404628753938141on_nat] :
          ( ! [Ys2: list_S5253404628753938141on_nat] :
              ( ( ord_less_nat @ ( size_s5686234912572693873on_nat @ Ys2 ) @ ( size_s5686234912572693873on_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_755_length__induct,axiom,
    ! [P: list_l8336264072933094989od_a_o > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ! [Xs2: list_l8336264072933094989od_a_o] :
          ( ! [Ys2: list_l8336264072933094989od_a_o] :
              ( ( ord_less_nat @ ( size_s3379185846251495609od_a_o @ Ys2 ) @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_756_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_757_length__induct,axiom,
    ! [P: list_a > $o,Xs: list_a] :
      ( ! [Xs2: list_a] :
          ( ! [Ys2: list_a] :
              ( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_758_length__induct,axiom,
    ! [P: list_l8999537460041914605on_nat > $o,Xs: list_l8999537460041914605on_nat] :
      ( ! [Xs2: list_l8999537460041914605on_nat] :
          ( ! [Ys2: list_l8999537460041914605on_nat] :
              ( ( ord_less_nat @ ( size_s650155532264187265on_nat @ Ys2 ) @ ( size_s650155532264187265on_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_759_length__induct,axiom,
    ! [P: list_l1726306699111145479on_nat > $o,Xs: list_l1726306699111145479on_nat] :
      ( ! [Xs2: list_l1726306699111145479on_nat] :
          ( ! [Ys2: list_l1726306699111145479on_nat] :
              ( ( ord_less_nat @ ( size_s129639936549468787on_nat @ Ys2 ) @ ( size_s129639936549468787on_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_760_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_761_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_762_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_763_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
          | ( M4 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_764_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_765_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
          & ( M4 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_766_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_767_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_768_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_769_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_770_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_771_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_772_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_773_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_774_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_775_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_776_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_777_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_778_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_779_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_780_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_781_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_782_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_783_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_784_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_785_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_786_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_787_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_788_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_789_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_790_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_791_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_792_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K3 )
               => ~ ( P @ I3 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_793_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_794_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( ( ord_less_nat @ M3 @ N2 )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_795_nth__equalityI,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o] :
      ( ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_s1727999245659094451od_a_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s1727999245659094451od_a_o @ Xs ) )
           => ( ( nth_Product_prod_a_o @ Xs @ I2 )
              = ( nth_Product_prod_a_o @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_796_nth__equalityI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( ( nth_Su8471564416988646406on_nat @ Xs @ I2 )
              = ( nth_Su8471564416988646406on_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_797_nth__equalityI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( ( nth_li3248255784598043854od_a_o @ Xs @ I2 )
              = ( nth_li3248255784598043854od_a_o @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_798_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_799_nth__equalityI,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
           => ( ( nth_a @ Xs @ I2 )
              = ( nth_a @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_800_nth__equalityI,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
        = ( size_s650155532264187265on_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s650155532264187265on_nat @ Xs ) )
           => ( ( nth_li4091121329102240790on_nat @ Xs @ I2 )
              = ( nth_li4091121329102240790on_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_801_nth__equalityI,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( ( size_s129639936549468787on_nat @ Xs )
        = ( size_s129639936549468787on_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Xs ) )
           => ( ( nth_li1302981970230039432on_nat @ Xs @ I2 )
              = ( nth_li1302981970230039432on_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_802_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > product_prod_a_o > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X5: product_prod_a_o] : ( P @ I4 @ X5 ) ) )
      = ( ? [Xs3: list_P3196869577824546119od_a_o] :
            ( ( ( size_s1727999245659094451od_a_o @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_Product_prod_a_o @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_803_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > sum_su4907927656612468301on_nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X5: sum_su4907927656612468301on_nat] : ( P @ I4 @ X5 ) ) )
      = ( ? [Xs3: list_S5253404628753938141on_nat] :
            ( ( ( size_s5686234912572693873on_nat @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_Su8471564416988646406on_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_804_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_P3196869577824546119od_a_o > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X5: list_P3196869577824546119od_a_o] : ( P @ I4 @ X5 ) ) )
      = ( ? [Xs3: list_l8336264072933094989od_a_o] :
            ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_li3248255784598043854od_a_o @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_805_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X5: nat] : ( P @ I4 @ X5 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_806_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > a > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X5: a] : ( P @ I4 @ X5 ) ) )
      = ( ? [Xs3: list_a] :
            ( ( ( size_size_list_a @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_a @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_807_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_S5253404628753938141on_nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X5: list_S5253404628753938141on_nat] : ( P @ I4 @ X5 ) ) )
      = ( ? [Xs3: list_l8999537460041914605on_nat] :
            ( ( ( size_s650155532264187265on_nat @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_li4091121329102240790on_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_808_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_P9127870862658512897on_nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X5: list_P9127870862658512897on_nat] : ( P @ I4 @ X5 ) ) )
      = ( ? [Xs3: list_l1726306699111145479on_nat] :
            ( ( ( size_s129639936549468787on_nat @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_li1302981970230039432on_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_809_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_P3196869577824546119od_a_o,Z: list_P3196869577824546119od_a_o] : ( Y3 = Z ) )
    = ( ^ [Xs3: list_P3196869577824546119od_a_o,Ys3: list_P3196869577824546119od_a_o] :
          ( ( ( size_s1727999245659094451od_a_o @ Xs3 )
            = ( size_s1727999245659094451od_a_o @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s1727999245659094451od_a_o @ Xs3 ) )
             => ( ( nth_Product_prod_a_o @ Xs3 @ I4 )
                = ( nth_Product_prod_a_o @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_810_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_S5253404628753938141on_nat,Z: list_S5253404628753938141on_nat] : ( Y3 = Z ) )
    = ( ^ [Xs3: list_S5253404628753938141on_nat,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
             => ( ( nth_Su8471564416988646406on_nat @ Xs3 @ I4 )
                = ( nth_Su8471564416988646406on_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_811_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_l8336264072933094989od_a_o,Z: list_l8336264072933094989od_a_o] : ( Y3 = Z ) )
    = ( ^ [Xs3: list_l8336264072933094989od_a_o,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
             => ( ( nth_li3248255784598043854od_a_o @ Xs3 @ I4 )
                = ( nth_li3248255784598043854od_a_o @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_812_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I4 )
                = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_813_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_a,Z: list_a] : ( Y3 = Z ) )
    = ( ^ [Xs3: list_a,Ys3: list_a] :
          ( ( ( size_size_list_a @ Xs3 )
            = ( size_size_list_a @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs3 ) )
             => ( ( nth_a @ Xs3 @ I4 )
                = ( nth_a @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_814_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_l8999537460041914605on_nat,Z: list_l8999537460041914605on_nat] : ( Y3 = Z ) )
    = ( ^ [Xs3: list_l8999537460041914605on_nat,Ys3: list_l8999537460041914605on_nat] :
          ( ( ( size_s650155532264187265on_nat @ Xs3 )
            = ( size_s650155532264187265on_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s650155532264187265on_nat @ Xs3 ) )
             => ( ( nth_li4091121329102240790on_nat @ Xs3 @ I4 )
                = ( nth_li4091121329102240790on_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_815_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_l1726306699111145479on_nat,Z: list_l1726306699111145479on_nat] : ( Y3 = Z ) )
    = ( ^ [Xs3: list_l1726306699111145479on_nat,Ys3: list_l1726306699111145479on_nat] :
          ( ( ( size_s129639936549468787on_nat @ Xs3 )
            = ( size_s129639936549468787on_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s129639936549468787on_nat @ Xs3 ) )
             => ( ( nth_li1302981970230039432on_nat @ Xs3 @ I4 )
                = ( nth_li1302981970230039432on_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_816_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_817_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_818_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_819_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_820_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_821_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_822_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_823_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_824_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_825_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_826_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_827_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_828_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_829_map__equality__iff,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_830_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_a @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_831_map__equality__iff,axiom,
    ! [F: nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_a_a @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_a @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_832_map__equality__iff,axiom,
    ! [F: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
      ( ( ( map_a_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_833_map__equality__iff,axiom,
    ! [F: a > a,Xs: list_a,G: nat > a,Ys: list_nat] :
      ( ( ( map_a_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_834_map__equality__iff,axiom,
    ! [F: a > nat,Xs: list_a,G: a > nat,Ys: list_a] :
      ( ( ( map_a_nat @ F @ Xs )
        = ( map_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I4 ) )
              = ( G @ ( nth_a @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_835_map__equality__iff,axiom,
    ! [F: a > a,Xs: list_a,G: a > a,Ys: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = ( map_a_a @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I4 ) )
              = ( G @ ( nth_a @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_836_map__equality__iff,axiom,
    ! [F: product_prod_a_o > nat,Xs: list_P3196869577824546119od_a_o,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Pr3957037749107795149_o_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_s1727999245659094451od_a_o @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_Product_prod_a_o @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_837_map__equality__iff,axiom,
    ! [F: product_prod_a_o > a,Xs: list_P3196869577824546119od_a_o,G: nat > a,Ys: list_nat] :
      ( ( ( map_Pr8253109945893590849_a_o_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
      = ( ( ( size_s1727999245659094451od_a_o @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_Product_prod_a_o @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_838_sum__list__mono2,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Ys @ I2 ) ) )
       => ( ord_less_eq_int @ ( groups4559388385066561235st_int @ Xs ) @ ( groups4559388385066561235st_int @ Ys ) ) ) ) ).

% sum_list_mono2
thf(fact_839_sum__list__mono2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
       => ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).

% sum_list_mono2
thf(fact_840_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K3 @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K3 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_841_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).

% minf(8)
thf(fact_842_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X4 ) ) ).

% minf(8)
thf(fact_843_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ( ord_less_eq_nat @ X4 @ T ) ) ).

% minf(6)
thf(fact_844_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ( ord_less_eq_int @ X4 @ T ) ) ).

