%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP099^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:33 AM UTC 2026
% Result : Theorem 2.69s 1.68s
% Output : Refutation 2.69s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP099^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n029.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:26:29 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.32/0.46 ---- Original DTF file ---
% 0.32/0.46 thf(spec,logic,$$dhol).
% 0.32/0.46 %------------------------------------------------------------------------------
% 0.32/0.46 % File : ITP099^2 : TPTP v9.2.1. Released v7.5.0.
% 0.32/0.46 % Domain : Interactive Theorem Proving
% 0.32/0.46 % Problem : Sledgehammer ListInf problem prob_412__5414926_1
% 0.32/0.46 % Version : Especial.
% 0.32/0.46 % English :
% 0.32/0.46
% 0.32/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.32/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.32/0.46 % Source : [Des21]
% 0.32/0.46 % Names : ListInf/prob_412__5414926_1 [Des21]
% 0.32/0.46
% 0.32/0.46 % Status : Theorem
% 0.32/0.46 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.32/0.46 % Syntax : Number of formulae : 324 ( 53 unt; 42 typ; 0 def)
% 0.32/0.46 % Number of atoms : 946 ( 245 equ; 0 cnn)
% 0.32/0.46 % Maximal formula atoms : 13 ( 3 avg)
% 0.32/0.46 % Number of connectives : 3542 ( 80 ~; 25 |; 48 &;2865 @)
% 0.32/0.46 % ( 0 <=>; 524 =>; 0 <=; 0 <~>)
% 0.32/0.46 % Maximal formula depth : 21 ( 8 avg)
% 0.32/0.46 % Number of types : 3 ( 2 usr)
% 0.32/0.46 % Number of type conns : 138 ( 138 >; 0 *; 0 +; 0 <<)
% 0.32/0.46 % Number of symbols : 43 ( 40 usr; 4 con; 0-4 aty)
% 0.32/0.46 % Number of variables : 1085 ( 70 ^; 935 !; 44 ?;1085 :)
% 0.32/0.46 % ( 36 !>; 0 ?*; 0 @-; 0 @+)
% 0.32/0.46 % SPC : TH1_THM_EQU_NAR
% 0.32/0.46
% 0.32/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:17:00.338
% 0.32/0.46 %------------------------------------------------------------------------------
% 0.32/0.46 % Could-be-implicit typings (3)
% 0.32/0.46 thf(ty_t_List_Olist,type,
% 0.32/0.46 list: $tType > $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_t_Nat_Onat,type,
% 0.32/0.46 nat: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_tf_a,type,
% 0.32/0.46 a: $tType ).
% 0.32/0.46
% 0.32/0.46 % Explicit typings (39)
% 0.32/0.46 thf(sy_cl_HOL_Otype,type,
% 0.32/0.46 type:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Nat_Osize,type,
% 0.32/0.46 size:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.32/0.46 minus:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.32/0.46 ord:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.32/0.46 order:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.32/0.46 group_add:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.32/0.46 no_bot:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.32/0.46 no_top:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/0.46 linorder:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/0.46 preorder:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.32/0.46 wellorder:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.32/0.46 dense_order:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.32/0.46 dense_linorder:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.32/0.46 cancel_semigroup_add:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.32/0.46 ordered_ab_group_add:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.32/0.46 cancel146912293up_add:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.32/0.46 ordere236663937imp_le:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.32/0.46 condit1656338222tinuum:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.32/0.46 condit1037483654norder:
% 0.32/0.46 !>[A: $tType] : $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.32/0.46 minus_minus:
% 0.32/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.32/0.46 plus_plus:
% 0.32/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_If,type,
% 0.32/0.46 if:
% 0.32/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_List2_Olist__asc,type,
% 0.32/0.46 list_asc:
% 0.32/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_List2_Olist__desc,type,
% 0.32/0.46 list_desc:
% 0.32/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_List2_Olist__strict__asc,type,
% 0.32/0.46 list_strict_asc:
% 0.32/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_List2_Olist__strict__desc,type,
% 0.32/0.46 list_strict_desc:
% 0.32/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
% 0.32/0.46 listIn521021761append:
% 0.32/0.46 !>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__drop,type,
% 0.32/0.46 listIn1569887318i_drop:
% 0.32/0.46 !>[A: $tType] : ( nat > ( nat > A ) > nat > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__take,type,
% 0.32/0.46 listIn1033672622i_take:
% 0.32/0.46 !>[A: $tType] : ( nat > ( nat > A ) > ( list @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_List_Oappend,type,
% 0.32/0.46 append:
% 0.32/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_List_Odrop,type,
% 0.32/0.46 drop:
% 0.32/0.46 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_List_Olast,type,
% 0.32/0.46 last:
% 0.32/0.46 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_List_Onth,type,
% 0.32/0.46 nth:
% 0.32/0.46 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.32/0.46 size_size:
% 0.32/0.46 !>[A: $tType] : ( A > nat ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.32/0.46 ord_less:
% 0.32/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/0.46 ord_less_eq:
% 0.32/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.46
% 0.32/0.46 thf(sy_v_f,type,
% 0.32/0.46 f: nat > a ).
% 0.32/0.46
% 0.32/0.46 thf(sy_v_n,type,
% 0.32/0.46 n: nat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_v_xs,type,
% 0.32/0.46 xs: list @ a ).
% 0.32/0.46
% 0.32/0.46 % Relevant facts (256)
% 0.32/0.46 thf(fact_0_same__i__append__eq,axiom,
% 0.32/0.46 ! [A: $tType,Xs: list @ A,F: nat > A,G: nat > A] :
% 0.32/0.46 ( ( ( listIn521021761append @ A @ Xs @ F )
% 0.32/0.46 = ( listIn521021761append @ A @ Xs @ G ) )
% 0.32/0.46 = ( F = G ) ) ).
% 0.32/0.46
% 0.32/0.46 % same_i_append_eq
% 0.32/0.46 thf(fact_1_i__append__eq__i__append__conv,axiom,
% 0.32/0.46 ! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
% 0.32/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.32/0.46 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.32/0.46 => ( ( ( listIn521021761append @ A @ Xs @ F )
% 0.32/0.46 = ( listIn521021761append @ A @ Ys @ G ) )
% 0.32/0.46 = ( ( Xs = Ys )
% 0.32/0.46 & ( F = G ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % i_append_eq_i_append_conv
% 0.32/0.46 thf(fact_2_length__drop,axiom,
% 0.32/0.46 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.32/0.46 ( ( size_size @ ( list @ A ) @ ( drop @ A @ N @ Xs ) )
% 0.32/0.46 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 0.32/0.46
% 0.32/0.46 % length_drop
% 0.32/0.46 thf(fact_3_minus__apply,axiom,
% 0.32/0.46 ! [B: $tType,A: $tType] :
% 0.32/0.46 ( ( minus @ B )
% 0.32/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.32/0.46 = ( ^ [A2: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A2 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % minus_apply
% 0.32/0.46 thf(fact_4_diff__less__mono2,axiom,
% 0.32/0.46 ! [M: nat,N: nat,L: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ M @ N )
% 0.32/0.46 => ( ( ord_less @ nat @ M @ L )
% 0.32/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % diff_less_mono2
% 0.32/0.46 thf(fact_5_less__imp__diff__less,axiom,
% 0.32/0.46 ! [J: nat,K: nat,N: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ J @ K )
% 0.32/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 0.32/0.46
% 0.32/0.46 % less_imp_diff_less
% 0.32/0.46 thf(fact_6_less__diff__imp__less,axiom,
% 0.32/0.46 ! [I: nat,J: nat,M: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
% 0.32/0.46 => ( ord_less @ nat @ I @ J ) ) ).
% 0.32/0.46
% 0.32/0.46 % less_diff_imp_less
% 0.32/0.46 thf(fact_7_nat__diff__left__cancel__eq1,axiom,
% 0.32/0.46 ! [K: nat,M: nat,N: nat] :
% 0.32/0.46 ( ( ( minus_minus @ nat @ K @ M )
% 0.32/0.46 = ( minus_minus @ nat @ K @ N ) )
% 0.32/0.46 => ( ( ord_less @ nat @ M @ K )
% 0.32/0.46 => ( M = N ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % nat_diff_left_cancel_eq1
% 0.32/0.46 thf(fact_8_nat__diff__left__cancel__eq2,axiom,
% 0.32/0.46 ! [K: nat,M: nat,N: nat] :
% 0.32/0.46 ( ( ( minus_minus @ nat @ K @ M )
% 0.32/0.46 = ( minus_minus @ nat @ K @ N ) )
% 0.32/0.46 => ( ( ord_less @ nat @ N @ K )
% 0.32/0.46 => ( M = N ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % nat_diff_left_cancel_eq2
% 0.32/0.46 thf(fact_9_nat__diff__left__cancel__less,axiom,
% 0.32/0.46 ! [K: nat,M: nat,N: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
% 0.32/0.46 => ( ord_less @ nat @ N @ M ) ) ).
% 0.32/0.46
% 0.32/0.46 % nat_diff_left_cancel_less
% 0.32/0.46 thf(fact_10_nat__diff__right__cancel__eq1,axiom,
% 0.32/0.46 ! [M: nat,K: nat,N: nat] :
% 0.32/0.46 ( ( ( minus_minus @ nat @ M @ K )
% 0.32/0.46 = ( minus_minus @ nat @ N @ K ) )
% 0.32/0.46 => ( ( ord_less @ nat @ K @ M )
% 0.32/0.46 => ( M = N ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % nat_diff_right_cancel_eq1
% 0.32/0.46 thf(fact_11_nat__diff__right__cancel__eq2,axiom,
% 0.32/0.46 ! [M: nat,K: nat,N: nat] :
% 0.32/0.46 ( ( ( minus_minus @ nat @ M @ K )
% 0.32/0.46 = ( minus_minus @ nat @ N @ K ) )
% 0.32/0.46 => ( ( ord_less @ nat @ K @ N )
% 0.32/0.46 => ( M = N ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % nat_diff_right_cancel_eq2
% 0.32/0.46 thf(fact_12_nat__diff__right__cancel__less,axiom,
% 0.32/0.46 ! [N: nat,K: nat,M: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) )
% 0.32/0.46 => ( ord_less @ nat @ N @ M ) ) ).
% 0.32/0.46
% 0.32/0.46 % nat_diff_right_cancel_less
% 0.32/0.46 thf(fact_13_length__induct,axiom,
% 0.32/0.46 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.32/0.46 ( ! [Xs2: list @ A] :
% 0.32/0.46 ( ! [Ys2: list @ A] :
% 0.32/0.46 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.32/0.46 => ( P @ Ys2 ) )
% 0.32/0.46 => ( P @ Xs2 ) )
% 0.32/0.46 => ( P @ Xs ) ) ).
% 0.32/0.46
% 0.32/0.46 % length_induct
% 0.32/0.46 thf(fact_14_measure__induct__rule,axiom,
% 0.32/0.46 ! [B: $tType,A: $tType] :
% 0.32/0.46 ( ( wellorder @ B )
% 0.32/0.46 => ! [F: A > B,P: A > $o,A3: A] :
% 0.32/0.46 ( ! [X2: A] :
% 0.32/0.46 ( ! [Y: A] :
% 0.32/0.46 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 0.32/0.46 => ( P @ Y ) )
% 0.32/0.46 => ( P @ X2 ) )
% 0.32/0.46 => ( P @ A3 ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % measure_induct_rule
% 0.32/0.46 thf(fact_15_measure__induct,axiom,
% 0.32/0.46 ! [B: $tType,A: $tType] :
% 0.32/0.46 ( ( wellorder @ B )
% 0.32/0.46 => ! [F: A > B,P: A > $o,A3: A] :
% 0.32/0.46 ( ! [X2: A] :
% 0.32/0.46 ( ! [Y: A] :
% 0.32/0.46 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 0.32/0.46 => ( P @ Y ) )
% 0.32/0.46 => ( P @ X2 ) )
% 0.32/0.46 => ( P @ A3 ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % measure_induct
% 0.32/0.46 thf(fact_16_fun__diff__def,axiom,
% 0.32/0.46 ! [B: $tType,A: $tType] :
% 0.32/0.46 ( ( minus @ B )
% 0.32/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.32/0.46 = ( ^ [A2: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A2 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % fun_diff_def
% 0.32/0.46 thf(fact_17_infinite__descent__measure,axiom,
% 0.32/0.46 ! [A: $tType,P: A > $o,V: A > nat,X3: A] :
% 0.32/0.46 ( ! [X2: A] :
% 0.32/0.46 ( ~ ( P @ X2 )
% 0.32/0.46 => ? [Y: A] :
% 0.32/0.46 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 0.32/0.46 & ~ ( P @ Y ) ) )
% 0.32/0.46 => ( P @ X3 ) ) ).
% 0.32/0.46
% 0.32/0.46 % infinite_descent_measure
% 0.32/0.46 thf(fact_18_linorder__neqE__nat,axiom,
% 0.32/0.46 ! [X3: nat,Y2: nat] :
% 0.32/0.46 ( ( X3 != Y2 )
% 0.32/0.46 => ( ~ ( ord_less @ nat @ X3 @ Y2 )
% 0.32/0.46 => ( ord_less @ nat @ Y2 @ X3 ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % linorder_neqE_nat
% 0.32/0.46 thf(fact_19_infinite__descent,axiom,
% 0.32/0.46 ! [P: nat > $o,N: nat] :
% 0.32/0.46 ( ! [N2: nat] :
% 0.32/0.46 ( ~ ( P @ N2 )
% 0.32/0.46 => ? [M2: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.32/0.46 & ~ ( P @ M2 ) ) )
% 0.32/0.46 => ( P @ N ) ) ).
% 0.32/0.46
% 0.32/0.46 % infinite_descent
% 0.32/0.46 thf(fact_20_nat__less__induct,axiom,
% 0.32/0.46 ! [P: nat > $o,N: nat] :
% 0.32/0.46 ( ! [N2: nat] :
% 0.32/0.46 ( ! [M2: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.32/0.46 => ( P @ M2 ) )
% 0.32/0.46 => ( P @ N2 ) )
% 0.32/0.46 => ( P @ N ) ) ).
% 0.32/0.46
% 0.32/0.46 % nat_less_induct
% 0.32/0.46 thf(fact_21_less__irrefl__nat,axiom,
% 0.32/0.46 ! [N: nat] :
% 0.32/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.32/0.46
% 0.32/0.46 % less_irrefl_nat
% 0.32/0.46 thf(fact_22_less__not__refl3,axiom,
% 0.32/0.46 ! [S: nat,T: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ S @ T )
% 0.32/0.46 => ( S != T ) ) ).