% minf(6)
thf(fact_845_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ( ord_less_eq_nat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_846_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ( ord_less_eq_int @ T @ X4 ) ) ).

% pinf(8)
thf(fact_847_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_848_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ~ ( ord_less_eq_int @ X4 @ T ) ) ).

% pinf(6)
thf(fact_849_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: nat] :
              ( ( ord_less_eq_nat @ A @ C2 )
              & ( ord_less_eq_nat @ C2 @ B )
              & ! [X4: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X4 )
                    & ( ord_less_nat @ X4 @ C2 ) )
                 => ( P @ X4 ) )
              & ! [D2: nat] :
                  ( ! [X2: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X2 )
                        & ( ord_less_nat @ X2 @ D2 ) )
                     => ( P @ X2 ) )
                 => ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_850_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: int] :
              ( ( ord_less_eq_int @ A @ C2 )
              & ( ord_less_eq_int @ C2 @ B )
              & ! [X4: int] :
                  ( ( ( ord_less_eq_int @ A @ X4 )
                    & ( ord_less_int @ X4 @ C2 ) )
                 => ( P @ X4 ) )
              & ! [D2: int] :
                  ( ! [X2: int] :
                      ( ( ( ord_less_eq_int @ A @ X2 )
                        & ( ord_less_int @ X2 @ D2 ) )
                     => ( P @ X2 ) )
                 => ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_851_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_P3196869577824546119od_a_o,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( ( nth_Pr3559203214505555487od_a_o @ ( enumer6454736120857440038od_a_o @ N @ Xs ) @ M )
        = ( produc5340308163291612696od_a_o @ ( plus_plus_nat @ N @ M ) @ ( nth_Product_prod_a_o @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_852_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_S5253404628753938141on_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( nth_Pr5135637237369297925on_nat @ ( enumer7657777930897539368on_nat @ N @ Xs ) @ M )
        = ( produc8397911583705546934on_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_Su8471564416988646406on_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_853_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_l8336264072933094989od_a_o,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_Pr8565389872998445861od_a_o @ ( enumer7178362783773462060od_a_o @ N @ Xs ) @ M )
        = ( produc3946064260745446430od_a_o @ ( plus_plus_nat @ N @ M ) @ ( nth_li3248255784598043854od_a_o @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_854_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_855_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_a,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_a @ Xs ) )
     => ( ( nth_Pr2476257081389315164_nat_a @ ( enumerate_a @ N @ Xs ) @ M )
        = ( product_Pair_nat_a @ ( plus_plus_nat @ N @ M ) @ ( nth_a @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_856_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_l8999537460041914605on_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s650155532264187265on_nat @ Xs ) )
     => ( ( nth_Pr2104752557224806933on_nat @ ( enumer9047912371274807352on_nat @ N @ Xs ) @ M )
        = ( produc4312340823571519686on_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_li4091121329102240790on_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_857_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_l1726306699111145479on_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s129639936549468787on_nat @ Xs ) )
     => ( ( nth_Pr2260264872863644895on_nat @ ( enumer6979154339162643686on_nat @ N @ Xs ) @ M )
        = ( produc3379708602565778136on_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_li1302981970230039432on_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_858_list__ex__length,axiom,
    ( list_e12574053215267589od_a_o
    = ( ^ [P4: product_prod_a_o > $o,Xs3: list_P3196869577824546119od_a_o] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s1727999245659094451od_a_o @ Xs3 ) )
          & ( P4 @ ( nth_Product_prod_a_o @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_859_list__ex__length,axiom,
    ( list_e3769318529992910857on_nat
    = ( ^ [P4: sum_su4907927656612468301on_nat > $o,Xs3: list_S5253404628753938141on_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
          & ( P4 @ ( nth_Su8471564416988646406on_nat @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_860_list__ex__length,axiom,
    ( list_e5403782194016407179od_a_o
    = ( ^ [P4: list_P3196869577824546119od_a_o > $o,Xs3: list_l8336264072933094989od_a_o] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
          & ( P4 @ ( nth_li3248255784598043854od_a_o @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_861_list__ex__length,axiom,
    ( list_ex_nat
    = ( ^ [P4: nat > $o,Xs3: list_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) )
          & ( P4 @ ( nth_nat @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_862_list__ex__length,axiom,
    ( list_ex_a
    = ( ^ [P4: a > $o,Xs3: list_a] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs3 ) )
          & ( P4 @ ( nth_a @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_863_list__ex__length,axiom,
    ( list_e511240961062746777on_nat
    = ( ^ [P4: list_S5253404628753938141on_nat > $o,Xs3: list_l8999537460041914605on_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s650155532264187265on_nat @ Xs3 ) )
          & ( P4 @ ( nth_li4091121329102240790on_nat @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_864_list__ex__length,axiom,
    ( list_e7608443322527242565on_nat
    = ( ^ [P4: list_P9127870862658512897on_nat > $o,Xs3: list_l1726306699111145479on_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s129639936549468787on_nat @ Xs3 ) )
          & ( P4 @ ( nth_li1302981970230039432on_nat @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_865_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_866_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_867_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_868_sorted__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_869_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_870_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_871_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_872_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_873_list__ex__append,axiom,
    ! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( list_ex_nat @ P @ ( append_nat @ Xs @ Ys ) )
      = ( ( list_ex_nat @ P @ Xs )
        | ( list_ex_nat @ P @ Ys ) ) ) ).

% list_ex_append
thf(fact_874_list__ex__append,axiom,
    ! [P: sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( list_e3769318529992910857on_nat @ P @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( ( list_e3769318529992910857on_nat @ P @ Xs )
        | ( list_e3769318529992910857on_nat @ P @ Ys ) ) ) ).

% list_ex_append
thf(fact_875_list__ex__append,axiom,
    ! [P: produc1313295348606329723on_nat > $o,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( list_e8248363684495001279on_nat @ P @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( ( list_e8248363684495001279on_nat @ P @ Xs )
        | ( list_e8248363684495001279on_nat @ P @ Ys ) ) ) ).

% list_ex_append
thf(fact_876_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_877_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_878_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_879_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_880_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_881_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_882_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_883_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_884_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_885_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_886_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_887_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_888_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_889_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_890_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_891_sorted__wrt__filter,axiom,
    ! [F: nat > nat > $o,Xs: list_nat,P: nat > $o] :
      ( ( sorted_wrt_nat @ F @ Xs )
     => ( sorted_wrt_nat @ F @ ( filter_nat @ P @ Xs ) ) ) ).

% sorted_wrt_filter
thf(fact_892_sorted__wrt__drop,axiom,
    ! [F: nat > nat > $o,Xs: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ F @ Xs )
     => ( sorted_wrt_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% sorted_wrt_drop
thf(fact_893_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_894_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_895_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_896_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z2: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).

% zadd_int_left
thf(fact_897_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_898_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_899_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_900_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_901_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_902_sorted__drop,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N @ Xs ) ) ) ).

% sorted_drop
thf(fact_903_sorted__drop,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( drop_int @ N @ Xs ) ) ) ).

% sorted_drop
thf(fact_904_sorted__remdups,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( remdups_nat @ Xs ) ) ) ).

% sorted_remdups
thf(fact_905_sorted__remdups,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( remdups_int @ Xs ) ) ) ).

% sorted_remdups
thf(fact_906_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_907_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_908_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_909_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_910_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_911_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_912_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_913_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_914_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_915_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_916_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_917_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_918_sorted__filter,axiom,
    ! [F: nat > nat,Xs: list_nat,P: nat > $o] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs ) )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).

% sorted_filter
thf(fact_919_sorted__filter,axiom,
    ! [F: a > nat,Xs: list_a,P: a > $o] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_a_nat @ F @ Xs ) )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_a_nat @ F @ ( filter_a @ P @ Xs ) ) ) ) ).

% sorted_filter
thf(fact_920_sorted__wrt__iff__nth__less,axiom,
    ( sorted7645611771639492654od_a_o
    = ( ^ [P4: product_prod_a_o > product_prod_a_o > $o,Xs3: list_P3196869577824546119od_a_o] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s1727999245659094451od_a_o @ Xs3 ) )
           => ( P4 @ ( nth_Product_prod_a_o @ Xs3 @ I4 ) @ ( nth_Product_prod_a_o @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_921_sorted__wrt__iff__nth__less,axiom,
    ( sorted63530893423634976on_nat
    = ( ^ [P4: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs3: list_S5253404628753938141on_nat] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
           => ( P4 @ ( nth_Su8471564416988646406on_nat @ Xs3 @ I4 ) @ ( nth_Su8471564416988646406on_nat @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_922_sorted__wrt__iff__nth__less,axiom,
    ( sorted770596854460197684od_a_o
    = ( ^ [P4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs3: list_l8336264072933094989od_a_o] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
           => ( P4 @ ( nth_li3248255784598043854od_a_o @ Xs3 @ I4 ) @ ( nth_li3248255784598043854od_a_o @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_923_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_nat
    = ( ^ [P4: nat > nat > $o,Xs3: list_nat] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
           => ( P4 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_924_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_a
    = ( ^ [P4: a > a > $o,Xs3: list_a] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs3 ) )
           => ( P4 @ ( nth_a @ Xs3 @ I4 ) @ ( nth_a @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_925_sorted__wrt__iff__nth__less,axiom,
    ( sorted3751386010246432048on_nat
    = ( ^ [P4: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Xs3: list_l8999537460041914605on_nat] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s650155532264187265on_nat @ Xs3 ) )
           => ( P4 @ ( nth_li4091121329102240790on_nat @ Xs3 @ I4 ) @ ( nth_li4091121329102240790on_nat @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_926_sorted__wrt__iff__nth__less,axiom,
    ( sorted8381134733921163758on_nat
    = ( ^ [P4: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o,Xs3: list_l1726306699111145479on_nat] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s129639936549468787on_nat @ Xs3 ) )
           => ( P4 @ ( nth_li1302981970230039432on_nat @ Xs3 @ I4 ) @ ( nth_li1302981970230039432on_nat @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_927_sorted__wrt__nth__less,axiom,
    ! [P: product_prod_a_o > product_prod_a_o > $o,Xs: list_P3196869577824546119od_a_o,I: nat,J: nat] :
      ( ( sorted7645611771639492654od_a_o @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s1727999245659094451od_a_o @ Xs ) )
         => ( P @ ( nth_Product_prod_a_o @ Xs @ I ) @ ( nth_Product_prod_a_o @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_928_sorted__wrt__nth__less,axiom,
    ! [P: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,I: nat,J: nat] :
      ( ( sorted63530893423634976on_nat @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
         => ( P @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) @ ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_929_sorted__wrt__nth__less,axiom,
    ! [P: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,I: nat,J: nat] :
      ( ( sorted770596854460197684od_a_o @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s3379185846251495609od_a_o @ Xs ) )
         => ( P @ ( nth_li3248255784598043854od_a_o @ Xs @ I ) @ ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_930_sorted__wrt__nth__less,axiom,
    ! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_931_sorted__wrt__nth__less,axiom,
    ! [P: a > a > $o,Xs: list_a,I: nat,J: nat] :
      ( ( sorted_wrt_a @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
         => ( P @ ( nth_a @ Xs @ I ) @ ( nth_a @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_932_sorted__wrt__nth__less,axiom,
    ! [P: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Xs: list_l8999537460041914605on_nat,I: nat,J: nat] :
      ( ( sorted3751386010246432048on_nat @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s650155532264187265on_nat @ Xs ) )
         => ( P @ ( nth_li4091121329102240790on_nat @ Xs @ I ) @ ( nth_li4091121329102240790on_nat @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_933_sorted__wrt__nth__less,axiom,
    ! [P: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o,Xs: list_l1726306699111145479on_nat,I: nat,J: nat] :
      ( ( sorted8381134733921163758on_nat @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s129639936549468787on_nat @ Xs ) )
         => ( P @ ( nth_li1302981970230039432on_nat @ Xs @ I ) @ ( nth_li1302981970230039432on_nat @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_934_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_935_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_936_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_937_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
       => ( ( nth_Pr2476257081389315164_nat_a @ ( zip_nat_a @ Xs @ Ys ) @ I )
          = ( product_Pair_nat_a @ ( nth_nat @ Xs @ I ) @ ( nth_a @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_938_nth__zip,axiom,
    ! [I: nat,Xs: list_a,Ys: list_M7834926499894101121tation] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s3663592329813318933tation @ Ys ) )
       => ( ( nth_Pr7991355429936432087tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) @ I )
          = ( produc7247797765490398872tation @ ( nth_a @ Xs @ I ) @ ( nth_Mu116543695464807850tation @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_939_nth__zip,axiom,
    ! [I: nat,Xs: list_a,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr8461465654520414006_a_nat @ ( zip_a_nat @ Xs @ Ys ) @ I )
          = ( product_Pair_a_nat @ ( nth_a @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_940_nth__zip,axiom,
    ! [I: nat,Xs: list_a,Ys: list_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
       => ( ( nth_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) @ I )
          = ( product_Pair_a_a @ ( nth_a @ Xs @ I ) @ ( nth_a @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_941_nth__zip,axiom,
    ! [I: nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s8208510060688613859at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr4952906083868132727at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I )
          = ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_942_nth__zip,axiom,
    ! [I: nat,Xs: list_P3196869577824546119od_a_o,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr4383229733491484665_o_nat @ ( zip_Pr7459748363961616666_o_nat @ Xs @ Ys ) @ I )
          = ( produc8040076513301755962_o_nat @ ( nth_Product_prod_a_o @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_943_nth__zip,axiom,
    ! [I: nat,Xs: list_P3196869577824546119od_a_o,Ys: list_a] :
      ( ( ord_less_nat @ I @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
       => ( ( nth_Pr7891897355475409579_a_o_a @ ( zip_Pr6655751374316219444_a_o_a @ Xs @ Ys ) @ I )
          = ( produc8966842053957846292_a_o_a @ ( nth_Product_prod_a_o @ Xs @ I ) @ ( nth_a @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_944_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s1727999245659094451od_a_o @ Ys ) )
       => ( ( nth_Pr3559203214505555487od_a_o @ ( zip_na4759980013951473400od_a_o @ Xs @ Ys ) @ I )
          = ( produc5340308163291612696od_a_o @ ( nth_nat @ Xs @ I ) @ ( nth_Product_prod_a_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_945_nth__zip,axiom,
    ! [I: nat,Xs: list_a,Ys: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s1727999245659094451od_a_o @ Ys ) )
       => ( ( nth_Pr3445859635990675825od_a_o @ ( zip_a_8763575823207437946od_a_o @ Xs @ Ys ) @ I )
          = ( produc1851294465994288986od_a_o @ ( nth_a @ Xs @ I ) @ ( nth_Product_prod_a_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_946_sorted__rev__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ J3 ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).

% sorted_rev_iff_nth_mono
thf(fact_947_sorted__rev__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ J3 ) @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).

% sorted_rev_iff_nth_mono
thf(fact_948_sorted__rev__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).

% sorted_rev_nth_mono
thf(fact_949_sorted__rev__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs @ J ) @ ( nth_int @ Xs @ I ) ) ) ) ) ).

% sorted_rev_nth_mono
thf(fact_950_sorted__insort__insert__key,axiom,
    ! [F: nat > nat,Xs: list_nat,X: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs ) )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord1921536354676448932at_nat @ F @ X @ Xs ) ) ) ) ).

% sorted_insort_insert_key
thf(fact_951_sorted__insort__insert__key,axiom,
    ! [F: a > nat,Xs: list_a,X: a] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_a_nat @ F @ Xs ) )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_a_nat @ F @ ( linord1046132949341221836_a_nat @ F @ X @ Xs ) ) ) ) ).

% sorted_insort_insert_key
thf(fact_952_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_953_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_954_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_955_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_956_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_957_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_958_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_959_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_960_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_961_length__rev,axiom,
    ! [Xs: list_P3196869577824546119od_a_o] :
      ( ( size_s1727999245659094451od_a_o @ ( rev_Product_prod_a_o @ Xs ) )
      = ( size_s1727999245659094451od_a_o @ Xs ) ) ).

% length_rev
thf(fact_962_length__rev,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( rev_Su2558329242186917479on_nat @ Xs ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_rev
thf(fact_963_length__rev,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( rev_li7470600241620887981od_a_o @ Xs ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_rev
thf(fact_964_length__rev,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rev_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rev
thf(fact_965_length__rev,axiom,
    ! [Xs: list_a] :
      ( ( size_size_list_a @ ( rev_a @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_rev
thf(fact_966_length__rev,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( size_s650155532264187265on_nat @ ( rev_li8757231381177283063on_nat @ Xs ) )
      = ( size_s650155532264187265on_nat @ Xs ) ) ).

% length_rev
thf(fact_967_length__rev,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( size_s129639936549468787on_nat @ ( rev_li931470434339634279on_nat @ Xs ) )
      = ( size_s129639936549468787on_nat @ Xs ) ) ).

% length_rev
thf(fact_968_rev__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( rev_nat @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( rev_nat @ Ys ) @ ( rev_nat @ Xs ) ) ) ).

% rev_append
thf(fact_969_rev__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( rev_Su2558329242186917479on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( rev_Su2558329242186917479on_nat @ Ys ) @ ( rev_Su2558329242186917479on_nat @ Xs ) ) ) ).

% rev_append
thf(fact_970_rev__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( rev_Pr7970673129963250401on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( rev_Pr7970673129963250401on_nat @ Ys ) @ ( rev_Pr7970673129963250401on_nat @ Xs ) ) ) ).

% rev_append
thf(fact_971_length__concat__rev,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( size_s7490768528479217773on_nat @ ( concat937493773106420796on_nat @ ( rev_li931470434339634279on_nat @ Xs ) ) )
      = ( size_s7490768528479217773on_nat @ ( concat937493773106420796on_nat @ Xs ) ) ) ).

% length_concat_rev
thf(fact_972_length__concat__rev,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( size_s1727999245659094451od_a_o @ ( concat7947350394041715842od_a_o @ ( rev_li7470600241620887981od_a_o @ Xs ) ) )
      = ( size_s1727999245659094451od_a_o @ ( concat7947350394041715842od_a_o @ Xs ) ) ) ).

% length_concat_rev
thf(fact_973_length__concat__rev,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( concat6161509665823917772on_nat @ ( rev_li8757231381177283063on_nat @ Xs ) ) )
      = ( size_s5686234912572693873on_nat @ ( concat6161509665823917772on_nat @ Xs ) ) ) ).

% length_concat_rev
thf(fact_974_length__concat__rev,axiom,
    ! [Xs: list_l7431997615882138323od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( concat3529464442532842888od_a_o @ ( rev_li5279380913480878003od_a_o @ Xs ) ) )
      = ( size_s3379185846251495609od_a_o @ ( concat3529464442532842888od_a_o @ Xs ) ) ) ).

% length_concat_rev
thf(fact_975_length__concat__rev,axiom,
    ! [Xs: list_list_nat] :
      ( ( size_size_list_nat @ ( concat_nat @ ( rev_list_nat @ Xs ) ) )
      = ( size_size_list_nat @ ( concat_nat @ Xs ) ) ) ).

% length_concat_rev
thf(fact_976_length__concat__rev,axiom,
    ! [Xs: list_list_a] :
      ( ( size_size_list_a @ ( concat_a @ ( rev_list_a @ Xs ) ) )
      = ( size_size_list_a @ ( concat_a @ Xs ) ) ) ).