% 0.32/0.46
% 0.32/0.46 % less_not_refl3
% 0.32/0.46 thf(fact_23_less__not__refl2,axiom,
% 0.32/0.46 ! [N: nat,M: nat] :
% 0.32/0.46 ( ( ord_less @ nat @ N @ M )
% 0.32/0.46 => ( M != N ) ) ).
% 0.32/0.46
% 0.32/0.46 % less_not_refl2
% 0.32/0.46 thf(fact_24_less__not__refl,axiom,
% 0.32/0.46 ! [N: nat] :
% 0.32/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.32/0.46
% 0.32/0.46 % less_not_refl
% 0.32/0.46 thf(fact_25_nat__neq__iff,axiom,
% 0.32/0.46 ! [M: nat,N: nat] :
% 0.32/0.46 ( ( M != N )
% 0.32/0.46 = ( ( ord_less @ nat @ M @ N )
% 0.32/0.46 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % nat_neq_iff
% 0.32/0.46 thf(fact_26_size__neq__size__imp__neq,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( size @ A )
% 0.32/0.46 => ! [X3: A,Y2: A] :
% 0.32/0.46 ( ( ( size_size @ A @ X3 )
% 0.32/0.46 != ( size_size @ A @ Y2 ) )
% 0.32/0.46 => ( X3 != Y2 ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % size_neq_size_imp_neq
% 0.32/0.46 thf(fact_27_neq__if__length__neq,axiom,
% 0.32/0.46 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.32/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.32/0.46 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.32/0.46 => ( Xs != Ys ) ) ).
% 0.32/0.46
% 0.32/0.46 % neq_if_length_neq
% 0.32/0.46 thf(fact_28_Ex__list__of__length,axiom,
% 0.32/0.46 ! [A: $tType,N: nat] :
% 0.32/0.46 ? [Xs2: list @ A] :
% 0.32/0.46 ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.32/0.46 = N ) ).
% 0.32/0.46
% 0.32/0.46 % Ex_list_of_length
% 0.32/0.46 thf(fact_29_eq__imp__diff__eq,axiom,
% 0.32/0.46 ! [M: nat,N: nat,K: nat] :
% 0.32/0.46 ( ( M = N )
% 0.32/0.47 => ( ( minus_minus @ nat @ M @ K )
% 0.32/0.47 = ( minus_minus @ nat @ N @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_imp_diff_eq
% 0.32/0.47 thf(fact_30_diff__commute,axiom,
% 0.32/0.47 ! [I: nat,J: nat,K: nat] :
% 0.32/0.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.32/0.47 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_commute
% 0.32/0.47 thf(fact_31_diff__strict__right__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_strict_right_mono
% 0.32/0.47 thf(fact_32_diff__strict__left__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [B3: A,A3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.32/0.47 => ( ord_less @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_strict_left_mono
% 0.32/0.47 thf(fact_33_diff__eq__diff__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A,D: A] :
% 0.32/0.47 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.32/0.47 = ( minus_minus @ A @ C @ D ) )
% 0.32/0.47 => ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_eq_diff_less
% 0.32/0.47 thf(fact_34_diff__strict__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A,D: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ D @ C )
% 0.32/0.47 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_strict_mono
% 0.32/0.47 thf(fact_35_i__append__i__drop__eq1,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ( ( listIn1569887318i_drop @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
% 0.32/0.47 = ( listIn521021761append @ A @ ( drop @ A @ N @ Xs ) @ F ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_i_drop_eq1
% 0.32/0.47 thf(fact_36_i__append__i__drop__eq2,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 0.32/0.47 => ( ( listIn1569887318i_drop @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
% 0.32/0.47 = ( listIn1569887318i_drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ F ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_i_drop_eq2
% 0.32/0.47 thf(fact_37_last__drop,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ( ( last @ A @ ( drop @ A @ N @ Xs ) )
% 0.32/0.47 = ( last @ A @ Xs ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % last_drop
% 0.32/0.47 thf(fact_38_i__append__nth2,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 0.32/0.47 => ( ( listIn521021761append @ A @ Xs @ F @ N )
% 0.32/0.47 = ( F @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_nth2
% 0.32/0.47 thf(fact_39_i__append__nth,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( listIn521021761append @ A )
% 0.32/0.47 = ( ^ [Xs3: list @ A,F2: nat > A,N3: nat] : ( if @ A @ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ ( nth @ A @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_nth
% 0.32/0.47 thf(fact_40_i__append__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( listIn521021761append @ A )
% 0.32/0.47 = ( ^ [Xs3: list @ A,F2: nat > A,N3: nat] : ( if @ A @ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ ( nth @ A @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_def
% 0.32/0.47 thf(fact_41_drop__append,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
% 0.32/0.47 ( ( drop @ A @ N @ ( append @ A @ Xs @ Ys ) )
% 0.32/0.47 = ( append @ A @ ( drop @ A @ N @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % drop_append
% 0.32/0.47 thf(fact_42_append_Oassoc,axiom,
% 0.32/0.47 ! [A: $tType,A3: list @ A,B3: list @ A,C: list @ A] :
% 0.32/0.47 ( ( append @ A @ ( append @ A @ A3 @ B3 ) @ C )
% 0.32/0.47 = ( append @ A @ A3 @ ( append @ A @ B3 @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % append.assoc
% 0.32/0.47 thf(fact_43_append__assoc,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 0.32/0.47 ( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
% 0.32/0.47 = ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % append_assoc
% 0.32/0.47 thf(fact_44_append__same__eq,axiom,
% 0.32/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
% 0.32/0.47 ( ( ( append @ A @ Ys @ Xs )
% 0.32/0.47 = ( append @ A @ Zs @ Xs ) )
% 0.32/0.47 = ( Ys = Zs ) ) ).
% 0.32/0.47
% 0.32/0.47 % append_same_eq
% 0.32/0.47 thf(fact_45_ext,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.32/0.47 ( ! [X2: A] :
% 0.32/0.47 ( ( F @ X2 )
% 0.32/0.47 = ( G @ X2 ) )
% 0.32/0.47 => ( F = G ) ) ).
% 0.32/0.47
% 0.32/0.47 % ext
% 0.32/0.47 thf(fact_46_same__append__eq,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 0.32/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.32/0.47 = ( append @ A @ Xs @ Zs ) )
% 0.32/0.47 = ( Ys = Zs ) ) ).
% 0.32/0.47
% 0.32/0.47 % same_append_eq
% 0.32/0.47 thf(fact_47_diff__diff__cancel,axiom,
% 0.32/0.47 ! [I: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ N )
% 0.32/0.47 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 0.32/0.47 = I ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_diff_cancel
% 0.32/0.47 thf(fact_48_append__eq__append__conv,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
% 0.32/0.47 ( ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.32/0.47 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.32/0.47 | ( ( size_size @ ( list @ A ) @ Us )
% 0.32/0.47 = ( size_size @ ( list @ A ) @ Vs ) ) )
% 0.32/0.47 => ( ( ( append @ A @ Xs @ Us )
% 0.32/0.47 = ( append @ A @ Ys @ Vs ) )
% 0.32/0.47 = ( ( Xs = Ys )
% 0.32/0.47 & ( Us = Vs ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % append_eq_append_conv
% 0.32/0.47 thf(fact_49_i__append__assoc,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A] :
% 0.32/0.47 ( ( listIn521021761append @ A @ Xs @ ( listIn521021761append @ A @ Ys @ F ) )
% 0.32/0.47 = ( listIn521021761append @ A @ ( append @ A @ Xs @ Ys ) @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_assoc
% 0.32/0.47 thf(fact_50_i__append__nth1,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ( ( listIn521021761append @ A @ Xs @ F @ N )
% 0.32/0.47 = ( nth @ A @ Xs @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_nth1
% 0.32/0.47 thf(fact_51_le__refl,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.32/0.47
% 0.32/0.47 % le_refl
% 0.32/0.47 thf(fact_52_le__trans,axiom,
% 0.32/0.47 ! [I: nat,J: nat,K: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.32/0.47 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_trans
% 0.32/0.47 thf(fact_53_eq__imp__le,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( M = N )
% 0.32/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_imp_le
% 0.32/0.47 thf(fact_54_le__antisym,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.32/0.47 => ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_antisym
% 0.32/0.47 thf(fact_55_nat__le__linear,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_le_linear
% 0.32/0.47 thf(fact_56_append__eq__appendI,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
% 0.32/0.47 ( ( ( append @ A @ Xs @ Xs1 )
% 0.32/0.47 = Zs )
% 0.32/0.47 => ( ( Ys
% 0.32/0.47 = ( append @ A @ Xs1 @ Us ) )
% 0.32/0.47 => ( ( append @ A @ Xs @ Ys )
% 0.32/0.47 = ( append @ A @ Zs @ Us ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % append_eq_appendI
% 0.32/0.47 thf(fact_57_Nat_Oex__has__greatest__nat,axiom,
% 0.32/0.47 ! [P: nat > $o,K: nat,B3: nat] :
% 0.32/0.47 ( ( P @ K )
% 0.32/0.47 => ( ! [Y3: nat] :
% 0.32/0.47 ( ( P @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ nat @ Y3 @ B3 ) )
% 0.32/0.47 => ? [X2: nat] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 & ! [Y: nat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq @ nat @ Y @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nat.ex_has_greatest_nat
% 0.32/0.47 thf(fact_58_append__eq__append__conv2,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
% 0.32/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.32/0.47 = ( append @ A @ Zs @ Ts ) )
% 0.32/0.47 = ( ? [Us2: list @ A] :
% 0.32/0.47 ( ( ( Xs
% 0.32/0.47 = ( append @ A @ Zs @ Us2 ) )
% 0.32/0.47 & ( ( append @ A @ Us2 @ Ys )
% 0.32/0.47 = Ts ) )
% 0.32/0.47 | ( ( ( append @ A @ Xs @ Us2 )
% 0.32/0.47 = Zs )
% 0.32/0.47 & ( Ys
% 0.32/0.47 = ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % append_eq_append_conv2
% 0.32/0.47 thf(fact_59_diff__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A,D: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ D @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_mono
% 0.32/0.47 thf(fact_60_diff__left__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [B3: A,A3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_left_mono
% 0.32/0.47 thf(fact_61_diff__right__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_right_mono
% 0.32/0.47 thf(fact_62_diff__eq__diff__less__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A,D: A] :
% 0.32/0.47 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.32/0.47 = ( minus_minus @ A @ C @ D ) )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_eq_diff_less_eq
% 0.32/0.47 thf(fact_63_i__append__eq__i__append__conv2__aux,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,F: nat > A,Ys: list @ A,G: nat > A] :
% 0.32/0.47 ( ( ( listIn521021761append @ A @ Xs @ F )
% 0.32/0.47 = ( listIn521021761append @ A @ Ys @ G ) )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
% 0.32/0.47 => ? [Zs2: list @ A] :
% 0.32/0.47 ( ( ( append @ A @ Xs @ Zs2 )
% 0.32/0.47 = Ys )
% 0.32/0.47 & ( F
% 0.32/0.47 = ( listIn521021761append @ A @ Zs2 @ G ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_eq_i_append_conv2_aux
% 0.32/0.47 thf(fact_64_ge__less__neq__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A4: A,N3: A] :
% 0.32/0.47 ! [X: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ A4 )
% 0.32/0.47 => ( N3 != X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ge_less_neq_conv
% 0.32/0.47 thf(fact_65_less__ge__neq__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [N3: A,A4: A] :
% 0.32/0.47 ! [X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ X )
% 0.32/0.47 => ( N3 != X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_ge_neq_conv
% 0.32/0.47 thf(fact_66_greater__le__neq__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [A4: A,N3: A] :
% 0.32/0.47 ! [X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ A4 )
% 0.32/0.47 => ( N3 != X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % greater_le_neq_conv
% 0.32/0.47 thf(fact_67_le__greater__neq__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [N3: A,A4: A] :
% 0.32/0.47 ! [X: A] :
% 0.32/0.47 ( ( ord_less @ A @ A4 @ X )
% 0.32/0.47 => ( N3 != X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_greater_neq_conv
% 0.32/0.47 thf(fact_68_nat__less__le,axiom,
% 0.32/0.47 ( ( ord_less @ nat )
% 0.32/0.47 = ( ^ [M3: nat,N3: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 0.32/0.47 & ( M3 != N3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_less_le
% 0.32/0.47 thf(fact_69_less__imp__le__nat,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_le_nat
% 0.32/0.47 thf(fact_70_le__eq__less__or__eq,axiom,
% 0.32/0.47 ( ( ord_less_eq @ nat )
% 0.32/0.47 = ( ^ [M3: nat,N3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.32/0.47 | ( M3 = N3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_eq_less_or_eq
% 0.32/0.47 thf(fact_71_less__or__eq__imp__le,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 | ( M = N ) )
% 0.32/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_or_eq_imp_le
% 0.32/0.47 thf(fact_72_le__neq__implies__less,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ( M != N )
% 0.32/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_neq_implies_less
% 0.32/0.47 thf(fact_73_less__mono__imp__le__mono,axiom,
% 0.32/0.47 ! [F: nat > nat,I: nat,J: nat] :
% 0.32/0.47 ( ! [I2: nat,J2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ J2 )
% 0.32/0.47 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_mono_imp_le_mono
% 0.32/0.47 thf(fact_74_eq__diff__iff,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ( ( minus_minus @ nat @ M @ K )
% 0.32/0.47 = ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 = ( M = N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_diff_iff
% 0.32/0.47 thf(fact_75_le__diff__iff,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_diff_iff
% 0.32/0.47 thf(fact_76_Nat_Odiff__diff__eq,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nat.diff_diff_eq
% 0.32/0.47 thf(fact_77_diff__le__mono,axiom,
% 0.32/0.47 ! [M: nat,N: nat,L: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_le_mono
% 0.32/0.47 thf(fact_78_diff__le__self,axiom,
% 0.32/0.47 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.32/0.47
% 0.32/0.47 % diff_le_self
% 0.32/0.47 thf(fact_79_le__diff__iff_H,axiom,
% 0.32/0.47 ! [A3: nat,C: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ A3 @ C )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ B3 @ C )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A3 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_diff_iff'
% 0.32/0.47 thf(fact_80_diff__le__mono2,axiom,
% 0.32/0.47 ! [M: nat,N: nat,L: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_le_mono2
% 0.32/0.47 thf(fact_81_le__diff__swap,axiom,
% 0.32/0.47 ! [I: nat,K: nat,J: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ K )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ J ) @ I )
% 0.32/0.47 = ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ I ) @ J ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_diff_swap
% 0.32/0.47 thf(fact_82_le__diff__imp__le,axiom,
% 0.32/0.47 ! [I: nat,J: nat,M: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
% 0.32/0.47 => ( ord_less_eq @ nat @ I @ J ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_diff_imp_le
% 0.32/0.47 thf(fact_83_le__imp__diff__le,axiom,
% 0.32/0.47 ! [J: nat,K: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ J @ K )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_imp_diff_le
% 0.32/0.47 thf(fact_84_eq__diff__left__iff,axiom,
% 0.32/0.47 ! [M: nat,K: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ K )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ N @ K )
% 0.32/0.47 => ( ( ( minus_minus @ nat @ K @ M )
% 0.32/0.47 = ( minus_minus @ nat @ K @ N ) )
% 0.32/0.47 = ( M = N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_diff_left_iff
% 0.32/0.47 thf(fact_85_le__diff__le__imp__le,axiom,
% 0.32/0.47 ! [I: nat,J: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.32/0.47 => ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_diff_le_imp_le
% 0.32/0.47 thf(fact_86_nat__diff__left__cancel__le2,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ N @ K )
% 0.32/0.47 => ( ord_less_eq @ nat @ N @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_diff_left_cancel_le2
% 0.32/0.47 thf(fact_87_nat__diff__right__cancel__le2,axiom,
% 0.32/0.47 ! [M: nat,K: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_diff_right_cancel_le2
% 0.32/0.47 thf(fact_88_i__append__eq__i__appendI,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,Xs4: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
% 0.32/0.47 ( ( ( append @ A @ Xs @ Xs4 )
% 0.32/0.47 = Ys )
% 0.32/0.47 => ( ( F
% 0.32/0.47 = ( listIn521021761append @ A @ Xs4 @ G ) )
% 0.32/0.47 => ( ( listIn521021761append @ A @ Xs @ F )
% 0.32/0.47 = ( listIn521021761append @ A @ Ys @ G ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_eq_i_appendI
% 0.32/0.47 thf(fact_89_i__append__eq__i__append__conv2,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,F: nat > A,Ys: list @ A,G: nat > A] :
% 0.32/0.47 ( ( ( listIn521021761append @ A @ Xs @ F )
% 0.32/0.47 = ( listIn521021761append @ A @ Ys @ G ) )
% 0.32/0.47 = ( ? [Zs3: list @ A] :
% 0.32/0.47 ( ( ( Xs
% 0.32/0.47 = ( append @ A @ Ys @ Zs3 ) )
% 0.32/0.47 & ( ( listIn521021761append @ A @ Zs3 @ F )
% 0.32/0.47 = G ) )
% 0.32/0.47 | ( ( ( append @ A @ Xs @ Zs3 )
% 0.32/0.47 = Ys )
% 0.32/0.47 & ( F
% 0.32/0.47 = ( listIn521021761append @ A @ Zs3 @ G ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_eq_i_append_conv2
% 0.32/0.47 thf(fact_90_nth__append,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
% 0.32/0.47 ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
% 0.32/0.47 = ( nth @ A @ Xs @ N ) ) )
% 0.32/0.47 & ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
% 0.32/0.47 = ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nth_append
% 0.32/0.47 thf(fact_91_nth__equalityI,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.32/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.32/0.47 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.32/0.47 => ( ! [I2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ( ( nth @ A @ Xs @ I2 )
% 0.32/0.47 = ( nth @ A @ Ys @ I2 ) ) )
% 0.32/0.47 => ( Xs = Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nth_equalityI
% 0.32/0.47 thf(fact_92_Skolem__list__nth,axiom,