% length_concat_rev
thf(fact_977_length__concat__rev,axiom,
    ! [Xs: list_l8365940747029150845on_nat] :
      ( ( size_s650155532264187265on_nat @ ( concat4774721536428481500on_nat @ ( rev_li3223746489240878855on_nat @ Xs ) ) )
      = ( size_s650155532264187265on_nat @ ( concat4774721536428481500on_nat @ Xs ) ) ) ).

% length_concat_rev
thf(fact_978_length__concat__rev,axiom,
    ! [Xs: list_l6719245731233385357on_nat] :
      ( ( size_s129639936549468787on_nat @ ( concat4341727598008994882on_nat @ ( rev_li6897998627854010733on_nat @ Xs ) ) )
      = ( size_s129639936549468787on_nat @ ( concat4341727598008994882on_nat @ Xs ) ) ) ).

% length_concat_rev
thf(fact_979_sum__list__rev,axiom,
    ! [Xs: list_nat] :
      ( ( groups4561878855575611511st_nat @ ( rev_nat @ Xs ) )
      = ( groups4561878855575611511st_nat @ Xs ) ) ).

% sum_list_rev
thf(fact_980_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_981_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_982_zip__append,axiom,
    ! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_983_zip__append,axiom,
    ! [Xs: list_nat,Us: list_a,Ys: list_nat,Vs: list_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Us ) )
     => ( ( zip_nat_a @ ( append_nat @ Xs @ Ys ) @ ( append_a @ Us @ Vs ) )
        = ( append1694031006427026248_nat_a @ ( zip_nat_a @ Xs @ Us ) @ ( zip_nat_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_984_zip__append,axiom,
    ! [Xs: list_a,Us: list_nat,Ys: list_a,Vs: list_nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_a_nat @ ( append_a @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append7679239579558125090_a_nat @ ( zip_a_nat @ Xs @ Us ) @ ( zip_a_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_985_zip__append,axiom,
    ! [Xs: list_a,Us: list_a,Ys: list_a,Vs: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Us ) )
     => ( ( zip_a_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Us @ Vs ) )
        = ( append5335208819046833346od_a_a @ ( zip_a_a @ Xs @ Us ) @ ( zip_a_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_986_zip__append,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Us: list_nat,Ys: list_P3196869577824546119od_a_o,Vs: list_nat] :
      ( ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_Pr7459748363961616666_o_nat @ ( append2462165029785658588od_a_o @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append1777253338276409573_o_nat @ ( zip_Pr7459748363961616666_o_nat @ Xs @ Us ) @ ( zip_Pr7459748363961616666_o_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_987_zip__append,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Us: list_a,Ys: list_P3196869577824546119od_a_o,Vs: list_a] :
      ( ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_size_list_a @ Us ) )
     => ( ( zip_Pr6655751374316219444_a_o_a @ ( append2462165029785658588od_a_o @ Xs @ Ys ) @ ( append_a @ Us @ Vs ) )
        = ( append2889585718960403647_a_o_a @ ( zip_Pr6655751374316219444_a_o_a @ Xs @ Us ) @ ( zip_Pr6655751374316219444_a_o_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_988_zip__append,axiom,
    ! [Xs: list_nat,Us: list_P3196869577824546119od_a_o,Ys: list_nat,Vs: list_P3196869577824546119od_a_o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1727999245659094451od_a_o @ Us ) )
     => ( ( zip_na4759980013951473400od_a_o @ ( append_nat @ Xs @ Ys ) @ ( append2462165029785658588od_a_o @ Us @ Vs ) )
        = ( append953226819290480395od_a_o @ ( zip_na4759980013951473400od_a_o @ Xs @ Us ) @ ( zip_na4759980013951473400od_a_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_989_zip__append,axiom,
    ! [Xs: list_a,Us: list_P3196869577824546119od_a_o,Ys: list_a,Vs: list_P3196869577824546119od_a_o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s1727999245659094451od_a_o @ Us ) )
     => ( ( zip_a_8763575823207437946od_a_o @ ( append_a @ Xs @ Ys ) @ ( append2462165029785658588od_a_o @ Us @ Vs ) )
        = ( append7666920036330445701od_a_o @ ( zip_a_8763575823207437946od_a_o @ Xs @ Us ) @ ( zip_a_8763575823207437946od_a_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_990_zip__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Us: list_nat,Ys: list_l8336264072933094989od_a_o,Vs: list_nat] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_li1039786862263726752_o_nat @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append4227299948903008363_o_nat @ ( zip_li1039786862263726752_o_nat @ Xs @ Us ) @ ( zip_li1039786862263726752_o_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_991_zip__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Us: list_a,Ys: list_l8336264072933094989od_a_o,Vs: list_a] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_a @ Us ) )
     => ( ( zip_li7249964909781669934_a_o_a @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( append_a @ Us @ Vs ) )
        = ( append5952328758920950457_a_o_a @ ( zip_li7249964909781669934_a_o_a @ Xs @ Us ) @ ( zip_li7249964909781669934_a_o_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_992_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_993_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_994_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_995_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_996_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_997_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_998_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_999_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ N )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1000_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_1001_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1002_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ N )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ I4 ) ) ) ) ).

% All_less_Suc
thf(fact_1003_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M6: nat] :
            ( ( M
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1004_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_1005_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_1006_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1007_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I2 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1008_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1009_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1010_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_1011_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_1012_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_1013_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [A2: nat,B2: nat] :
          ( ( semiri1314217659103216013at_int @ A2 )
          = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_1014_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_1015_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1016_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( zip_nat_nat @ ( rev_nat @ Xs ) @ ( rev_nat @ Ys ) )
        = ( rev_Pr6102188148953555047at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1017_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( zip_nat_a @ ( rev_nat @ Xs ) @ ( rev_a @ Ys ) )
        = ( rev_Pr4566615044306411965_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1018_zip__rev,axiom,
    ! [Xs: list_a,Ys: list_nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( zip_a_nat @ ( rev_a @ Xs ) @ ( rev_nat @ Ys ) )
        = ( rev_Pr1328451580582734999_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1019_zip__rev,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( zip_a_a @ ( rev_a @ Xs ) @ ( rev_a @ Ys ) )
        = ( rev_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1020_zip__rev,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_nat] :
      ( ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( zip_Pr7459748363961616666_o_nat @ ( rev_Product_prod_a_o @ Xs ) @ ( rev_nat @ Ys ) )
        = ( rev_Pr2316947648935570650_o_nat @ ( zip_Pr7459748363961616666_o_nat @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1021_zip__rev,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_a] :
      ( ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( zip_Pr6655751374316219444_a_o_a @ ( rev_Product_prod_a_o @ Xs ) @ ( rev_a @ Ys ) )
        = ( rev_Pr1805073168067529738_a_o_a @ ( zip_Pr6655751374316219444_a_o_a @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1022_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_P3196869577824546119od_a_o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1727999245659094451od_a_o @ Ys ) )
     => ( ( zip_na4759980013951473400od_a_o @ ( rev_nat @ Xs ) @ ( rev_Product_prod_a_o @ Ys ) )
        = ( rev_Pr1492921129949641472od_a_o @ ( zip_na4759980013951473400od_a_o @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1023_zip__rev,axiom,
    ! [Xs: list_a,Ys: list_P3196869577824546119od_a_o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s1727999245659094451od_a_o @ Ys ) )
     => ( ( zip_a_8763575823207437946od_a_o @ ( rev_a @ Xs ) @ ( rev_Product_prod_a_o @ Ys ) )
        = ( rev_Pr6582407485437571792od_a_o @ ( zip_a_8763575823207437946od_a_o @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1024_zip__rev,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( zip_li1039786862263726752_o_nat @ ( rev_li7470600241620887981od_a_o @ Xs ) @ ( rev_nat @ Ys ) )
        = ( rev_Pr7814164308232144352_o_nat @ ( zip_li1039786862263726752_o_nat @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1025_zip__rev,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_a] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( zip_li7249964909781669934_a_o_a @ ( rev_li7470600241620887981od_a_o @ Xs ) @ ( rev_a @ Ys ) )
        = ( rev_Pr5954160277411162244_a_o_a @ ( zip_li7249964909781669934_a_o_a @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_1026_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_1027_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_1028_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_1029_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1030_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1031_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1032_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X2: nat] : ( R @ X2 @ X2 )
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z3 )
               => ( R @ X2 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1033_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1034_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
             => ( P @ M5 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1035_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1036_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1037_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1038_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
     => ? [M3: nat] :
          ( M7
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_1039_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1040_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1041_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1042_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_1043_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1044_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1045_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1046_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1047_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
     => ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
       => ( ! [X2: nat,Y2: nat] :
              ( ( P @ X2 @ Y2 )
             => ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1048_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1049_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1050_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1051_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1052_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1053_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1054_drop__zip,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( drop_P8868858903918902087at_nat @ N @ ( zip_nat_nat @ Xs @ Ys ) )
      = ( zip_nat_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_1055_rev__map,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( rev_li931470434339634279on_nat @ ( map_na8715451351012060651on_nat @ F @ Xs ) )
      = ( map_na8715451351012060651on_nat @ F @ ( rev_nat @ Xs ) ) ) ).

% rev_map
thf(fact_1056_rev__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( rev_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( rev_nat @ Xs ) ) ) ).

% rev_map
thf(fact_1057_rev__map,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( rev_Su2558329242186917479on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( map_a_2711072540680412167on_nat @ F @ ( rev_a @ Xs ) ) ) ).

% rev_map
thf(fact_1058_rev__map,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( rev_li931470434339634279on_nat @ ( map_a_7431629901395152583on_nat @ F @ Xs ) )
      = ( map_a_7431629901395152583on_nat @ F @ ( rev_a @ Xs ) ) ) ).

% rev_map
thf(fact_1059_rev__map,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( rev_li8757231381177283063on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( map_a_3887198048842487703on_nat @ F @ ( rev_a @ Xs ) ) ) ).

% rev_map
thf(fact_1060_rev__map,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( rev_a @ ( map_nat_a @ F @ Xs ) )
      = ( map_nat_a @ F @ ( rev_nat @ Xs ) ) ) ).

% rev_map
thf(fact_1061_rev__map,axiom,
    ! [F: a > product_prod_a_o,Xs: list_a] :
      ( ( rev_Product_prod_a_o @ ( map_a_1137562357930033543od_a_o @ F @ Xs ) )
      = ( map_a_1137562357930033543od_a_o @ F @ ( rev_a @ Xs ) ) ) ).

% rev_map
thf(fact_1062_rev__map,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,Xs: list_a] :
      ( ( rev_li7470600241620887981od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ Xs ) )
      = ( map_a_5058921920454310925od_a_o @ F @ ( rev_a @ Xs ) ) ) ).

% rev_map
thf(fact_1063_rev__map,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( rev_nat @ ( map_a_nat @ F @ Xs ) )
      = ( map_a_nat @ F @ ( rev_a @ Xs ) ) ) ).

% rev_map
thf(fact_1064_rev__map,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( rev_a @ ( map_a_a @ F @ Xs ) )
      = ( map_a_a @ F @ ( rev_a @ Xs ) ) ) ).

% rev_map
thf(fact_1065_rev__concat,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( rev_Pr7970673129963250401on_nat @ ( concat937493773106420796on_nat @ Xs ) )
      = ( concat937493773106420796on_nat @ ( map_li815642509948034692on_nat @ rev_Pr7970673129963250401on_nat @ ( rev_li931470434339634279on_nat @ Xs ) ) ) ) ).

% rev_concat
thf(fact_1066_rev__concat,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( rev_Su2558329242186917479on_nat @ ( concat6161509665823917772on_nat @ Xs ) )
      = ( concat6161509665823917772on_nat @ ( map_li4295621887133137792on_nat @ rev_Su2558329242186917479on_nat @ ( rev_li8757231381177283063on_nat @ Xs ) ) ) ) ).

% rev_concat
thf(fact_1067_imp__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P5: $o] :
      ( ( X = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P5 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
           => P5 ) ) ) ) ).

% imp_le_cong
thf(fact_1068_conj__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P5: $o] :
      ( ( X = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P5 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
            & P5 ) ) ) ) ).

% conj_le_cong
thf(fact_1069_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1070_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1071_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1072_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1073_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1074_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1075_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1076_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1077_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_1078_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_1079_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1080_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M4: nat] :
            ( N
            = ( suc @ M4 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1081_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1082_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_1083_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1084_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1085_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1086_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1087_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1088_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1089_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1090_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1091_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1092_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1093_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1094_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1095_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1096_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N3: nat] :
        ? [K2: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1097_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1098_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1099_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q2: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).

% less_natE
thf(fact_1100_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I4 ) ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).

% sorted_rev_iff_nth_Suc
thf(fact_1101_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ ( suc @ I4 ) ) @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).

% sorted_rev_iff_nth_Suc
thf(fact_1102_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K3 )
               => ~ ( P @ I3 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1103_assms_I1_J,axiom,
    ( ( multis4221440410396274297_cnf_a @ cnf )
    = ( produc1545487600272464053on_nat @ xs @ ( produc4176415733022414961on_nat @ ys @ ( produc7152764266822961111on_nat @ s @ ns ) ) ) ) ).

% assms(1)
thf(fact_1104_inf__concat__simple_Ocases,axiom,
    ! [X: produc8199716216217303280at_nat] :
      ( ! [F3: nat > nat] :
          ( X
         != ( produc72220940542539688at_nat @ F3 @ zero_zero_nat ) )
     => ~ ! [F3: nat > nat,N2: nat] :
            ( X
           != ( produc72220940542539688at_nat @ F3 @ ( suc @ N2 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_1105_sorted__transpose,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li381573783031109843_o_nat @ size_s1727999245659094451od_a_o @ ( transp726662699833356113od_a_o @ Xs ) ) ) ) ).

% sorted_transpose
thf(fact_1106_sorted__transpose,axiom,
    ! [Xs: list_l8999537460041914605on_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li2142278717278591475at_nat @ size_s5686234912572693873on_nat @ ( transp8163977000467346109on_nat @ Xs ) ) ) ) ).

% sorted_transpose
thf(fact_1107_sorted__transpose,axiom,
    ! [Xs: list_l7431997615882138323od_a_o] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li999134049962033241_o_nat @ size_s3379185846251495609od_a_o @ ( transp4609918424245124311od_a_o @ Xs ) ) ) ) ).

% sorted_transpose
thf(fact_1108_sorted__transpose,axiom,
    ! [Xs: list_list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( transpose_nat @ Xs ) ) ) ) ).

% sorted_transpose
thf(fact_1109_sorted__transpose,axiom,
    ! [Xs: list_list_a] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_a_nat @ size_size_list_a @ ( transpose_a @ Xs ) ) ) ) ).

% sorted_transpose
thf(fact_1110_sorted__transpose,axiom,
    ! [Xs: list_l8365940747029150845on_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li3435095097222882691at_nat @ size_s650155532264187265on_nat @ ( transp8770635966524267341on_nat @ Xs ) ) ) ) ).

% sorted_transpose
thf(fact_1111_sorted__transpose,axiom,
    ! [Xs: list_l6719245731233385357on_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li8636112849402572307at_nat @ size_s129639936549468787on_nat @ ( transp6957102645529343889on_nat @ Xs ) ) ) ) ).