% 0.32/0.47 ! [A: $tType,K: nat,P: nat > A > $o] :
% 0.32/0.47 ( ( ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ K )
% 0.32/0.47 => ? [X4: A] : ( P @ I3 @ X4 ) ) )
% 0.32/0.47 = ( ? [Xs3: list @ A] :
% 0.32/0.47 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.32/0.47 = K )
% 0.32/0.47 & ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ K )
% 0.32/0.47 => ( P @ I3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Skolem_list_nth
% 0.32/0.47 thf(fact_93_list__eq__iff__nth__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ^ [Y4: list @ A,Z: list @ A] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 0.32/0.47 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.32/0.47 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.32/0.47 & ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.32/0.47 => ( ( nth @ A @ Xs3 @ I3 )
% 0.32/0.47 = ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_eq_iff_nth_eq
% 0.32/0.47 thf(fact_94_less__diff__iff,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_diff_iff
% 0.32/0.47 thf(fact_95_diff__less__mono,axiom,
% 0.32/0.47 ! [A3: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ C @ A3 )
% 0.32/0.47 => ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_less_mono
% 0.32/0.47 thf(fact_96_less__diff__le__imp__less,axiom,
% 0.32/0.47 ! [I: nat,J: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.32/0.47 => ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_diff_le_imp_less
% 0.32/0.47 thf(fact_97_nat__diff__left__cancel__le1,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
% 0.32/0.47 => ( ( ord_less @ nat @ M @ K )
% 0.32/0.47 => ( ord_less_eq @ nat @ N @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_diff_left_cancel_le1
% 0.32/0.47 thf(fact_98_nat__diff__right__cancel__le1,axiom,
% 0.32/0.47 ! [M: nat,K: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 => ( ( ord_less @ nat @ K @ M )
% 0.32/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_diff_right_cancel_le1
% 0.32/0.47 thf(fact_99_i__drop__nth__sub,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,X3: nat,S: nat > A] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ N @ X3 )
% 0.32/0.47 => ( ( listIn1569887318i_drop @ A @ N @ S @ ( minus_minus @ nat @ X3 @ N ) )
% 0.32/0.47 = ( S @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_drop_nth_sub
% 0.32/0.47 thf(fact_100_diff__eq__diff__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A,D: A] :
% 0.32/0.47 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.32/0.47 = ( minus_minus @ A @ C @ D ) )
% 0.32/0.47 => ( ( A3 = B3 )
% 0.32/0.47 = ( C = D ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_eq_diff_eq
% 0.32/0.47 thf(fact_101_diff__right__commute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel146912293up_add @ A )
% 0.32/0.47 => ! [A3: A,C: A,B3: A] :
% 0.32/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B3 )
% 0.32/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_right_commute
% 0.32/0.47 thf(fact_102_nth__append2,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,N: nat,Ys: list @ A] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 0.32/0.47 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
% 0.32/0.47 = ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nth_append2
% 0.32/0.47 thf(fact_103_nth__append1,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
% 0.32/0.47 = ( nth @ A @ Xs @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nth_append1
% 0.32/0.47 thf(fact_104_order__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_refl
% 0.32/0.47 thf(fact_105_list__desc__trans__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ( ( list_desc @ A )
% 0.32/0.47 = ( ^ [Xs3: list @ A] :
% 0.32/0.47 ! [J3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.32/0.47 => ! [I3: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I3 @ J3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_desc_trans_le
% 0.32/0.47 thf(fact_106_nat__descend__induct,axiom,
% 0.32/0.47 ! [N: nat,P: nat > $o,M: nat] :
% 0.32/0.47 ( ! [K2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ K2 )
% 0.32/0.47 => ( P @ K2 ) )
% 0.32/0.47 => ( ! [K2: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.32/0.47 => ( ! [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ K2 @ I4 )
% 0.32/0.47 => ( P @ I4 ) )
% 0.32/0.47 => ( P @ K2 ) ) )
% 0.32/0.47 => ( P @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_descend_induct
% 0.32/0.47 thf(fact_107_Lattices__Big_Oex__has__greatest__nat,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
% 0.32/0.47 ( ( P @ K )
% 0.32/0.47 => ( ! [Y3: A] :
% 0.32/0.47 ( ( P @ Y3 )
% 0.32/0.47 => ( ord_less @ nat @ ( F @ Y3 ) @ B3 ) )
% 0.32/0.47 => ? [X2: A] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 & ! [Y: A] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( F @ Y ) @ ( F @ X2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Lattices_Big.ex_has_greatest_nat
% 0.32/0.47 thf(fact_108_list__asc__trans__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ( ( list_asc @ A )
% 0.32/0.47 = ( ^ [Xs3: list @ A] :
% 0.32/0.47 ! [J3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.32/0.47 => ! [I3: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I3 @ J3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_asc_trans_le
% 0.32/0.47 thf(fact_109_list__strict__asc__trans__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [Xs: list @ A] :
% 0.32/0.47 ( ( list_strict_asc @ A @ Xs )
% 0.32/0.47 => ! [J4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ! [I4: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I4 @ J4 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_strict_asc_trans_le
% 0.32/0.47 thf(fact_110_list__strict__asc__imp__list__asc,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [Xs: list @ A] :
% 0.32/0.47 ( ( list_strict_asc @ A @ Xs )
% 0.32/0.47 => ( list_asc @ A @ Xs ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_strict_asc_imp_list_asc
% 0.32/0.47 thf(fact_111_le__funD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G: A > B,X3: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funD
% 0.32/0.47 thf(fact_112_le__funE,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G: A > B,X3: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funE
% 0.32/0.47 thf(fact_113_le__funI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G: A > B] :
% 0.32/0.47 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funI
% 0.32/0.47 thf(fact_114_le__fun__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.32/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.32/0.47 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_fun_def
% 0.32/0.47 thf(fact_115_order__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_116_order__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C2: $tType] :
% 0.32/0.47 ( ( ( order @ C2 )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,B3: A,F: A > C2,C: C2] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_117_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_118_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A3: A,B3: A,F: A > B,C: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ( F @ B3 )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_119_eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [X: A,Y5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.32/0.47 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_120_antisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_121_linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 | ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_122_eq__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( X3 = Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_123_le__cases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ~ ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_124_order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_125_le__cases3,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_126_antisym__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [Y2: A,X3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_127_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_128_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( A3 = B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_129_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( B3 = C )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_130_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.32/0.47 => ( A3 = B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_131_order__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_132_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_133_linorder__wlog,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.32/0.47 ( ! [A5: A,B5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.32/0.47 => ( P @ A5 @ B5 ) )
% 0.32/0.47 => ( ! [A5: A,B5: A] :
% 0.32/0.47 ( ( P @ B5 @ A5 )
% 0.32/0.47 => ( P @ A5 @ B5 ) )
% 0.32/0.47 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_134_dual__order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_135_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.32/0.47 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_136_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( A3 = B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_137_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A3: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.32/0.47 => ( A3 != B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_implies_not_eq
% 0.32/0.47 thf(fact_138_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( A3 != B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_implies_not_eq
% 0.32/0.47 thf(fact_139_not__less__iff__gr__or__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.32/0.47 = ( ( ord_less @ A @ Y2 @ X3 )
% 0.32/0.47 | ( X3 = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less_iff_gr_or_eq
% 0.32/0.47 thf(fact_140_dual__order_Ostrict__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_less @ A @ C @ B3 )
% 0.32/0.47 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_trans
% 0.32/0.47 thf(fact_141_linorder__less__wlog,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.32/0.47 ( ! [A5: A,B5: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B5 )
% 0.32/0.47 => ( P @ A5 @ B5 ) )
% 0.32/0.47 => ( ! [A5: A] : ( P @ A5 @ A5 )
% 0.32/0.47 => ( ! [A5: A,B5: A] :
% 0.32/0.47 ( ( P @ B5 @ A5 )
% 0.32/0.47 => ( P @ A5 @ B5 ) )
% 0.32/0.47 => ( P @ A3 @ B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_less_wlog
% 0.32/0.47 thf(fact_142_exists__least__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( wellorder @ A )
% 0.32/0.47 => ( ( ^ [P2: A > $o] :
% 0.32/0.47 ? [X4: A] : ( P2 @ X4 ) )
% 0.32/0.47 = ( ^ [P3: A > $o] :
% 0.32/0.47 ? [N3: A] :
% 0.32/0.47 ( ( P3 @ N3 )
% 0.32/0.47 & ! [M3: A] :
% 0.32/0.47 ( ( ord_less @ A @ M3 @ N3 )
% 0.32/0.47 => ~ ( P3 @ M3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % exists_least_iff
% 0.32/0.47 thf(fact_143_less__imp__not__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_less
% 0.32/0.47 thf(fact_144_order_Ostrict__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_trans
% 0.32/0.47 thf(fact_145_dual__order_Oirrefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A] :
% 0.32/0.47 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.irrefl
% 0.32/0.47 thf(fact_146_linorder__cases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( X3 != Y2 )
% 0.32/0.47 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_cases
% 0.32/0.47 thf(fact_147_less__imp__triv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A,P: $o] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ord_less @ A @ Y2 @ X3 )
% 0.32/0.47 => P ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_triv
% 0.32/0.47 thf(fact_148_less__imp__not__eq2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( Y2 != X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_eq2
% 0.32/0.47 thf(fact_149_antisym__conv3,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [Y2: A,X3: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ Y2 @ X3 )
% 0.32/0.47 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv3
% 0.32/0.47 thf(fact_150_less__induct,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( wellorder @ A )
% 0.32/0.47 => ! [P: A > $o,A3: A] :
% 0.32/0.47 ( ! [X2: A] :
% 0.32/0.47 ( ! [Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ Y @ X2 )
% 0.32/0.47 => ( P @ Y ) )
% 0.32/0.47 => ( P @ X2 ) )
% 0.32/0.47 => ( P @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_induct
% 0.32/0.47 thf(fact_151_less__not__sym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_not_sym
% 0.32/0.47 thf(fact_152_less__imp__not__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( X3 != Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_eq
% 0.32/0.47 thf(fact_153_dual__order_Oasym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A3: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.32/0.47 => ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.asym
% 0.32/0.47 thf(fact_154_ord__less__eq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( B3 = C )
% 0.32/0.47 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_less_eq_trans
% 0.32/0.47 thf(fact_155_ord__eq__less__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( A3 = B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_less_trans
% 0.32/0.47 thf(fact_156_less__irrefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A] :
% 0.32/0.47 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_irrefl
% 0.32/0.47 thf(fact_157_less__linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 | ( X3 = Y2 )
% 0.32/0.47 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_linear
% 0.32/0.47 thf(fact_158_less__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.32/0.47 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_trans
% 0.32/0.47 thf(fact_159_less__asym_H,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_asym'
% 0.32/0.47 thf(fact_160_less__asym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_asym
% 0.32/0.47 thf(fact_161_less__imp__neq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( X3 != Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_neq
% 0.32/0.47 thf(fact_162_dense,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_order @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ? [Z3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Z3 )
% 0.32/0.47 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense
% 0.32/0.47 thf(fact_163_order_Oasym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.asym
% 0.32/0.47 thf(fact_164_neq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( X3 != Y2 )
% 0.32/0.47 = ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 | ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neq_iff
% 0.32/0.47 thf(fact_165_neqE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( X3 != Y2 )
% 0.32/0.47 => ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neqE
% 0.32/0.47 thf(fact_166_gt__ex,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( no_top @ A )
% 0.32/0.47 => ! [X3: A] :
% 0.32/0.47 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% 0.32/0.47
% 0.32/0.47 % gt_ex
% 0.32/0.47 thf(fact_167_lt__ex,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( no_bot @ A )
% 0.32/0.47 => ! [X3: A] :
% 0.32/0.47 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % lt_ex
% 0.32/0.47 thf(fact_168_order__less__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C2: $tType] :
% 0.32/0.47 ( ( ( order @ C2 )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,B3: A,F: A > C2,C: C2] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_subst2
% 0.32/0.47 thf(fact_169_order__less__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_subst1
% 0.32/0.47 thf(fact_170_ord__less__eq__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A3: A,B3: A,F: A > B,C: B] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ( F @ B3 )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_less_eq_subst
% 0.32/0.47 thf(fact_171_ord__eq__less__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_less_subst
% 0.32/0.47 thf(fact_172_ex__has__least__nat,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.32/0.47 ( ( P @ K )
% 0.32/0.47 => ? [X2: A] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 & ! [Y: A] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( M @ X2 ) @ ( M @ Y ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ex_has_least_nat
% 0.32/0.47 thf(fact_173_list__drop__eq__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ^ [Y4: list @ A,Z: list @ A] : ( Y4 = Z ) )
% 0.32/0.47 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 0.32/0.47 ! [N3: nat] :
% 0.32/0.47 ( ( drop @ A @ N3 @ Xs3 )
% 0.32/0.47 = ( drop @ A @ N3 @ Ys3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_drop_eq_conv
% 0.32/0.47 thf(fact_174_list__asc__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ( ( list_asc @ A )
% 0.32/0.47 = ( ^ [Xs3: list @ A] :
% 0.32/0.47 ! [J3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.32/0.47 => ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ J3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_asc_trans
% 0.32/0.47 thf(fact_175_list__strict__asc__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ( ( list_strict_asc @ A )
% 0.32/0.47 = ( ^ [Xs3: list @ A] :
% 0.32/0.47 ! [J3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.32/0.47 => ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ J3 )
% 0.32/0.47 => ( ord_less @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_strict_asc_trans
% 0.32/0.47 thf(fact_176_list__desc__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ( ( list_desc @ A )
% 0.32/0.47 = ( ^ [Xs3: list @ A] :
% 0.32/0.47 ! [J3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.32/0.47 => ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ J3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_desc_trans
% 0.32/0.47 thf(fact_177_order_Onot__eq__order__implies__strict,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( A3 != B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.not_eq_order_implies_strict