% sorted_transpose
thf(fact_1112_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1113_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_a,R2: set_Pr4193341848836149977_nat_a] :
      ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ Ys ) @ ( listrel_nat_a @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ ( nth_nat @ Xs @ N3 ) @ ( nth_a @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1114_listrel__iff__nth,axiom,
    ! [Xs: list_a,Ys: list_M7834926499894101121tation,R2: set_Pr3525538344199662036tation] :
      ( ( member1315478125161133003tation @ ( produc8106702317871607470tation @ Xs @ Ys ) @ ( listre7259538626121459390tation @ R2 ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_s3663592329813318933tation @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
           => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ ( nth_a @ Xs @ N3 ) @ ( nth_Mu116543695464807850tation @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1115_listrel__iff__nth,axiom,
    ! [Xs: list_a,Ys: list_nat,R2: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R2 ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
           => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ ( nth_a @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1116_listrel__iff__nth,axiom,
    ! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N3 ) @ ( nth_a @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1117_listrel__iff__nth,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,R2: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R2 ) )
      = ( ( ( size_s8208510060688613859at_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s8208510060688613859at_nat @ Xs ) )
           => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1118_listrel__iff__nth,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_nat,R2: set_Pr203297967321116406_o_nat] :
      ( ( member4901634147397032557st_nat @ ( produc5808336593651053776st_nat @ Xs @ Ys ) @ ( listre6563109718979866208_o_nat @ R2 ) )
      = ( ( ( size_s1727999245659094451od_a_o @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s1727999245659094451od_a_o @ Xs ) )
           => ( member6380975538497366487_o_nat @ ( produc8040076513301755962_o_nat @ ( nth_Product_prod_a_o @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1119_listrel__iff__nth,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_a,R2: set_Pr7970750561109311876_a_o_a] :
      ( ( member782963794189141453list_a @ ( produc890980275325801876list_a @ Xs @ Ys ) @ ( listre2383635182559186030_a_o_a @ R2 ) )
      = ( ( ( size_s1727999245659094451od_a_o @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s1727999245659094451od_a_o @ Xs ) )
           => ( member5411179314388669517_a_o_a @ ( produc8966842053957846292_a_o_a @ ( nth_Product_prod_a_o @ Xs @ N3 ) @ ( nth_a @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1120_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_P3196869577824546119od_a_o,R2: set_Pr1737513652403227164od_a_o] :
      ( ( member4548588291273015283od_a_o @ ( produc863210786870130958od_a_o @ Xs @ Ys ) @ ( listre3863341368969722942od_a_o @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s1727999245659094451od_a_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member5556949019511437309od_a_o @ ( produc5340308163291612696od_a_o @ ( nth_nat @ Xs @ N3 ) @ ( nth_Product_prod_a_o @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1121_listrel__iff__nth,axiom,
    ! [Xs: list_a,Ys: list_P3196869577824546119od_a_o,R2: set_Pr168325451112473930od_a_o] :
      ( ( member760206308082160659od_a_o @ ( produc3649760268003600218od_a_o @ Xs @ Ys ) @ ( listre4491459631450404532od_a_o @ R2 ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_s1727999245659094451od_a_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
           => ( member965141594903935763od_a_o @ ( produc1851294465994288986od_a_o @ ( nth_a @ Xs @ N3 ) @ ( nth_Product_prod_a_o @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1122_map__tailrec__rev,axiom,
    ( map_ta7223986904443112186on_nat
    = ( ^ [F2: nat > list_P9127870862658512897on_nat,As: list_nat] : ( append3839213209832495004on_nat @ ( rev_li931470434339634279on_nat @ ( map_na8715451351012060651on_nat @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1123_map__tailrec__rev,axiom,
    ( map_ta7164188454487880599at_nat
    = ( ^ [F2: nat > nat,As: list_nat] : ( append_nat @ ( rev_nat @ ( map_nat_nat @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1124_map__tailrec__rev,axiom,
    ( map_ta1893281715583187990on_nat
    = ( ^ [F2: a > sum_su4907927656612468301on_nat,As: list_a] : ( append3523812819975775986on_nat @ ( rev_Su2558329242186917479on_nat @ ( map_a_2711072540680412167on_nat @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1125_map__tailrec__rev,axiom,
    ( map_ta8846586008205793144on_nat
    = ( ^ [F2: a > list_P9127870862658512897on_nat,As: list_a] : ( append3839213209832495004on_nat @ ( rev_li931470434339634279on_nat @ ( map_a_7431629901395152583on_nat @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1126_map__tailrec__rev,axiom,
    ( map_ta797854582910477350on_nat
    = ( ^ [F2: a > list_S5253404628753938141on_nat,As: list_a] : ( append6174841615030841602on_nat @ ( rev_li8757231381177283063on_nat @ ( map_a_3887198048842487703on_nat @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1127_map__tailrec__rev,axiom,
    ( map_ta3519391893248468727_nat_a
    = ( ^ [F2: nat > a,As: list_nat] : ( append_a @ ( rev_a @ ( map_nat_a @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1128_map__tailrec__rev,axiom,
    ( map_ta4243852458646442680od_a_o
    = ( ^ [F2: a > product_prod_a_o,As: list_a] : ( append2462165029785658588od_a_o @ ( rev_Product_prod_a_o @ ( map_a_1137562357930033543od_a_o @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1129_map__tailrec__rev,axiom,
    ( map_ta6072168029591339198od_a_o
    = ( ^ [F2: a > list_P3196869577824546119od_a_o,As: list_a] : ( append2159120467191820514od_a_o @ ( rev_li7470600241620887981od_a_o @ ( map_a_5058921920454310925od_a_o @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1130_map__tailrec__rev,axiom,
    ( map_ta8710832428924958105_a_nat
    = ( ^ [F2: a > nat,As: list_a] : ( append_nat @ ( rev_nat @ ( map_a_nat @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1131_map__tailrec__rev,axiom,
    ( map_tailrec_rev_a_a
    = ( ^ [F2: a > a,As: list_a] : ( append_a @ ( rev_a @ ( map_a_a @ F2 @ As ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1132_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1133_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_a,R2: set_Pr4193341848836149977_nat_a] :
      ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ Ys ) @ ( listrel_nat_a @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1134_listrel__eq__len,axiom,
    ! [Xs: list_a,Ys: list_nat,R2: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R2 ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1135_listrel__eq__len,axiom,
    ! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1136_listrel__eq__len,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_nat,R2: set_Pr203297967321116406_o_nat] :
      ( ( member4901634147397032557st_nat @ ( produc5808336593651053776st_nat @ Xs @ Ys ) @ ( listre6563109718979866208_o_nat @ R2 ) )
     => ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1137_listrel__eq__len,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_a,R2: set_Pr7970750561109311876_a_o_a] :
      ( ( member782963794189141453list_a @ ( produc890980275325801876list_a @ Xs @ Ys ) @ ( listre2383635182559186030_a_o_a @ R2 ) )
     => ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1138_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_P3196869577824546119od_a_o,R2: set_Pr1737513652403227164od_a_o] :
      ( ( member4548588291273015283od_a_o @ ( produc863210786870130958od_a_o @ Xs @ Ys ) @ ( listre3863341368969722942od_a_o @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s1727999245659094451od_a_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1139_listrel__eq__len,axiom,
    ! [Xs: list_a,Ys: list_P3196869577824546119od_a_o,R2: set_Pr168325451112473930od_a_o] :
      ( ( member760206308082160659od_a_o @ ( produc3649760268003600218od_a_o @ Xs @ Ys ) @ ( listre4491459631450404532od_a_o @ R2 ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_s1727999245659094451od_a_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1140_listrel__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,R2: set_Pr8850484174313502844_o_nat] :
      ( ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ Xs @ Ys ) @ ( listre90293961589800166_o_nat @ R2 ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1141_listrel__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_a,R2: set_Pr6398239925682376062_a_o_a] :
      ( ( member7130946049884474439list_a @ ( produc5675822972178626830list_a @ Xs @ Ys ) @ ( listre7037198679601288040_a_o_a @ R2 ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1142_transpose__map__map,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_list_nat] :
      ( ( transp6957102645529343889on_nat @ ( map_li3981902705288052833on_nat @ ( map_na8715451351012060651on_nat @ F ) @ Xs ) )
      = ( map_li3981902705288052833on_nat @ ( map_na8715451351012060651on_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1143_transpose__map__map,axiom,
    ! [F: nat > nat,Xs: list_list_nat] :
      ( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) )
      = ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1144_transpose__map__map,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_list_a] :
      ( ( transp8163977000467346109on_nat @ ( map_li3793676233746214813on_nat @ ( map_a_2711072540680412167on_nat @ F ) @ Xs ) )
      = ( map_li3793676233746214813on_nat @ ( map_a_2711072540680412167on_nat @ F ) @ ( transpose_a @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1145_transpose__map__map,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_list_a] :
      ( ( transp6957102645529343889on_nat @ ( map_li4634365374901557063on_nat @ ( map_a_7431629901395152583on_nat @ F ) @ Xs ) )
      = ( map_li4634365374901557063on_nat @ ( map_a_7431629901395152583on_nat @ F ) @ ( transpose_a @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1146_transpose__map__map,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_list_a] :
      ( ( transp8770635966524267341on_nat @ ( map_li7277942993266323885on_nat @ ( map_a_3887198048842487703on_nat @ F ) @ Xs ) )
      = ( map_li7277942993266323885on_nat @ ( map_a_3887198048842487703on_nat @ F ) @ ( transpose_a @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1147_transpose__map__map,axiom,
    ! [F: nat > a,Xs: list_list_nat] :
      ( ( transpose_a @ ( map_list_nat_list_a @ ( map_nat_a @ F ) @ Xs ) )
      = ( map_list_nat_list_a @ ( map_nat_a @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1148_transpose__map__map,axiom,
    ! [F: a > product_prod_a_o,Xs: list_list_a] :
      ( ( transp726662699833356113od_a_o @ ( map_li1480699347688408199od_a_o @ ( map_a_1137562357930033543od_a_o @ F ) @ Xs ) )
      = ( map_li1480699347688408199od_a_o @ ( map_a_1137562357930033543od_a_o @ F ) @ ( transpose_a @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1149_transpose__map__map,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,Xs: list_list_a] :
      ( ( transp4609918424245124311od_a_o @ ( map_li7794534110530448781od_a_o @ ( map_a_5058921920454310925od_a_o @ F ) @ Xs ) )
      = ( map_li7794534110530448781od_a_o @ ( map_a_5058921920454310925od_a_o @ F ) @ ( transpose_a @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1150_transpose__map__map,axiom,
    ! [F: a > nat,Xs: list_list_a] :
      ( ( transpose_nat @ ( map_list_a_list_nat @ ( map_a_nat @ F ) @ Xs ) )
      = ( map_list_a_list_nat @ ( map_a_nat @ F ) @ ( transpose_a @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1151_transpose__map__map,axiom,
    ! [F: a > a,Xs: list_list_a] :
      ( ( transpose_a @ ( map_list_a_list_a @ ( map_a_a @ F ) @ Xs ) )
      = ( map_list_a_list_a @ ( map_a_a @ F ) @ ( transpose_a @ Xs ) ) ) ).

% transpose_map_map
thf(fact_1152_length__transpose__sorted,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li8334353970798598029at_nat @ size_s7490768528479217773on_nat @ Xs ) ) )
     => ( ( ( Xs = nil_li2682182165152866823on_nat )
         => ( ( size_s129639936549468787on_nat @ ( transp5043004568955403019on_nat @ Xs ) )
            = zero_zero_nat ) )
        & ( ( Xs != nil_li2682182165152866823on_nat )
         => ( ( size_s129639936549468787on_nat @ ( transp5043004568955403019on_nat @ Xs ) )
            = ( size_s7490768528479217773on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ zero_zero_nat ) ) ) ) ) ) ).

% length_transpose_sorted
thf(fact_1153_length__transpose__sorted,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li381573783031109843_o_nat @ size_s1727999245659094451od_a_o @ Xs ) ) )
     => ( ( ( Xs = nil_li1372968520647675725od_a_o )
         => ( ( size_s3379185846251495609od_a_o @ ( transp726662699833356113od_a_o @ Xs ) )
            = zero_zero_nat ) )
        & ( ( Xs != nil_li1372968520647675725od_a_o )
         => ( ( size_s3379185846251495609od_a_o @ ( transp726662699833356113od_a_o @ Xs ) )
            = ( size_s1727999245659094451od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ zero_zero_nat ) ) ) ) ) ) ).

% length_transpose_sorted
thf(fact_1154_length__transpose__sorted,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li2142278717278591475at_nat @ size_s5686234912572693873on_nat @ Xs ) ) )
     => ( ( ( Xs = nil_li5563851662744376407on_nat )
         => ( ( size_s650155532264187265on_nat @ ( transp8163977000467346109on_nat @ Xs ) )
            = zero_zero_nat ) )
        & ( ( Xs != nil_li5563851662744376407on_nat )
         => ( ( size_s650155532264187265on_nat @ ( transp8163977000467346109on_nat @ Xs ) )
            = ( size_s5686234912572693873on_nat @ ( nth_li4091121329102240790on_nat @ Xs @ zero_zero_nat ) ) ) ) ) ) ).

% length_transpose_sorted
thf(fact_1155_length__transpose__sorted,axiom,
    ! [Xs: list_l7431997615882138323od_a_o] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li999134049962033241_o_nat @ size_s3379185846251495609od_a_o @ Xs ) ) )
     => ( ( ( Xs = nil_li6902169125192129875od_a_o )
         => ( ( size_s1509257606055081791od_a_o @ ( transp4609918424245124311od_a_o @ Xs ) )
            = zero_zero_nat ) )
        & ( ( Xs != nil_li6902169125192129875od_a_o )
         => ( ( size_s1509257606055081791od_a_o @ ( transp4609918424245124311od_a_o @ Xs ) )
            = ( size_s3379185846251495609od_a_o @ ( nth_li2142833064033982036od_a_o @ Xs @ zero_zero_nat ) ) ) ) ) ) ).