% 0.32/0.47 thf(fact_178_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A3: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.32/0.47 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_implies_order
% 0.32/0.47 thf(fact_179_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [B4: A,A4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.32/0.47 & ( A4 != B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_iff_order
% 0.32/0.47 thf(fact_180_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [B4: A,A4: A] :
% 0.32/0.47 ( ( ord_less @ A @ B4 @ A4 )
% 0.32/0.47 | ( A4 = B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.order_iff_strict
% 0.32/0.47 thf(fact_181_order_Ostrict__implies__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_implies_order
% 0.32/0.47 thf(fact_182_dense__le__bounded,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ! [W: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ W )
% 0.32/0.47 => ( ( ord_less @ A @ W @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense_le_bounded
% 0.32/0.47 thf(fact_183_dense__ge__bounded,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_linorder @ A )
% 0.32/0.47 => ! [Z2: A,X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/0.47 => ( ! [W: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z2 @ W )
% 0.32/0.47 => ( ( ord_less @ A @ W @ X3 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense_ge_bounded
% 0.32/0.47 thf(fact_184_dual__order_Ostrict__trans2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.32/0.47 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_trans2
% 0.32/0.47 thf(fact_185_dual__order_Ostrict__trans1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.32/0.47 => ( ( ord_less @ A @ C @ B3 )
% 0.32/0.47 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_trans1
% 0.32/0.47 thf(fact_186_order_Ostrict__iff__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 & ( A4 != B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_iff_order
% 0.32/0.47 thf(fact_187_order_Oorder__iff__strict,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.32/0.47 | ( A4 = B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.order_iff_strict
% 0.32/0.47 thf(fact_188_order_Ostrict__trans2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_trans2
% 0.32/0.47 thf(fact_189_order_Ostrict__trans1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_trans1
% 0.32/0.47 thf(fact_190_not__le__imp__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [Y2: A,X3: A] :
% 0.32/0.47 ( ~ ( ord_less_eq @ A @ Y2 @ X3 )
% 0.32/0.47 => ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_le_imp_less
% 0.32/0.47 thf(fact_191_less__le__not__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [X: A,Y5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.32/0.47 & ~ ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_le_not_le
% 0.32/0.47 thf(fact_192_le__imp__less__or__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 | ( X3 = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_imp_less_or_eq
% 0.32/0.47 thf(fact_193_le__less__linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_less_linear
% 0.32/0.47 thf(fact_194_dense__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_linorder @ A )
% 0.32/0.47 => ! [Y2: A,Z2: A] :
% 0.32/0.47 ( ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense_le
% 0.32/0.47 thf(fact_195_dense__ge,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_linorder @ A )
% 0.32/0.47 => ! [Z2: A,Y2: A] :
% 0.32/0.47 ( ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z2 @ X2 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ X2 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense_ge
% 0.32/0.47 thf(fact_196_less__le__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.32/0.47 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_le_trans
% 0.32/0.47 thf(fact_197_le__less__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.32/0.47 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_less_trans
% 0.32/0.47 thf(fact_198_less__imp__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_le
% 0.32/0.47 thf(fact_199_antisym__conv2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv2
% 0.32/0.47 thf(fact_200_antisym__conv1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv1
% 0.32/0.47 thf(fact_201_le__neq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( A3 != B3 )
% 0.32/0.47 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_neq_trans
% 0.32/0.47 thf(fact_202_not__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less
% 0.32/0.47 thf(fact_203_not__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
% 0.32/0.47 = ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_le
% 0.32/0.47 thf(fact_204_order__less__le__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C2: $tType] :
% 0.32/0.47 ( ( ( order @ C2 )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,B3: A,F: A > C2,C: C2] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_le_subst2
% 0.32/0.47 thf(fact_205_order__less__le__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_le_subst1
% 0.32/0.47 thf(fact_206_order__le__less__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C2: $tType] :
% 0.32/0.47 ( ( ( order @ C2 )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,B3: A,F: A > C2,C: C2] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_le_less_subst2
% 0.32/0.47 thf(fact_207_order__le__less__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_le_less_subst1
% 0.32/0.47 thf(fact_208_less__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [X: A,Y5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.32/0.47 & ( X != Y5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_le
% 0.32/0.47 thf(fact_209_le__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [X: A,Y5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y5 )
% 0.32/0.47 | ( X = Y5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_less
% 0.32/0.47 thf(fact_210_leI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X3: A,Y2: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % leI
% 0.32/0.47 thf(fact_211_leD,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [Y2: A,X3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.32/0.47 => ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % leD
% 0.32/0.47 thf(fact_212_list__strict__desc__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ( ( list_strict_desc @ A )
% 0.32/0.47 = ( ^ [Xs3: list @ A] :
% 0.32/0.47 ! [J3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.32/0.47 => ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ J3 )
% 0.32/0.47 => ( ord_less @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_strict_desc_trans
% 0.32/0.47 thf(fact_213_complete__interval,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( condit1037483654norder @ A )
% 0.32/0.47 => ! [A3: A,B3: A,P: A > $o] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 => ( ( P @ A3 )
% 0.32/0.47 => ( ~ ( P @ B3 )
% 0.32/0.47 => ? [C3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ C3 )
% 0.32/0.47 & ( ord_less_eq @ A @ C3 @ B3 )
% 0.32/0.47 & ! [X5: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ A3 @ X5 )
% 0.32/0.47 & ( ord_less @ A @ X5 @ C3 ) )
% 0.32/0.47 => ( P @ X5 ) )
% 0.32/0.47 & ! [D2: A] :
% 0.32/0.47 ( ! [X2: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ A3 @ X2 )
% 0.32/0.47 & ( ord_less @ A @ X2 @ D2 ) )
% 0.32/0.47 => ( P @ X2 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % complete_interval
% 0.32/0.47 thf(fact_214_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.47 ! [B: $tType] :
% 0.32/0.47 ( ( linorder @ B )
% 0.32/0.47 => ! [B6: B,A6: B] :
% 0.32/0.47 ( ( ~ ( ord_less_eq @ B @ B6 @ A6 ) )
% 0.32/0.47 = ( ord_less @ B @ A6 @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % verit_comp_simplify1(3)
% 0.32/0.47 thf(fact_215_pinf_I6_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(6)
% 0.32/0.47 thf(fact_216_pinf_I8_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.32/0.47 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(8)
% 0.32/0.47 thf(fact_217_less__fun__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ( ( ord_less @ ( A > B ) )
% 0.32/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.32/0.47 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_fun_def
% 0.32/0.47 thf(fact_218_verit__la__disequality,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( A3 = B3 )
% 0.32/0.47 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 0.32/0.47 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % verit_la_disequality
% 0.32/0.47 thf(fact_219_minf_I11_J,axiom,
% 0.32/0.47 ! [C2: $tType,D3: $tType] :
% 0.32/0.47 ( ( ord @ C2 )
% 0.32/0.47 => ! [F3: D3] :
% 0.32/0.47 ? [Z3: C2] :
% 0.32/0.47 ! [X5: C2] :
% 0.32/0.47 ( ( ord_less @ C2 @ X5 @ Z3 )
% 0.32/0.47 => ( F3 = F3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(11)
% 0.32/0.47 thf(fact_220_minf_I7_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.32/0.47 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(7)
% 0.32/0.47 thf(fact_221_minf_I5_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.32/0.47 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(5)
% 0.32/0.47 thf(fact_222_minf_I4_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.32/0.47 => ( X5 != T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(4)
% 0.32/0.47 thf(fact_223_minf_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.32/0.47 => ( X5 != T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(3)
% 0.32/0.47 thf(fact_224_minf_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/0.47 ( ? [Z4: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z4 )
% 0.32/0.47 => ( ( P @ X2 )
% 0.32/0.47 = ( P4 @ X2 ) ) )
% 0.32/0.47 => ( ? [Z4: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z4 )
% 0.32/0.47 => ( ( Q @ X2 )
% 0.32/0.47 = ( Q2 @ X2 ) ) )
% 0.32/0.47 => ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.32/0.47 => ( ( ( P @ X5 )
% 0.32/0.47 | ( Q @ X5 ) )
% 0.32/0.47 = ( ( P4 @ X5 )
% 0.32/0.47 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(2)
% 0.32/0.47 thf(fact_225_minf_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/0.47 ( ? [Z4: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z4 )
% 0.32/0.47 => ( ( P @ X2 )
% 0.32/0.47 = ( P4 @ X2 ) ) )
% 0.32/0.47 => ( ? [Z4: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z4 )
% 0.32/0.47 => ( ( Q @ X2 )
% 0.32/0.47 = ( Q2 @ X2 ) ) )
% 0.32/0.47 => ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.32/0.47 => ( ( ( P @ X5 )
% 0.32/0.47 & ( Q @ X5 ) )
% 0.32/0.47 = ( ( P4 @ X5 )
% 0.32/0.47 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(1)
% 0.32/0.47 thf(fact_226_pinf_I11_J,axiom,
% 0.32/0.47 ! [C2: $tType,D3: $tType] :
% 0.32/0.47 ( ( ord @ C2 )
% 0.32/0.47 => ! [F3: D3] :
% 0.32/0.47 ? [Z3: C2] :
% 0.32/0.47 ! [X5: C2] :
% 0.32/0.47 ( ( ord_less @ C2 @ Z3 @ X5 )
% 0.32/0.47 => ( F3 = F3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(11)
% 0.32/0.47 thf(fact_227_pinf_I7_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.32/0.47 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(7)
% 0.32/0.47 thf(fact_228_pinf_I5_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.32/0.47 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(5)
% 0.32/0.47 thf(fact_229_pinf_I4_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.32/0.47 => ( X5 != T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(4)
% 0.32/0.47 thf(fact_230_pinf_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.32/0.47 => ( X5 != T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(3)
% 0.32/0.47 thf(fact_231_pinf_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/0.47 ( ? [Z4: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.32/0.47 => ( ( P @ X2 )
% 0.32/0.47 = ( P4 @ X2 ) ) )
% 0.32/0.47 => ( ? [Z4: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.32/0.47 => ( ( Q @ X2 )
% 0.32/0.47 = ( Q2 @ X2 ) ) )
% 0.32/0.47 => ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.32/0.47 => ( ( ( P @ X5 )
% 0.32/0.47 | ( Q @ X5 ) )
% 0.32/0.47 = ( ( P4 @ X5 )
% 0.32/0.47 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(2)
% 0.32/0.47 thf(fact_232_pinf_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/0.47 ( ? [Z4: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.32/0.47 => ( ( P @ X2 )
% 0.32/0.47 = ( P4 @ X2 ) ) )
% 0.32/0.47 => ( ? [Z4: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.32/0.47 => ( ( Q @ X2 )
% 0.32/0.47 = ( Q2 @ X2 ) ) )
% 0.32/0.47 => ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.32/0.47 => ( ( ( P @ X5 )
% 0.32/0.47 & ( Q @ X5 ) )
% 0.32/0.47 = ( ( P4 @ X5 )
% 0.32/0.47 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(1)
% 0.32/0.47 thf(fact_233_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A] :
% 0.32/0.47 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % verit_comp_simplify1(1)
% 0.32/0.47 thf(fact_234_ex__gt__or__lt,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( condit1656338222tinuum @ A )
% 0.32/0.47 => ! [A3: A] :
% 0.32/0.47 ? [B5: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B5 )
% 0.32/0.47 | ( ord_less @ A @ B5 @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ex_gt_or_lt
% 0.32/0.47 thf(fact_235_list__strict__desc__imp__list__desc,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [Xs: list @ A] :
% 0.32/0.47 ( ( list_strict_desc @ A @ Xs )
% 0.32/0.47 => ( list_desc @ A @ Xs ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % list_strict_desc_imp_list_desc
% 0.32/0.47 thf(fact_236_minf_I8_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(8)
% 0.32/0.47 thf(fact_237_minf_I6_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z3: A] :
% 0.32/0.47 ! [X5: A] :
% 0.32/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.32/0.47 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(6)
% 0.32/0.47 thf(fact_238_i__append__i__take__eq2,axiom,
% 0.32/0.47 ! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 0.32/0.47 => ( ( listIn1033672622i_take @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
% 0.32/0.47 = ( append @ A @ Xs @ ( listIn1033672622i_take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ F ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % i_append_i_take_eq2
% 0.32/0.47 thf(fact_239_nth__drop,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,Xs: list @ A,I: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.32/0.47 => ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
% 0.32/0.47 = ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nth_drop
% 0.32/0.47 thf(fact_240_add__right__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel_semigroup_add @ A )
% 0.32/0.47 => ! [B3: A,A3: A,C: A] :
% 0.32/0.47 ( ( ( plus_plus @ A @ B3 @ A3 )
% 0.32/0.47 = ( plus_plus @ A @ C @ A3 ) )
% 0.32/0.47 = ( B3 = C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_right_cancel
% 0.32/0.47 thf(fact_241_add__left__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel_semigroup_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A,C: A] :
% 0.32/0.47 ( ( ( plus_plus @ A @ A3 @ B3 )
% 0.32/0.47 = ( plus_plus @ A @ A3 @ C ) )
% 0.32/0.47 = ( B3 = C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_left_cancel
% 0.32/0.47 thf(fact_242_add__le__cancel__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [A3: A,C: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_cancel_right
% 0.32/0.47 thf(fact_243_add__le__cancel__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [C: A,A3: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_cancel_left
% 0.32/0.47 thf(fact_244_add__less__cancel__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [C: A,A3: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
% 0.32/0.47 = ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_less_cancel_left
% 0.32/0.47 thf(fact_245_add__less__cancel__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [A3: A,C: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 0.32/0.47 = ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_less_cancel_right
% 0.32/0.47 thf(fact_246_add__diff__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
% 0.32/0.47 = A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_diff_cancel
% 0.32/0.47 thf(fact_247_diff__add__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
% 0.32/0.47 = A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_add_cancel
% 0.32/0.47 thf(fact_248_add__diff__cancel__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel146912293up_add @ A )
% 0.32/0.47 => ! [C: A,A3: A,B3: A] :
% 0.32/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
% 0.32/0.47 = ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_diff_cancel_left
% 0.32/0.47 thf(fact_249_add__diff__cancel__left_H,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel146912293up_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ A3 )
% 0.32/0.47 = B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_diff_cancel_left'
% 0.32/0.47 thf(fact_250_add__diff__cancel__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel146912293up_add @ A )
% 0.32/0.47 => ! [A3: A,C: A,B3: A] :
% 0.32/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 0.32/0.47 = ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_diff_cancel_right
% 0.32/0.47 thf(fact_251_add__diff__cancel__right_H,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel146912293up_add @ A )
% 0.32/0.47 => ! [A3: A,B3: A] :
% 0.32/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
% 0.32/0.47 = A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_diff_cancel_right'
% 0.32/0.47 thf(fact_252_nat__add__left__cancel__less,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.32/0.47 = ( ord_less @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_add_left_cancel_less
% 0.32/0.47 thf(fact_253_nat__add__left__cancel__le,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_add_left_cancel_le
% 0.32/0.47 thf(fact_254_diff__diff__left,axiom,
% 0.32/0.47 ! [I: nat,J: nat,K: nat] :
% 0.32/0.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.32/0.47 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_diff_left
% 0.32/0.47 thf(fact_255_drop__drop,axiom,
% 0.32/0.47 ! [A: $tType,N: nat,M: nat,Xs: list @ A] :
% 0.32/0.47 ( ( drop @ A @ N @ ( drop @ A @ M @ Xs ) )
% 0.32/0.47 = ( drop @ A @ ( plus_plus @ nat @ N @ M ) @ Xs ) ) ).