% length_transpose_sorted
thf(fact_1156_length__transpose__sorted,axiom,
    ! [Xs: list_list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs ) ) )
     => ( ( ( Xs = nil_list_nat )
         => ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
            = zero_zero_nat ) )
        & ( ( Xs != nil_list_nat )
         => ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
            = ( size_size_list_nat @ ( nth_list_nat @ Xs @ zero_zero_nat ) ) ) ) ) ) ).

% length_transpose_sorted
thf(fact_1157_length__transpose__sorted,axiom,
    ! [Xs: list_list_a] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_a_nat @ size_size_list_a @ Xs ) ) )
     => ( ( ( Xs = nil_list_a )
         => ( ( size_s349497388124573686list_a @ ( transpose_a @ Xs ) )
            = zero_zero_nat ) )
        & ( ( Xs != nil_list_a )
         => ( ( size_s349497388124573686list_a @ ( transpose_a @ Xs ) )
            = ( size_size_list_a @ ( nth_list_a @ Xs @ zero_zero_nat ) ) ) ) ) ) ).

% length_transpose_sorted
thf(fact_1158_length__transpose__sorted,axiom,
    ! [Xs: list_l8365940747029150845on_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li3435095097222882691at_nat @ size_s650155532264187265on_nat @ Xs ) ) )
     => ( ( ( Xs = nil_li3323240303193190759on_nat )
         => ( ( size_s8175741288642522897on_nat @ ( transp8770635966524267341on_nat @ Xs ) )
            = zero_zero_nat ) )
        & ( ( Xs != nil_li3323240303193190759on_nat )
         => ( ( size_s8175741288642522897on_nat @ ( transp8770635966524267341on_nat @ Xs ) )
            = ( size_s650155532264187265on_nat @ ( nth_li1677731029046685094on_nat @ Xs @ zero_zero_nat ) ) ) ) ) ) ).

% length_transpose_sorted
thf(fact_1159_length__transpose__sorted,axiom,
    ! [Xs: list_l6719245731233385357on_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li8636112849402572307at_nat @ size_s129639936549468787on_nat @ Xs ) ) )
     => ( ( ( Xs = nil_li985476221562969869on_nat )
         => ( ( size_s6092282996036616185on_nat @ ( transp6957102645529343889on_nat @ Xs ) )
            = zero_zero_nat ) )
        & ( ( Xs != nil_li985476221562969869on_nat )
         => ( ( size_s6092282996036616185on_nat @ ( transp6957102645529343889on_nat @ Xs ) )
            = ( size_s129639936549468787on_nat @ ( nth_li6929086678380665870on_nat @ Xs @ zero_zero_nat ) ) ) ) ) ) ).

% length_transpose_sorted
thf(fact_1160_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr8202701763255893447on_nat > set_Pr8202701763255893447on_nat > set_Pr8586234676504894535on_nat,Ns_list: set_Pr8202701763255893447on_nat > set_Pr8202701763255893447on_nat > set_Pr8586234676504894535on_nat,As2: list_P9127870862658512897on_nat,Bs: list_P9127870862658512897on_nat,NS: set_Pr8202701763255893447on_nat,S2: set_Pr8202701763255893447on_nat] :
      ( ( list_l5400313399564392903on_nat @ S_list @ Ns_list )
     => ( ( ( size_s7490768528479217773on_nat @ As2 )
          = ( size_s7490768528479217773on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Bs ) )
             => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ As2 @ I2 ) @ ( nth_Pr3336893728888831618on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As2 @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_1161_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr1246879249572150087od_a_o > set_Pr1246879249572150087od_a_o > set_Pr7420778282784443591od_a_o,Ns_list: set_Pr1246879249572150087od_a_o > set_Pr1246879249572150087od_a_o > set_Pr7420778282784443591od_a_o,As2: list_P3196869577824546119od_a_o,Bs: list_P3196869577824546119od_a_o,NS: set_Pr1246879249572150087od_a_o,S2: set_Pr1246879249572150087od_a_o] :
      ( ( list_l4384394330513415693od_a_o @ S_list @ Ns_list )
     => ( ( ( size_s1727999245659094451od_a_o @ As2 )
          = ( size_s1727999245659094451od_a_o @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s1727999245659094451od_a_o @ Bs ) )
             => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ ( nth_Product_prod_a_o @ As2 @ I2 ) @ ( nth_Product_prod_a_o @ Bs @ I2 ) ) @ NS ) )
         => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ As2 @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_1162_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,As2: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat,NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( list_l4998409290742832897on_nat @ S_list @ Ns_list )
     => ( ( ( size_s5686234912572693873on_nat @ As2 )
          = ( size_s5686234912572693873on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Bs ) )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As2 @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As2 @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_1163_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr7420778282784443591od_a_o > set_Pr7420778282784443591od_a_o > set_Pr6888711208267297863od_a_o,Ns_list: set_Pr7420778282784443591od_a_o > set_Pr7420778282784443591od_a_o > set_Pr6888711208267297863od_a_o,As2: list_l8336264072933094989od_a_o,Bs: list_l8336264072933094989od_a_o,NS: set_Pr7420778282784443591od_a_o,S2: set_Pr7420778282784443591od_a_o] :
      ( ( list_l1016356893908098451od_a_o @ S_list @ Ns_list )
     => ( ( ( size_s3379185846251495609od_a_o @ As2 )
          = ( size_s3379185846251495609od_a_o @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Bs ) )
             => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As2 @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Bs @ I2 ) ) @ NS ) )
         => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As2 @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_1164_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,As2: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ( ( size_size_list_nat @ As2 )
          = ( size_size_list_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As2 @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_1165_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,Ns_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,As2: list_a,Bs: list_a,NS: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( list_l5019108106899435338sion_a @ S_list @ Ns_list )
     => ( ( ( size_size_list_a @ As2 )
          = ( size_size_list_a @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Bs ) )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ As2 @ I2 ) @ ( nth_a @ Bs @ I2 ) ) @ NS ) )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As2 @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_1166_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,Ns_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,As2: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat,NS: set_Pr7828957961435280891on_nat,S2: set_Pr7828957961435280891on_nat] :
      ( ( list_l1107520425274447249on_nat @ S_list @ Ns_list )
     => ( ( ( size_s650155532264187265on_nat @ As2 )
          = ( size_s650155532264187265on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s650155532264187265on_nat @ Bs ) )
             => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( nth_li4091121329102240790on_nat @ As2 @ I2 ) @ ( nth_li4091121329102240790on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As2 @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_1167_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr8586234676504894535on_nat > set_Pr8586234676504894535on_nat > set_Pr9215882266573312199on_nat,Ns_list: set_Pr8586234676504894535on_nat > set_Pr8586234676504894535on_nat > set_Pr9215882266573312199on_nat,As2: list_l1726306699111145479on_nat,Bs: list_l1726306699111145479on_nat,NS: set_Pr8586234676504894535on_nat,S2: set_Pr8586234676504894535on_nat] :
      ( ( list_l2763215806797416525on_nat @ S_list @ Ns_list )
     => ( ( ( size_s129639936549468787on_nat @ As2 )
          = ( size_s129639936549468787on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Bs ) )
             => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ As2 @ I2 ) @ ( nth_li1302981970230039432on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ As2 @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_1168_transpose__column,axiom,
    ! [Xs: list_l1726306699111145479on_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li8334353970798598029at_nat @ size_s7490768528479217773on_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Xs ) )
       => ( ( map_li5165994537674389374on_nat
            @ ^ [Ys3: list_P9127870862658512897on_nat] : ( nth_Pr3336893728888831618on_nat @ Ys3 @ I )
            @ ( filter6410360991914645630on_nat
              @ ^ [Ys3: list_P9127870862658512897on_nat] : ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Ys3 ) )
              @ ( transp5043004568955403019on_nat @ Xs ) ) )
          = ( nth_li1302981970230039432on_nat @ Xs @ I ) ) ) ) ).

% transpose_column
thf(fact_1169_transpose__column,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li381573783031109843_o_nat @ size_s1727999245659094451od_a_o @ Xs ) ) )
     => ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( ( map_li1096681790294202110od_a_o
            @ ^ [Ys3: list_P3196869577824546119od_a_o] : ( nth_Product_prod_a_o @ Ys3 @ I )
            @ ( filter2863170042043607492od_a_o
              @ ^ [Ys3: list_P3196869577824546119od_a_o] : ( ord_less_nat @ I @ ( size_s1727999245659094451od_a_o @ Ys3 ) )
              @ ( transp726662699833356113od_a_o @ Xs ) ) )
          = ( nth_li3248255784598043854od_a_o @ Xs @ I ) ) ) ) ).

% transpose_column
thf(fact_1170_transpose__column,axiom,
    ! [Xs: list_l8999537460041914605on_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li2142278717278591475at_nat @ size_s5686234912572693873on_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I @ ( size_s650155532264187265on_nat @ Xs ) )
       => ( ( map_li5270162858998351984on_nat
            @ ^ [Ys3: list_S5253404628753938141on_nat] : ( nth_Su8471564416988646406on_nat @ Ys3 @ I )
            @ ( filter2112716456448801440on_nat
              @ ^ [Ys3: list_S5253404628753938141on_nat] : ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ys3 ) )
              @ ( transp8163977000467346109on_nat @ Xs ) ) )
          = ( nth_li4091121329102240790on_nat @ Xs @ I ) ) ) ) ).

% transpose_column
thf(fact_1171_transpose__column,axiom,
    ! [Xs: list_l7431997615882138323od_a_o,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li999134049962033241_o_nat @ size_s3379185846251495609od_a_o @ Xs ) ) )
     => ( ( ord_less_nat @ I @ ( size_s1509257606055081791od_a_o @ Xs ) )
       => ( ( map_li1518855448487388670od_a_o
            @ ^ [Ys3: list_l8336264072933094989od_a_o] : ( nth_li3248255784598043854od_a_o @ Ys3 @ I )
            @ ( filter6355613211903447114od_a_o
              @ ^ [Ys3: list_l8336264072933094989od_a_o] : ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Ys3 ) )
              @ ( transp4609918424245124311od_a_o @ Xs ) ) )
          = ( nth_li2142833064033982036od_a_o @ Xs @ I ) ) ) ) ).