% 0.32/0.47
% 0.32/0.47 % drop_drop
% 0.32/0.47
% 0.32/0.47 % Type constructors (22)
% 0.32/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( preorder @ A8 )
% 0.32/0.47 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( order @ A8 )
% 0.32/0.47 => ( order @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ord @ A8 )
% 0.32/0.47 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Groups_Ominus,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( minus @ A8 )
% 0.32/0.47 => ( minus @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.32/0.47 condit1037483654norder @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.32/0.47 ordere236663937imp_le @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.32/0.47 cancel146912293up_add @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.32/0.47 cancel_semigroup_add @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.32/0.47 wellorder @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.32/0.47 preorder @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.32/0.47 linorder @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.32/0.47 no_top @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.32/0.47 order @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.32/0.47 ord @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
% 0.32/0.47 minus @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.32/0.47 size @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
% 0.32/0.47 preorder @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
% 0.32/0.47 linorder @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
% 0.32/0.47 order @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 0.32/0.47 ord @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
% 0.32/0.47 minus @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_List_Olist___Nat_Osize_10,axiom,
% 0.32/0.47 ! [A7: $tType] : ( size @ ( list @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Helper facts (3)
% 0.32/0.47 thf(help_If_3_1_T,axiom,
% 0.32/0.47 ! [P: $o] :
% 0.32/0.47 ( ( P = $true )
% 0.32/0.47 | ( P = $false ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(help_If_2_1_T,axiom,
% 0.32/0.47 ! [A: $tType,X3: A,Y2: A] :
% 0.32/0.47 ( ( if @ A @ $false @ X3 @ Y2 )
% 0.32/0.47 = Y2 ) ).
% 0.32/0.47
% 0.32/0.47 thf(help_If_1_1_T,axiom,
% 0.32/0.47 ! [A: $tType,X3: A,Y2: A] :
% 0.32/0.47 ( ( if @ A @ $true @ X3 @ Y2 )
% 0.32/0.47 = X3 ) ).
% 0.32/0.47
% 0.32/0.47 % Conjectures (1)
% 0.32/0.47 thf(conj_0,conjecture,
% 0.32/0.47 ( ( ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
% 0.32/0.47 => ( ( listIn1569887318i_drop @ a @ n @ ( listIn521021761append @ a @ xs @ f ) )
% 0.32/0.47 = ( listIn521021761append @ a @ ( drop @ a @ n @ xs ) @ f ) ) )
% 0.32/0.47 & ( ~ ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
% 0.32/0.47 => ( ( listIn1569887318i_drop @ a @ n @ ( listIn521021761append @ a @ xs @ f ) )
% 0.32/0.47 = ( listIn1569887318i_drop @ a @ ( minus_minus @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) ) @ f ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 2.59/1.52 ---- Embedded in THF ---
% 2.59/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.59/1.53 %%% Generated on Tue Feb 24 08:26:30 EST 2026
% 2.59/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.59/1.53 %%% Logic specification used:
% 2.59/1.53 %%% thf(spec, logic, $$dhol).
% 2.59/1.53
% 2.59/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.rtTF0N8tiU/DTF2DTF11420.p
% 2.59/1.53 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.59/1.53 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.59/1.53 thf(ty_tf_a, type, a: $tType).
% 2.59/1.53 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 2.59/1.53 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.59/1.53 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.59/1.53 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.59/1.53 thf(sy_c_List2_Olist__asc, type, list_asc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.59/1.53 thf(sy_c_List2_Olist__desc, type, list_desc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.59/1.53 thf(sy_c_List2_Olist__strict__asc, type, list_strict_asc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.59/1.53 thf(sy_c_List2_Olist__strict__desc, type, list_strict_desc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.59/1.53 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append, type, listIn521021761append: (!> [A:$tType]: (((list @ A) > ((nat > A) > (nat > A)))))).
% 2.59/1.53 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__drop, type, listIn1569887318i_drop: (!> [A:$tType]: ((nat > ((nat > A) > (nat > A)))))).
% 2.59/1.53 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__take, type, listIn1033672622i_take: (!> [A:$tType]: ((nat > ((nat > A) > (list @ A)))))).
% 2.59/1.53 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.59/1.53 thf(sy_c_List_Odrop, type, drop: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))).
% 2.59/1.53 thf(sy_c_List_Olast, type, last: (!> [A:$tType]: (((list @ A) > A)))).
% 2.59/1.53 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 2.59/1.53 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.59/1.53 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.59/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.59/1.53 thf(sy_v_f, type, f: (nat > a)).
% 2.59/1.53 thf(sy_v_n, type, n: nat).
% 2.59/1.53 thf(sy_v_xs, type, xs: (list @ a)).
% 2.59/1.53 thf(fact_0_same__i__append__eq, axiom, (! [A:$tType,Xs:(list @ A),F:(nat > A),G:(nat > A)]: ((((((listIn521021761append @ A) @ Xs) @ F) = (((listIn521021761append @ A) @ Xs) @ G)) = (F = G))))).
% 2.59/1.53 thf(fact_1_i__append__eq__i__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),F:(nat > A),G:(nat > A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => (((((listIn521021761append @ A) @ Xs) @ F) = (((listIn521021761append @ A) @ Ys) @ G)) = ((Xs = Ys) & (F = G))))))).
% 2.59/1.53 thf(fact_2_length__drop, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ (((drop @ A) @ N) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N))))).
% 2.59/1.53 thf(fact_3_minus__apply, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A2:(A > B),B2:(A > B),X:A]: ((((minus_minus @ B) @ (A2 @ X)) @ (B2 @ X))))))))).
% 2.59/1.53 thf(fact_4_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))))))).
% 2.59/1.53 thf(fact_5_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))))).
% 2.59/1.53 thf(fact_6_less__diff__imp__less, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ J) @ M)) => (((ord_less @ nat) @ I) @ J))))).
% 2.59/1.53 thf(fact_7_nat__diff__left__cancel__eq1, axiom, (! [K:nat,M:nat,N:nat]: ((((((minus_minus @ nat) @ K) @ M) = (((minus_minus @ nat) @ K) @ N)) => ((((ord_less @ nat) @ M) @ K) => (M = N)))))).
% 2.59/1.53 thf(fact_8_nat__diff__left__cancel__eq2, axiom, (! [K:nat,M:nat,N:nat]: ((((((minus_minus @ nat) @ K) @ M) = (((minus_minus @ nat) @ K) @ N)) => ((((ord_less @ nat) @ N) @ K) => (M = N)))))).
% 2.59/1.53 thf(fact_9_nat__diff__left__cancel__less, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((minus_minus @ nat) @ K) @ M)) @ (((minus_minus @ nat) @ K) @ N)) => (((ord_less @ nat) @ N) @ M))))).
% 2.59/1.53 thf(fact_10_nat__diff__right__cancel__eq1, axiom, (! [M:nat,K:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) => ((((ord_less @ nat) @ K) @ M) => (M = N)))))).
% 2.59/1.53 thf(fact_11_nat__diff__right__cancel__eq2, axiom, (! [M:nat,K:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) => ((((ord_less @ nat) @ K) @ N) => (M = N)))))).
% 2.59/1.53 thf(fact_12_nat__diff__right__cancel__less, axiom, (! [N:nat,K:nat,M:nat]: (((((ord_less @ nat) @ (((minus_minus @ nat) @ N) @ K)) @ (((minus_minus @ nat) @ M) @ K)) => (((ord_less @ nat) @ N) @ M))))).
% 2.59/1.53 thf(fact_13_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs2:(list @ A)]: (((! [Ys2:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys2)) @ ((size_size @ (list @ A)) @ Xs2)) => (P @ Ys2)))) => (P @ Xs2)))) => (P @ Xs))))).
% 2.59/1.53 thf(fact_14_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A3:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ B) @ (F @ Y)) @ (F @ X2)) => (P @ Y)))) => (P @ X2)))) => (P @ A3)))))))).
% 2.59/1.53 thf(fact_15_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A3:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ B) @ (F @ Y)) @ (F @ X2)) => (P @ Y)))) => (P @ X2)))) => (P @ A3)))))))).
% 2.59/1.53 thf(fact_16_fun__diff__def, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A2:(A > B),B2:(A > B),X:A]: ((((minus_minus @ B) @ (A2 @ X)) @ (B2 @ X))))))))).
% 2.59/1.53 thf(fact_17_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X3:A]: (((! [X2:A]: (((~ ((P @ X2))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X2)) & (~ ((P @ Y))))))))) => (P @ X3))))).
% 2.59/1.53 thf(fact_18_linorder__neqE__nat, axiom, (! [X3:nat,Y2:nat]: (((X3 != Y2) => ((~ ((((ord_less @ nat) @ X3) @ Y2))) => (((ord_less @ nat) @ Y2) @ X3)))))).
% 2.59/1.53 thf(fact_19_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2))))))))) => (P @ N))))).
% 2.59/1.53 thf(fact_20_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 2.59/1.53 thf(fact_21_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.59/1.53 thf(fact_22_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.59/1.53 thf(fact_23_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.59/1.53 thf(fact_24_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.59/1.53 thf(fact_25_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.59/1.53 thf(fact_26_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X3:A,Y2:A]: (((((size_size @ A) @ X3) != ((size_size @ A) @ Y2)) => (X3 != Y2)))))))).
% 2.59/1.53 thf(fact_27_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys)) => (Xs != Ys))))).
% 2.59/1.53 thf(fact_28_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs2:(list @ A)]: ((((size_size @ (list @ A)) @ Xs2) = N)))))).
% 2.59/1.53 thf(fact_29_eq__imp__diff__eq, axiom, (! [M:nat,N:nat,K:nat]: (((M = N) => ((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)))))).
% 2.59/1.53 thf(fact_30_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))))).
% 2.59/1.53 thf(fact_31_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.59/1.53 thf(fact_32_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A3:A,C:A]: (((((ord_less @ A) @ B3) @ A3) => (((ord_less @ A) @ (((minus_minus @ A) @ C) @ A3)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.59/1.53 thf(fact_33_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less @ A) @ A3) @ B3) = (((ord_less @ A) @ C) @ D))))))))).
% 2.59/1.53 thf(fact_34_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,D:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ A) @ D) @ C) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.59/1.53 thf(fact_35_i__append__i__drop__eq1, axiom, (! [A:$tType,N:nat,Xs:(list @ A),F:(nat > A)]: (((((ord_less_eq @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((listIn1569887318i_drop @ A) @ N) @ (((listIn521021761append @ A) @ Xs) @ F)) = (((listIn521021761append @ A) @ (((drop @ A) @ N) @ Xs)) @ F)))))).
% 2.59/1.53 thf(fact_36_i__append__i__drop__eq2, axiom, (! [A:$tType,Xs:(list @ A),N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N) => ((((listIn1569887318i_drop @ A) @ N) @ (((listIn521021761append @ A) @ Xs) @ F)) = (((listIn1569887318i_drop @ A) @ (((minus_minus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs))) @ F)))))).
% 2.59/1.53 thf(fact_37_last__drop, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((last @ A) @ (((drop @ A) @ N) @ Xs)) = ((last @ A) @ Xs)))))).
% 2.59/1.53 thf(fact_38_i__append__nth2, axiom, (! [A:$tType,Xs:(list @ A),N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N) => (((((listIn521021761append @ A) @ Xs) @ F) @ N) = (F @ (((minus_minus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)))))))).
% 2.59/1.53 thf(fact_39_i__append__nth, axiom, (! [A:$tType]: (((listIn521021761append @ A) = (^ [Xs3:(list @ A),F2:(nat > A),N3:nat]: (((((if @ A) @ (((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs3))) @ (((nth @ A) @ Xs3) @ N3)) @ (F2 @ (((minus_minus @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs3)))))))))).
% 2.59/1.53 thf(fact_40_i__append__def, axiom, (! [A:$tType]: (((listIn521021761append @ A) = (^ [Xs3:(list @ A),F2:(nat > A),N3:nat]: (((((if @ A) @ (((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs3))) @ (((nth @ A) @ Xs3) @ N3)) @ (F2 @ (((minus_minus @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs3)))))))))).
% 2.59/1.53 thf(fact_41_drop__append, axiom, (! [A:$tType,N:nat,Xs:(list @ A),Ys:(list @ A)]: (((((drop @ A) @ N) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ (((drop @ A) @ N) @ Xs)) @ (((drop @ A) @ (((minus_minus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs))) @ Ys)))))).
% 2.59/1.53 thf(fact_42_append_Oassoc, axiom, (! [A:$tType,A3:(list @ A),B3:(list @ A),C:(list @ A)]: (((((append @ A) @ (((append @ A) @ A3) @ B3)) @ C) = (((append @ A) @ A3) @ (((append @ A) @ B3) @ C)))))).
% 2.59/1.53 thf(fact_43_append__assoc, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs:(list @ A)]: (((((append @ A) @ (((append @ A) @ Xs) @ Ys)) @ Zs) = (((append @ A) @ Xs) @ (((append @ A) @ Ys) @ Zs)))))).
% 2.59/1.53 thf(fact_44_append__same__eq, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A),Zs:(list @ A)]: ((((((append @ A) @ Ys) @ Xs) = (((append @ A) @ Zs) @ Xs)) = (Ys = Zs))))).
% 2.59/1.53 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.59/1.53 thf(fact_46_same__append__eq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Xs) @ Zs)) = (Ys = Zs))))).
% 2.59/1.53 thf(fact_47_diff__diff__cancel, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I)) = I))))).
% 2.59/1.53 thf(fact_48_append__eq__append__conv, axiom, (! [A:$tType,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))))))).
% 2.59/1.53 thf(fact_49_i__append__assoc, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),F:(nat > A)]: (((((listIn521021761append @ A) @ Xs) @ (((listIn521021761append @ A) @ Ys) @ F)) = (((listIn521021761append @ A) @ (((append @ A) @ Xs) @ Ys)) @ F))))).
% 2.59/1.53 thf(fact_50_i__append__nth1, axiom, (! [A:$tType,N:nat,Xs:(list @ A),F:(nat > A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((((listIn521021761append @ A) @ Xs) @ F) @ N) = (((nth @ A) @ Xs) @ N)))))).
% 2.59/1.53 thf(fact_51_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.59/1.53 thf(fact_52_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)))))).
% 2.59/1.53 thf(fact_53_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.59/1.53 thf(fact_54_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.59/1.53 thf(fact_55_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.59/1.53 thf(fact_56_append__eq__appendI, axiom, (! [A:$tType,Xs:(list @ A),Xs1:(list @ A),Zs:(list @ A),Ys:(list @ A),Us:(list @ A)]: ((((((append @ A) @ Xs) @ Xs1) = Zs) => ((Ys = (((append @ A) @ Xs1) @ Us)) => ((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Zs) @ Us))))))).
% 2.59/1.53 thf(fact_57_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B3:nat]: (((P @ K) => ((! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ B3)))) => (? [X2:nat]: (((P @ X2) & (! [Y:nat]: (((P @ Y) => (((ord_less_eq @ nat) @ Y) @ X2)))))))))))).
% 2.59/1.53 thf(fact_58_append__eq__append__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs:(list @ A),Ts:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Zs) @ Ts)) = (? [Us2:(list @ A)]: ((((Xs = (((append @ A) @ Zs) @ Us2)) & ((((append @ A) @ Us2) @ Ys) = Ts)) | (((((append @ A) @ Xs) @ Us2) = Zs) & (Ys = (((append @ A) @ Us2) @ Ts)))))))))).
% 2.59/1.53 thf(fact_59_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,D:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ D) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.59/1.53 thf(fact_60_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A3)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.59/1.53 thf(fact_61_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.59/1.53 thf(fact_62_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less_eq @ A) @ A3) @ B3) = (((ord_less_eq @ A) @ C) @ D))))))))).
% 2.59/1.53 thf(fact_63_i__append__eq__i__append__conv2__aux, axiom, (! [A:$tType,Xs:(list @ A),F:(nat > A),Ys:(list @ A),G:(nat > A)]: ((((((listIn521021761append @ A) @ Xs) @ F) = (((listIn521021761append @ A) @ Ys) @ G)) => ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys)) => (? [Zs2:(list @ A)]: ((((((append @ A) @ Xs) @ Zs2) = Ys) & (F = (((listIn521021761append @ A) @ Zs2) @ G)))))))))).
% 2.59/1.53 thf(fact_64_ge__less__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A4:A,N3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ A4) => (N3 != X))))))))))).
% 2.59/1.53 thf(fact_65_less__ge__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [N3:A,A4:A]: ((! [X:A]: (((((ord_less_eq @ A) @ A4) @ X) => (N3 != X))))))))))).
% 2.59/1.53 thf(fact_66_greater__le__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [A4:A,N3:A]: ((! [X:A]: (((((ord_less_eq @ A) @ X) @ A4) => (N3 != X))))))))))).
% 2.59/1.53 thf(fact_67_le__greater__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [N3:A,A4:A]: ((! [X:A]: (((((ord_less @ A) @ A4) @ X) => (N3 != X))))))))))).
% 2.59/1.53 thf(fact_68_nat__less__le, axiom, ((ord_less @ nat) = (^ [M3:nat,N3:nat]: (((((ord_less_eq @ nat) @ M3) @ N3) & (M3 != N3)))))).
% 2.59/1.53 thf(fact_69_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.59/1.53 thf(fact_70_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M3:nat,N3:nat]: (((((ord_less @ nat) @ M3) @ N3) | (M3 = N3)))))).
% 2.59/1.53 thf(fact_71_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.59/1.53 thf(fact_72_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.59/1.53 thf(fact_73_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))))))).
% 2.59/1.53 thf(fact_74_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))))))).
% 2.59/1.53 thf(fact_75_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))))))).
% 2.59/1.53 thf(fact_76_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))))))).
% 2.59/1.53 thf(fact_77_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)))))).
% 2.59/1.53 thf(fact_78_diff__le__self, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))).
% 2.59/1.53 thf(fact_79_le__diff__iff_H, axiom, (! [A3:nat,C:nat,B3:nat]: (((((ord_less_eq @ nat) @ A3) @ C) => ((((ord_less_eq @ nat) @ B3) @ C) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C) @ A3)) @ (((minus_minus @ nat) @ C) @ B3)) = (((ord_less_eq @ nat) @ B3) @ A3))))))).
% 2.59/1.53 thf(fact_80_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)))))).
% 2.59/1.53 thf(fact_81_le__diff__swap, axiom, (! [I:nat,K:nat,J:nat]: (((((ord_less_eq @ nat) @ I) @ K) => ((((ord_less_eq @ nat) @ J) @ K) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ K) @ J)) @ I) = (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ K) @ I)) @ J))))))).
% 2.59/1.53 thf(fact_82_le__diff__imp__le, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ (((minus_minus @ nat) @ J) @ M)) => (((ord_less_eq @ nat) @ I) @ J))))).
% 2.59/1.53 thf(fact_83_le__imp__diff__le, axiom, (! [J:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ J) @ N)) @ K))))).
% 2.59/1.53 thf(fact_84_eq__diff__left__iff, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ K) => ((((ord_less_eq @ nat) @ N) @ K) => (((((minus_minus @ nat) @ K) @ M) = (((minus_minus @ nat) @ K) @ N)) = (M = N))))))).
% 2.59/1.53 thf(fact_85_le__diff__le__imp__le, axiom, (! [I:nat,J:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ (((minus_minus @ nat) @ J) @ M)) => ((((ord_less_eq @ nat) @ N) @ M) => (((ord_less_eq @ nat) @ I) @ (((minus_minus @ nat) @ J) @ N))))))).
% 2.59/1.53 thf(fact_86_nat__diff__left__cancel__le2, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ K) @ M)) @ (((minus_minus @ nat) @ K) @ N)) => ((((ord_less_eq @ nat) @ N) @ K) => (((ord_less_eq @ nat) @ N) @ M)))))).
% 2.59/1.53 thf(fact_87_nat__diff__right__cancel__le2, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) => ((((ord_less_eq @ nat) @ K) @ N) => (((ord_less_eq @ nat) @ M) @ N)))))).
% 2.59/1.53 thf(fact_88_i__append__eq__i__appendI, axiom, (! [A:$tType,Xs:(list @ A),Xs4:(list @ A),Ys:(list @ A),F:(nat > A),G:(nat > A)]: ((((((append @ A) @ Xs) @ Xs4) = Ys) => ((F = (((listIn521021761append @ A) @ Xs4) @ G)) => ((((listIn521021761append @ A) @ Xs) @ F) = (((listIn521021761append @ A) @ Ys) @ G))))))).
% 2.59/1.53 thf(fact_89_i__append__eq__i__append__conv2, axiom, (! [A:$tType,Xs:(list @ A),F:(nat > A),Ys:(list @ A),G:(nat > A)]: ((((((listIn521021761append @ A) @ Xs) @ F) = (((listIn521021761append @ A) @ Ys) @ G)) = (? [Zs3:(list @ A)]: ((((Xs = (((append @ A) @ Ys) @ Zs3)) & ((((listIn521021761append @ A) @ Zs3) @ F) = G)) | (((((append @ A) @ Xs) @ Zs3) = Ys) & (F = (((listIn521021761append @ A) @ Zs3) @ G)))))))))).
% 2.59/1.53 thf(fact_90_nth__append, axiom, (! [A:$tType,N:nat,Xs:(list @ A),Ys:(list @ A)]: ((((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((append @ A) @ Xs) @ Ys)) @ N) = (((nth @ A) @ Xs) @ N))) & ((~ ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)))) => ((((nth @ A) @ (((append @ A) @ Xs) @ Ys)) @ N) = (((nth @ A) @ Ys) @ (((minus_minus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs))))))))).
% 2.59/1.53 thf(fact_91_nth__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I2) = (((nth @ A) @ Ys) @ I2))))) => (Xs = Ys)))))).
% 2.59/1.53 thf(fact_92_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K) => (? [X4:A]: (((P @ I3) @ X4)))))) = (? [Xs3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = K) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ K) => ((P @ I3) @ (((nth @ A) @ Xs3) @ I3)))))))))))).
% 2.59/1.53 thf(fact_93_list__eq__iff__nth__eq, axiom, (! [A:$tType]: (((^ [Y4:(list @ A),Z:(list @ A)]: ((Y4 = Z))) = (^ [Xs3:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ A)) @ Ys3)) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs3)) => ((((nth @ A) @ Xs3) @ I3) = (((nth @ A) @ Ys3) @ I3)))))))))))).
% 2.59/1.53 thf(fact_94_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))))))).
% 2.59/1.53 thf(fact_95_diff__less__mono, axiom, (! [A3:nat,B3:nat,C:nat]: (((((ord_less @ nat) @ A3) @ B3) => ((((ord_less_eq @ nat) @ C) @ A3) => (((ord_less @ nat) @ (((minus_minus @ nat) @ A3) @ C)) @ (((minus_minus @ nat) @ B3) @ C))))))).
% 2.59/1.53 thf(fact_96_less__diff__le__imp__less, axiom, (! [I:nat,J:nat,M:nat,N:nat]: (((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ J) @ M)) => ((((ord_less_eq @ nat) @ N) @ M) => (((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ J) @ N))))))).
% 2.59/1.53 thf(fact_97_nat__diff__left__cancel__le1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ K) @ M)) @ (((minus_minus @ nat) @ K) @ N)) => ((((ord_less @ nat) @ M) @ K) => (((ord_less_eq @ nat) @ N) @ M)))))).
% 2.59/1.53 thf(fact_98_nat__diff__right__cancel__le1, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) => ((((ord_less @ nat) @ K) @ M) => (((ord_less_eq @ nat) @ M) @ N)))))).
% 2.59/1.53 thf(fact_99_i__drop__nth__sub, axiom, (! [A:$tType,N:nat,X3:nat,S:(nat > A)]: (((((ord_less_eq @ nat) @ N) @ X3) => (((((listIn1569887318i_drop @ A) @ N) @ S) @ (((minus_minus @ nat) @ X3) @ N)) = (S @ X3)))))).
% 2.59/1.53 thf(fact_100_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((A3 = B3) = (C = D))))))))).
% 2.59/1.53 thf(fact_101_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A3:A,C:A,B3:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ C)) @ B3) = (((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ B3)) @ C)))))))).
% 2.59/1.53 thf(fact_102_nth__append2, axiom, (! [A:$tType,Xs:(list @ A),N:nat,Ys:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N) => ((((nth @ A) @ (((append @ A) @ Xs) @ Ys)) @ N) = (((nth @ A) @ Ys) @ (((minus_minus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)))))))).
% 2.59/1.53 thf(fact_103_nth__append1, axiom, (! [A:$tType,N:nat,Xs:(list @ A),Ys:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((append @ A) @ Xs) @ Ys)) @ N) = (((nth @ A) @ Xs) @ N)))))).