% transpose_column
thf(fact_1172_transpose__column,axiom,
    ! [Xs: list_list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ( map_list_nat_nat
            @ ^ [Ys3: list_nat] : ( nth_nat @ Ys3 @ I )
            @ ( filter_list_nat
              @ ^ [Ys3: list_nat] : ( ord_less_nat @ I @ ( size_size_list_nat @ Ys3 ) )
              @ ( transpose_nat @ Xs ) ) )
          = ( nth_list_nat @ Xs @ I ) ) ) ) ).

% transpose_column
thf(fact_1173_transpose__column,axiom,
    ! [Xs: list_list_a,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_a_nat @ size_size_list_a @ Xs ) ) )
     => ( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
       => ( ( map_list_a_a
            @ ^ [Ys3: list_a] : ( nth_a @ Ys3 @ I )
            @ ( filter_list_a
              @ ^ [Ys3: list_a] : ( ord_less_nat @ I @ ( size_size_list_a @ Ys3 ) )
              @ ( transpose_a @ Xs ) ) )
          = ( nth_list_a @ Xs @ I ) ) ) ) ).

% transpose_column
thf(fact_1174_transpose__column,axiom,
    ! [Xs: list_l8365940747029150845on_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li3435095097222882691at_nat @ size_s650155532264187265on_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I @ ( size_s8175741288642522897on_nat @ Xs ) )
       => ( ( map_li3418768000576831248on_nat
            @ ^ [Ys3: list_l8999537460041914605on_nat] : ( nth_li4091121329102240790on_nat @ Ys3 @ I )
            @ ( filter4537381055655288624on_nat
              @ ^ [Ys3: list_l8999537460041914605on_nat] : ( ord_less_nat @ I @ ( size_s650155532264187265on_nat @ Ys3 ) )
              @ ( transp8770635966524267341on_nat @ Xs ) ) )
          = ( nth_li1677731029046685094on_nat @ Xs @ I ) ) ) ) ).

% transpose_column
thf(fact_1175_transpose__column,axiom,
    ! [Xs: list_l6719245731233385357on_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_li8636112849402572307at_nat @ size_s129639936549468787on_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I @ ( size_s6092282996036616185on_nat @ Xs ) )
       => ( ( map_li262107655887197054on_nat
            @ ^ [Ys3: list_l1726306699111145479on_nat] : ( nth_li1302981970230039432on_nat @ Ys3 @ I )
            @ ( filter2477471313034112516on_nat
              @ ^ [Ys3: list_l1726306699111145479on_nat] : ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Ys3 ) )
              @ ( transp6957102645529343889on_nat @ Xs ) ) )
          = ( nth_li6929086678380665870on_nat @ Xs @ I ) ) ) ) ).

% transpose_column
thf(fact_1176_map__ident,axiom,
    ( ( map_a_a
      @ ^ [X3: a] : X3 )
    = ( ^ [Xs3: list_a] : Xs3 ) ) ).

% map_ident
thf(fact_1177_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_leq_as_int
thf(fact_1178_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_1179_remdups__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( remdups_nat @ ( upt @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% remdups_upt
thf(fact_1180_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_1181_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).

% drop_upt
thf(fact_1182_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_1183_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_1184_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_1185_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_1186_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_1187_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_1188_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_1189_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_1190_S,axiom,
    ( s
    = ( concat937493773106420796on_nat
      @ ( map_na8715451351012060651on_nat
        @ ^ [I4: nat] :
            ( map_Pr6945619049609468478on_nat
            @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I4 ) )
            @ ( nth_li3248255784598043854od_a_o @ cnf @ I4 ) )
        @ ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) ) ) ).

% S
thf(fact_1191_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1192_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1193_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1194_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1195_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1196_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_1197_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1198_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1199_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1200_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1201_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1202_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1203_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_1204_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1205_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1206_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1207_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1208_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1209_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1210_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1211_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1212_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1213_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1214_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1215_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1216_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1217_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_1218_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_1219_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_1220_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_1221_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1222_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1223_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_1224_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_1225_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1226_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1227_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1228_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1229_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1230_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1231_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1232_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1233_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_1234_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1235_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1236_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1237_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1238_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1239_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_1240_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1241_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1242_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_1243__092_060open_062length_Ays_A_061_Alength_A_Imap_A_I_092_060lambda_062x_O_AInl_A_Ix_M_AUnsigned_J_J_Avs_A_064_Amap_AInr_A_0910_O_O_060length_Acnf_093_J_092_060close_062,axiom,
    ( ( size_s5686234912572693873on_nat @ ys )
    = ( size_s5686234912572693873on_nat
      @ ( append3523812819975775986on_nat
        @ ( map_a_2711072540680412167on_nat
          @ ^ [X3: a] : ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5512618279175645674signed ) )
          @ vs )
        @ ( map_na5706685718985795555on_nat @ sum_In255631617079920996tation @ ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) ) ) ) ).

% \<open>length ys = length (map (\<lambda>x. Inl (x, Unsigned)) vs @ map Inr [0..<length cnf])\<close>
thf(fact_1244_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_1245_Annotation_Osize_I4_J,axiom,
    ( ( size_s7536720479165623301tation @ multis5512618279175645674signed )
    = zero_zero_nat ) ).

% Annotation.size(4)
thf(fact_1246_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_1247_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_1248_res,axiom,
    ( ( ( concat6161509665823917772on_nat
        @ ( map_a_3887198048842487703on_nat
          @ ^ [X3: a] : ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis8267843437418295310sitive ) ) @ ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5977116398950053002gative ) ) @ nil_Su3078906344068544711on_nat ) )
          @ vs ) )
      = xs )
    & ( ( append3523812819975775986on_nat
        @ ( map_a_2711072540680412167on_nat
          @ ^ [X3: a] : ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5512618279175645674signed ) )
          @ vs )
        @ ( map_na5706685718985795555on_nat @ sum_In255631617079920996tation @ ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) )
      = ys )
    & ( ( concat937493773106420796on_nat
        @ ( map_na8715451351012060651on_nat
          @ ^ [I4: nat] :
              ( map_Pr6945619049609468478on_nat
              @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I4 ) )
              @ ( nth_li3248255784598043854od_a_o @ cnf @ I4 ) )
          @ ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) )
      = s )
    & ( ( append2865821678206098582on_nat
        @ ( concat937493773106420796on_nat
          @ ( map_a_7431629901395152583on_nat
            @ ^ [X3: a] : ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis8267843437418295310sitive ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5512618279175645674signed ) ) ) @ ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5977116398950053002gative ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5512618279175645674signed ) ) ) @ nil_Pr8614093721167239297on_nat ) )
            @ vs ) )
        @ ( concat937493773106420796on_nat
          @ ( map_na8715451351012060651on_nat
            @ ^ [I4: nat] :
                ( map_Pr6945619049609468478on_nat
                @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I4 ) )
                @ ( nth_li3248255784598043854od_a_o @ cnf @ I4 ) )
            @ ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) ) )
      = ns ) ) ).

% res
thf(fact_1249_xs,axiom,
    ( xs
    = ( concat6161509665823917772on_nat
      @ ( map_a_3887198048842487703on_nat
        @ ^ [X3: a] : ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis8267843437418295310sitive ) ) @ ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5977116398950053002gative ) ) @ nil_Su3078906344068544711on_nat ) )
        @ vs ) ) ) ).

% xs
thf(fact_1250_Annotation_Osize_I5_J,axiom,
    ( ( size_s7536720479165623301tation @ multis8267843437418295310sitive )
    = zero_zero_nat ) ).

% Annotation.size(5)
thf(fact_1251_Annotation_Osize_I6_J,axiom,
    ( ( size_s7536720479165623301tation @ multis5977116398950053002gative )
    = zero_zero_nat ) ).

% Annotation.size(6)
thf(fact_1252_Annotation_Oexhaust,axiom,
    ! [Y: multis8755981257395730801tation] :
      ( ( Y != multis5512618279175645674signed )
     => ( ( Y != multis8267843437418295310sitive )
       => ( Y = multis5977116398950053002gative ) ) ) ).

% Annotation.exhaust
thf(fact_1253_Annotation_Odistinct_I1_J,axiom,
    multis5512618279175645674signed != multis8267843437418295310sitive ).

% Annotation.distinct(1)
thf(fact_1254_Annotation_Odistinct_I3_J,axiom,
    multis5512618279175645674signed != multis5977116398950053002gative ).

% Annotation.distinct(3)
thf(fact_1255_Annotation_Odistinct_I5_J,axiom,
    multis8267843437418295310sitive != multis5977116398950053002gative ).

% Annotation.distinct(5)
thf(fact_1256_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q3 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_1257_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_1258_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_1259_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1260_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_1261_Annotation_Osize__gen_I3_J,axiom,
    ( ( multis6502627171228953925tation @ multis5977116398950053002gative )
    = zero_zero_nat ) ).

% Annotation.size_gen(3)
thf(fact_1262_Annotation_Osize__gen_I2_J,axiom,
    ( ( multis6502627171228953925tation @ multis8267843437418295310sitive )
    = zero_zero_nat ) ).

% Annotation.size_gen(2)
thf(fact_1263_Annotation_Osize__gen_I1_J,axiom,
    ( ( multis6502627171228953925tation @ multis5512618279175645674signed )
    = zero_zero_nat ) ).

% Annotation.size_gen(1)
thf(fact_1264_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_1265_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_li381573783031109843_o_nat @ size_s1727999245659094451od_a_o @ cnf ) ) @ s2 ).

%------------------------------------------------------------------------------