% 2.59/1.53 thf(fact_104_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.59/1.53 thf(fact_105_list__desc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => ((list_desc @ A) = (^ [Xs3:(list @ A)]: ((! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs3)) => (! [I3:nat]: (((((ord_less_eq @ nat) @ I3) @ J3) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs3) @ J3)) @ (((nth @ A) @ Xs3) @ I3))))))))))))))).
% 2.59/1.53 thf(fact_106_nat__descend__induct, axiom, (! [N:nat,P:(nat > $o),M:nat]: (((! [K2:nat]: (((((ord_less @ nat) @ N) @ K2) => (P @ K2)))) => ((! [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) => ((! [I4:nat]: (((((ord_less @ nat) @ K2) @ I4) => (P @ I4)))) => (P @ K2))))) => (P @ M)))))).
% 2.59/1.53 thf(fact_107_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),B3:nat]: (((P @ K) => ((! [Y3:A]: (((P @ Y3) => (((ord_less @ nat) @ (F @ Y3)) @ B3)))) => (? [X2:A]: (((P @ X2) & (! [Y:A]: (((P @ Y) => (((ord_less_eq @ nat) @ (F @ Y)) @ (F @ X2))))))))))))).
% 2.59/1.53 thf(fact_108_list__asc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => ((list_asc @ A) = (^ [Xs3:(list @ A)]: ((! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs3)) => (! [I3:nat]: (((((ord_less_eq @ nat) @ I3) @ J3) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs3) @ I3)) @ (((nth @ A) @ Xs3) @ J3))))))))))))))).
% 2.59/1.53 thf(fact_109_list__strict__asc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_asc @ A) @ Xs) => (! [J4:nat]: (((((ord_less @ nat) @ J4) @ ((size_size @ (list @ A)) @ Xs)) => (! [I4:nat]: (((((ord_less_eq @ nat) @ I4) @ J4) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I4)) @ (((nth @ A) @ Xs) @ J4))))))))))))))).
% 2.59/1.53 thf(fact_110_list__strict__asc__imp__list__asc, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_asc @ A) @ Xs) => ((list_asc @ A) @ Xs)))))))).
% 2.59/1.53 thf(fact_111_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.59/1.53 thf(fact_112_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.59/1.53 thf(fact_113_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.59/1.53 thf(fact_114_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))))).
% 2.59/1.53 thf(fact_115_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less_eq @ B) @ X2) @ Y3) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 2.59/1.53 thf(fact_116_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A3:A,B3:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C2) @ (F @ B3)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ C2) @ (F @ A3)) @ C)))))))))).
% 2.59/1.53 thf(fact_117_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less_eq @ B) @ X2) @ Y3) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 2.59/1.53 thf(fact_118_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C) => ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C)))))))))).
% 2.59/1.53 thf(fact_119_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X))))))))).
% 2.59/1.53 thf(fact_120_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X3) => (X3 = Y2))))))))).
% 2.59/1.53 thf(fact_121_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.59/1.53 thf(fact_122_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((X3 = Y2) => (((ord_less_eq @ A) @ X3) @ Y2)))))))).
% 2.59/1.53 thf(fact_123_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.59/1.53 thf(fact_124_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.59/1.53 thf(fact_125_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X3) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z2)))) => (((((ord_less_eq @ A) @ X3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X3)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y2))))))))))))))))).
% 2.59/1.53 thf(fact_126_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y2) = (X3 = Y2))))))))).
% 2.59/1.53 thf(fact_127_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))).
% 2.59/1.53 thf(fact_128_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.59/1.53 thf(fact_129_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.59/1.53 thf(fact_130_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.59/1.53 thf(fact_131_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X3) @ Z2))))))))).
% 2.59/1.53 thf(fact_132_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.59/1.53 thf(fact_133_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A3) @ B3))))))))).
% 2.59/1.53 thf(fact_134_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less_eq @ A) @ C) @ A3))))))))).
% 2.59/1.53 thf(fact_135_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))).
% 2.59/1.53 thf(fact_136_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.59/1.53 thf(fact_137_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (A3 != B3)))))))).
% 2.59/1.53 thf(fact_138_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (A3 != B3)))))))).
% 2.59/1.53 thf(fact_139_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) = ((((ord_less @ A) @ Y2) @ X3) | (X3 = Y2))))))))).
% 2.59/1.53 thf(fact_140_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C:A]: (((((ord_less @ A) @ B3) @ A3) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A3))))))))).
% 2.59/1.53 thf(fact_141_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B5:A]: (((((ord_less @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A]: (((P @ A5) @ A5))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A3) @ B3)))))))))).
% 2.59/1.53 thf(fact_142_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X4:A]: ((P2 @ X4))))) = (^ [P3:(A > $o)]: ((? [N3:A]: (((P3 @ N3) & (! [M3:A]: (((((ord_less @ A) @ M3) @ N3) => (~ ((P3 @ M3)))))))))))))))).
% 2.59/1.53 thf(fact_143_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.59/1.53 thf(fact_144_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.59/1.53 thf(fact_145_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 2.59/1.53 thf(fact_146_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => ((X3 != Y2) => (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.59/1.53 thf(fact_147_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A,P:$o]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ X3) => P)))))))).
% 2.59/1.53 thf(fact_148_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (Y2 != X3)))))))).
% 2.59/1.53 thf(fact_149_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y2:A,X3:A]: (((~ ((((ord_less @ A) @ Y2) @ X3))) => ((~ ((((ord_less @ A) @ X3) @ Y2))) = (X3 = Y2))))))))).
% 2.59/1.53 thf(fact_150_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A3:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ A) @ Y) @ X2) => (P @ Y)))) => (P @ X2)))) => (P @ A3)))))))).
% 2.59/1.53 thf(fact_151_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.59/1.53 thf(fact_152_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (X3 != Y2)))))))).
% 2.59/1.53 thf(fact_153_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (~ ((((ord_less @ A) @ A3) @ B3)))))))))).
% 2.59/1.53 thf(fact_154_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => ((B3 = C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.59/1.53 thf(fact_155_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C:A]: (((A3 = B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.59/1.53 thf(fact_156_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((~ ((((ord_less @ A) @ X3) @ X3))))))))).
% 2.59/1.53 thf(fact_157_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) | ((X3 = Y2) | (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.59/1.53 thf(fact_158_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 2.59/1.53 thf(fact_159_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (~ ((((ord_less @ A) @ B3) @ A3)))))))))).
% 2.59/1.53 thf(fact_160_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.59/1.53 thf(fact_161_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (X3 != Y2)))))))).
% 2.59/1.53 thf(fact_162_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (? [Z3:A]: (((((ord_less @ A) @ X3) @ Z3) & (((ord_less @ A) @ Z3) @ Y2))))))))))).
% 2.59/1.53 thf(fact_163_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (~ ((((ord_less @ A) @ B3) @ A3)))))))))).
% 2.59/1.53 thf(fact_164_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((X3 != Y2) = ((((ord_less @ A) @ X3) @ Y2) | (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.59/1.53 thf(fact_165_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((X3 != Y2) => ((~ ((((ord_less @ A) @ X3) @ Y2))) => (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.59/1.53 thf(fact_166_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X3:A]: ((? [X_1:A]: ((((ord_less @ A) @ X3) @ X_1))))))))).
% 2.59/1.53 thf(fact_167_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X3:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X3))))))))).
% 2.59/1.53 thf(fact_168_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A3:A,B3:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ C2) @ (F @ B3)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A3)) @ C)))))))))).
% 2.59/1.53 thf(fact_169_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A3) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less @ B) @ X2) @ Y3) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 2.59/1.53 thf(fact_170_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C:B]: (((((ord_less @ A) @ A3) @ B3) => (((F @ B3) = C) => ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ B) @ (F @ A3)) @ C)))))))))).
% 2.59/1.53 thf(fact_171_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C:B]: (((A3 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less @ B) @ X2) @ Y3) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 2.59/1.53 thf(fact_172_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (? [X2:A]: (((P @ X2) & (! [Y:A]: (((P @ Y) => (((ord_less_eq @ nat) @ (M @ X2)) @ (M @ Y)))))))))))).
% 2.59/1.53 thf(fact_173_list__drop__eq__conv, axiom, (! [A:$tType]: (((^ [Y4:(list @ A),Z:(list @ A)]: ((Y4 = Z))) = (^ [Xs3:(list @ A),Ys3:(list @ A)]: ((! [N3:nat]: (((((drop @ A) @ N3) @ Xs3) = (((drop @ A) @ N3) @ Ys3)))))))))).
% 2.59/1.53 thf(fact_174_list__asc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_asc @ A) = (^ [Xs3:(list @ A)]: ((! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs3)) => (! [I3:nat]: (((((ord_less @ nat) @ I3) @ J3) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs3) @ I3)) @ (((nth @ A) @ Xs3) @ J3))))))))))))))).
% 2.59/1.53 thf(fact_175_list__strict__asc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_strict_asc @ A) = (^ [Xs3:(list @ A)]: ((! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs3)) => (! [I3:nat]: (((((ord_less @ nat) @ I3) @ J3) => (((ord_less @ A) @ (((nth @ A) @ Xs3) @ I3)) @ (((nth @ A) @ Xs3) @ J3))))))))))))))).
% 2.59/1.53 thf(fact_176_list__desc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_desc @ A) = (^ [Xs3:(list @ A)]: ((! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs3)) => (! [I3:nat]: (((((ord_less @ nat) @ I3) @ J3) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs3) @ J3)) @ (((nth @ A) @ Xs3) @ I3))))))))))))))).
% 2.59/1.53 thf(fact_177_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((A3 != B3) => ((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less @ A) @ A3) @ B3))))))))).
% 2.59/1.53 thf(fact_178_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 2.59/1.53 thf(fact_179_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (A4 != B4))))))))).
% 2.59/1.53 thf(fact_180_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) | (A4 = B4))))))))).
% 2.59/1.53 thf(fact_181_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.59/1.53 thf(fact_182_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X3:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y2) => ((! [W:A]: (((((ord_less @ A) @ X3) @ W) => ((((ord_less @ A) @ W) @ Y2) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.59/1.53 thf(fact_183_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X3:A,Y2:A]: (((((ord_less @ A) @ Z2) @ X3) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X3) => (((ord_less_eq @ A) @ Y2) @ W))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.59/1.53 thf(fact_184_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C:A]: (((((ord_less @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A3))))))))).
% 2.59/1.53 thf(fact_185_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A3))))))))).
% 2.59/1.53 thf(fact_186_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (A4 != B4))))))))).
% 2.59/1.53 thf(fact_187_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) | (A4 = B4))))))))).
% 2.59/1.53 thf(fact_188_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.59/1.53 thf(fact_189_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.59/1.53 thf(fact_190_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y2:A,X3:A]: (((~ ((((ord_less_eq @ A) @ Y2) @ X3))) => (((ord_less @ A) @ X3) @ Y2)))))))).
% 2.59/1.53 thf(fact_191_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (~ ((((ord_less_eq @ A) @ Y5) @ X))))))))))).
% 2.59/1.53 thf(fact_192_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less @ A) @ X3) @ Y2) | (X3 = Y2))))))))).
% 2.59/1.53 thf(fact_193_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) | (((ord_less @ A) @ Y2) @ X3)))))))).
% 2.59/1.53 thf(fact_194_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y2:A,Z2:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less_eq @ A) @ X2) @ Z2)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.59/1.53 thf(fact_195_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,Y2:A]: (((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => (((ord_less_eq @ A) @ Y2) @ X2)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.59/1.53 thf(fact_196_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 2.59/1.53 thf(fact_197_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 2.59/1.53 thf(fact_198_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less_eq @ A) @ X3) @ Y2)))))))).
% 2.59/1.53 thf(fact_199_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((~ ((((ord_less @ A) @ X3) @ Y2))) = (X3 = Y2))))))))).
% 2.59/1.53 thf(fact_200_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => ((((ord_less_eq @ A) @ X3) @ Y2) = (X3 = Y2))))))))).
% 2.59/1.53 thf(fact_201_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((A3 != B3) => (((ord_less @ A) @ A3) @ B3))))))))).
% 2.59/1.53 thf(fact_202_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) = (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.59/1.53 thf(fact_203_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y2))) = (((ord_less @ A) @ Y2) @ X3)))))))).
% 2.59/1.53 thf(fact_204_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A3:A,B3:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less_eq @ C2) @ (F @ B3)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A3)) @ C)))))))))).
% 2.59/1.53 thf(fact_205_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less_eq @ B) @ X2) @ Y3) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 2.59/1.53 thf(fact_206_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A3:A,B3:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less @ C2) @ (F @ B3)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A3)) @ C)))))))))).
% 2.59/1.53 thf(fact_207_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less @ B) @ X2) @ Y3) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 2.59/1.53 thf(fact_208_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (X != Y5))))))))).
% 2.59/1.53 thf(fact_209_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X:A,Y5:A]: (((((ord_less @ A) @ X) @ Y5) | (X = Y5))))))))).
% 2.59/1.53 thf(fact_210_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.59/1.53 thf(fact_211_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ X3) => (~ ((((ord_less @ A) @ X3) @ Y2)))))))))).
% 2.59/1.53 thf(fact_212_list__strict__desc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_strict_desc @ A) = (^ [Xs3:(list @ A)]: ((! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs3)) => (! [I3:nat]: (((((ord_less @ nat) @ I3) @ J3) => (((ord_less @ A) @ (((nth @ A) @ Xs3) @ J3)) @ (((nth @ A) @ Xs3) @ I3))))))))))))))).
% 2.59/1.53 thf(fact_213_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A3:A,B3:A,P:(A > $o)]: (((((ord_less @ A) @ A3) @ B3) => ((P @ A3) => ((~ ((P @ B3))) => (? [C3:A]: (((((ord_less_eq @ A) @ A3) @ C3) & ((((ord_less_eq @ A) @ C3) @ B3) & ((! [X5:A]: ((((((ord_less_eq @ A) @ A3) @ X5) & (((ord_less @ A) @ X5) @ C3)) => (P @ X5)))) & (! [D2:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A3) @ X2) & (((ord_less @ A) @ X2) @ D2)) => (P @ X2)))) => (((ord_less_eq @ A) @ D2) @ C3)))))))))))))))))).
% 2.59/1.53 thf(fact_214_verit__comp__simplify1_I3_J, axiom, (! [B:$tType]: (((linorder @ B) => (! [B6:B,A6:B]: (((~ ((((ord_less_eq @ B) @ B6) @ A6))) = (((ord_less @ B) @ A6) @ B6)))))))).
% 2.59/1.53 thf(fact_215_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ T)))))))))))))).
% 2.59/1.53 thf(fact_216_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less_eq @ A) @ T) @ X5)))))))))))).
% 2.59/1.53 thf(fact_217_less__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F2))))))))))).
% 2.59/1.53 thf(fact_218_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B3:A]: (((A3 = B3) | ((~ ((((ord_less_eq @ A) @ A3) @ B3))) | (~ ((((ord_less_eq @ A) @ B3) @ A3))))))))))).
% 2.59/1.53 thf(fact_219_minf_I11_J, axiom, (! [C2:$tType,D3:$tType]: (((ord @ C2) => (! [F3:D3]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ X5) @ Z3) => (F3 = F3)))))))))))).
% 2.59/1.53 thf(fact_220_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less @ A) @ T) @ X5)))))))))))))).
% 2.59/1.53 thf(fact_221_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less @ A) @ X5) @ T)))))))))))).
% 2.59/1.53 thf(fact_222_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.59/1.53 thf(fact_223_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.59/1.53 thf(fact_224_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P4 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P @ X5) | (Q @ X5)) = ((P4 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.59/1.53 thf(fact_225_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P4 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P @ X5) & (Q @ X5)) = ((P4 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.59/1.53 thf(fact_226_pinf_I11_J, axiom, (! [C2:$tType,D3:$tType]: (((ord @ C2) => (! [F3:D3]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ Z3) @ X5) => (F3 = F3)))))))))))).
% 2.59/1.53 thf(fact_227_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less @ A) @ T) @ X5)))))))))))).
% 2.59/1.53 thf(fact_228_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less @ A) @ X5) @ T)))))))))))))).
% 2.59/1.53 thf(fact_229_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.59/1.53 thf(fact_230_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.59/1.53 thf(fact_231_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P4 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P @ X5) | (Q @ X5)) = ((P4 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.59/1.53 thf(fact_232_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P4 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P @ X5) & (Q @ X5)) = ((P4 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.59/1.53 thf(fact_233_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 2.59/1.53 thf(fact_234_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A3:A]: ((? [B5:A]: (((((ord_less @ A) @ A3) @ B5) | (((ord_less @ A) @ B5) @ A3)))))))))).
% 2.59/1.53 thf(fact_235_list__strict__desc__imp__list__desc, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_desc @ A) @ Xs) => ((list_desc @ A) @ Xs)))))))).
% 2.59/1.53 thf(fact_236_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X5)))))))))))))).
% 2.59/1.53 thf(fact_237_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less_eq @ A) @ X5) @ T)))))))))))).
% 2.59/1.53 thf(fact_238_i__append__i__take__eq2, axiom, (! [A:$tType,Xs:(list @ A),N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N) => ((((listIn1033672622i_take @ A) @ N) @ (((listIn521021761append @ A) @ Xs) @ F)) = (((append @ A) @ Xs) @ (((listIn1033672622i_take @ A) @ (((minus_minus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs))) @ F))))))).
% 2.59/1.53 thf(fact_239_nth__drop, axiom, (! [A:$tType,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))))))).
% 2.59/1.53 thf(fact_240_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B3:A,A3:A,C:A]: ((((((plus_plus @ A) @ B3) @ A3) = (((plus_plus @ A) @ C) @ A3)) = (B3 = C)))))))).
% 2.59/1.53 thf(fact_241_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,B3:A,C:A]: ((((((plus_plus @ A) @ A3) @ B3) = (((plus_plus @ A) @ A3) @ C)) = (B3 = C)))))))).
% 2.59/1.53 thf(fact_242_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A3:A,C:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C)) @ (((plus_plus @ A) @ B3) @ C)) = (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.59/1.53 thf(fact_243_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A3)) @ (((plus_plus @ A) @ C) @ B3)) = (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.59/1.53 thf(fact_244_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A3:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A3)) @ (((plus_plus @ A) @ C) @ B3)) = (((ord_less @ A) @ A3) @ B3)))))))).
% 2.59/1.53 thf(fact_245_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A3:A,C:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C)) @ (((plus_plus @ A) @ B3) @ C)) = (((ord_less @ A) @ A3) @ B3)))))))).
% 2.59/1.53 thf(fact_246_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B3)) @ B3) = A3))))))).
% 2.59/1.53 thf(fact_247_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B3:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A3) @ B3)) @ B3) = A3))))))).
% 2.59/1.53 thf(fact_248_add__diff__cancel__left, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [C:A,A3:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ C) @ A3)) @ (((plus_plus @ A) @ C) @ B3)) = (((minus_minus @ A) @ A3) @ B3)))))))).
% 2.59/1.53 thf(fact_249_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A3:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B3)) @ A3) = B3))))))).
% 2.59/1.53 thf(fact_250_add__diff__cancel__right, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A3:A,C:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ C)) @ (((plus_plus @ A) @ B3) @ C)) = (((minus_minus @ A) @ A3) @ B3)))))))).
% 2.59/1.53 thf(fact_251_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A3:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B3)) @ B3) = A3))))))).
% 2.59/1.53 thf(fact_252_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))))).
% 2.59/1.53 thf(fact_253_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))))).
% 2.59/1.53 thf(fact_254_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)))))).
% 2.59/1.53 thf(fact_255_drop__drop, axiom, (! [A:$tType,N:nat,M:nat,Xs:(list @ A)]: (((((drop @ A) @ N) @ (((drop @ A) @ M) @ Xs)) = (((drop @ A) @ (((plus_plus @ nat) @ N) @ M)) @ Xs))))).
% 2.59/1.53 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.59/1.53 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.59/1.53 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.59/1.53 thf(tcon_fun___Groups_Ominus, axiom, (! [A7:$tType,A8:$tType]: (((minus @ A8) => (minus @ (A7 > A8)))))).
% 2.59/1.53 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Groups_Ominus_4, axiom, (minus @ nat)).
% 2.59/1.53 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 2.59/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_5, axiom, (preorder @ $o)).
% 2.59/1.53 thf(tcon_HOL_Obool___Orderings_Olinorder_6, axiom, (linorder @ $o)).
% 2.59/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_7, axiom, (order @ $o)).
% 2.59/1.53 thf(tcon_HOL_Obool___Orderings_Oord_8, axiom, (ord @ $o)).
% 2.59/1.53 thf(tcon_HOL_Obool___Groups_Ominus_9, axiom, (minus @ $o)).
% 2.59/1.53 thf(tcon_List_Olist___Nat_Osize_10, axiom, (! [A7:$tType]: ((size @ (list @ A7))))).
% 2.59/1.53 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.59/1.53 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y2:A]: ((((((if @ A) @ $false) @ X3) @ Y2) = Y2)))).
% 2.59/1.53 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y2:A]: ((((((if @ A) @ $true) @ X3) @ Y2) = X3)))).
% 2.59/1.53 thf(conj_0, conjecture, (((((ord_less @ nat) @ n) @ ((size_size @ (list @ a)) @ xs)) => ((((listIn1569887318i_drop @ a) @ n) @ (((listIn521021761append @ a) @ xs) @ f)) = (((listIn521021761append @ a) @ (((drop @ a) @ n) @ xs)) @ f))) & ((~ ((((ord_less @ nat) @ n) @ ((size_size @ (list @ a)) @ xs)))) => ((((listIn1569887318i_drop @ a) @ n) @ (((listIn521021761append @ a) @ xs) @ f)) = (((listIn1569887318i_drop @ a) @ (((minus_minus @ nat) @ n) @ ((size_size @ (list @ a)) @ xs))) @ f))))).
% 2.59/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.rtTF0N8tiU/DTF2DTF11420.p
% 2.59/1.53 ------------------------
% 2.59/1.58 ---- Cleaned THF ---
% 2.59/1.58 thf(ty_t_List_Olist,type,
% 2.59/1.58 list: $tType > $tType ).
% 2.59/1.58
% 2.59/1.58 thf(ty_t_Nat_Onat,type,
% 2.59/1.58 nat: $tType ).
% 2.59/1.58
% 2.59/1.58 thf(ty_tf_a,type,
% 2.59/1.58 a: $tType ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_HOL_Otype,type,
% 2.59/1.58 type:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Nat_Osize,type,
% 2.59/1.58 size:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Groups_Ominus,type,
% 2.59/1.58 minus:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Oord,type,
% 2.59/1.58 ord:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Oorder,type,
% 2.59/1.58 order:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Groups_Ogroup__add,type,
% 2.59/1.58 group_add:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Ono__bot,type,
% 2.59/1.58 no_bot:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Ono__top,type,
% 2.59/1.58 no_top:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Olinorder,type,
% 2.59/1.58 linorder:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Opreorder,type,
% 2.59/1.58 preorder:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Owellorder,type,
% 2.59/1.58 wellorder:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Odense__order,type,
% 2.59/1.58 dense_order:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.59/1.58 dense_linorder:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.59/1.58 cancel_semigroup_add:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.59/1.58 ordered_ab_group_add:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.59/1.58 cancel146912293up_add:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.59/1.58 ordere236663937imp_le:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.59/1.58 condit1656338222tinuum:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.59/1.58 condit1037483654norder:
% 2.59/1.58 !>[A: $tType] : $o ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.59/1.58 minus_minus:
% 2.59/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.59/1.58 plus_plus:
% 2.59/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_If,type,
% 2.59/1.58 if:
% 2.59/1.58 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_List2_Olist__asc,type,
% 2.59/1.58 list_asc:
% 2.59/1.58 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_List2_Olist__desc,type,
% 2.59/1.58 list_desc:
% 2.59/1.58 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_List2_Olist__strict__asc,type,
% 2.59/1.58 list_strict_asc:
% 2.59/1.58 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_List2_Olist__strict__desc,type,
% 2.59/1.58 list_strict_desc:
% 2.59/1.58 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
% 2.59/1.58 listIn521021761append:
% 2.59/1.58 !>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__drop,type,
% 2.59/1.58 listIn1569887318i_drop:
% 2.59/1.58 !>[A: $tType] : ( nat > ( nat > A ) > nat > A ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__take,type,
% 2.59/1.58 listIn1033672622i_take:
% 2.59/1.58 !>[A: $tType] : ( nat > ( nat > A ) > ( list @ A ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_List_Oappend,type,
% 2.59/1.58 append:
% 2.59/1.58 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_List_Odrop,type,
% 2.59/1.58 drop:
% 2.59/1.58 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_List_Olast,type,
% 2.59/1.58 last:
% 2.59/1.58 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_List_Onth,type,
% 2.59/1.58 nth:
% 2.59/1.58 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.59/1.58 size_size:
% 2.59/1.58 !>[A: $tType] : ( A > nat ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.59/1.58 ord_less:
% 2.59/1.58 !>[A: $tType] : ( A > A > $o ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.59/1.58 ord_less_eq:
% 2.59/1.58 !>[A: $tType] : ( A > A > $o ) ).
% 2.59/1.58
% 2.59/1.58 thf(sy_v_f,type,
% 2.59/1.58 f: nat > a ).
% 2.59/1.58
% 2.59/1.58 thf(sy_v_n,type,
% 2.59/1.58 n: nat ).
% 2.59/1.58
% 2.59/1.58 thf(sy_v_xs,type,
% 2.59/1.58 xs: list @ a ).
% 2.59/1.58
% 2.59/1.58 thf(fact_0_same__i__append__eq,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,F: nat > A,G: nat > A] :
% 2.59/1.58 ( ( ( listIn521021761append @ A @ Xs @ F )
% 2.59/1.58 = ( listIn521021761append @ A @ Xs @ G ) )
% 2.59/1.58 = ( F = G ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_1_i__append__eq__i__append__conv,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
% 2.59/1.58 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.59/1.58 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.59/1.58 => ( ( ( listIn521021761append @ A @ Xs @ F )
% 2.59/1.58 = ( listIn521021761append @ A @ Ys @ G ) )
% 2.59/1.58 = ( ( Xs = Ys )
% 2.59/1.58 & ( F = G ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_2_length__drop,axiom,
% 2.59/1.58 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.59/1.58 ( ( size_size @ ( list @ A ) @ ( drop @ A @ N @ Xs ) )
% 2.59/1.58 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_3_minus__apply,axiom,
% 2.59/1.58 ! [B: $tType,A: $tType] :
% 2.59/1.58 ( ( minus @ B )
% 2.59/1.58 => ( ( minus_minus @ ( A > B ) )
% 2.59/1.58 = ( ^ [A2: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A2 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_4_diff__less__mono2,axiom,
% 2.59/1.58 ! [M: nat,N: nat,L: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ M @ N )
% 2.59/1.58 => ( ( ord_less @ nat @ M @ L )
% 2.59/1.58 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_5_less__imp__diff__less,axiom,
% 2.59/1.58 ! [J: nat,K: nat,N: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ J @ K )
% 2.59/1.58 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_6_less__diff__imp__less,axiom,
% 2.59/1.58 ! [I: nat,J: nat,M: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
% 2.59/1.58 => ( ord_less @ nat @ I @ J ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_7_nat__diff__left__cancel__eq1,axiom,
% 2.59/1.58 ! [K: nat,M: nat,N: nat] :
% 2.59/1.58 ( ( ( minus_minus @ nat @ K @ M )
% 2.59/1.58 = ( minus_minus @ nat @ K @ N ) )
% 2.59/1.58 => ( ( ord_less @ nat @ M @ K )
% 2.59/1.58 => ( M = N ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_8_nat__diff__left__cancel__eq2,axiom,
% 2.59/1.58 ! [K: nat,M: nat,N: nat] :
% 2.59/1.58 ( ( ( minus_minus @ nat @ K @ M )
% 2.59/1.58 = ( minus_minus @ nat @ K @ N ) )
% 2.59/1.58 => ( ( ord_less @ nat @ N @ K )
% 2.59/1.58 => ( M = N ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_9_nat__diff__left__cancel__less,axiom,
% 2.59/1.58 ! [K: nat,M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
% 2.59/1.58 => ( ord_less @ nat @ N @ M ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_10_nat__diff__right__cancel__eq1,axiom,
% 2.59/1.58 ! [M: nat,K: nat,N: nat] :
% 2.59/1.58 ( ( ( minus_minus @ nat @ M @ K )
% 2.59/1.58 = ( minus_minus @ nat @ N @ K ) )
% 2.59/1.58 => ( ( ord_less @ nat @ K @ M )
% 2.59/1.58 => ( M = N ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_11_nat__diff__right__cancel__eq2,axiom,
% 2.59/1.58 ! [M: nat,K: nat,N: nat] :
% 2.59/1.58 ( ( ( minus_minus @ nat @ M @ K )
% 2.59/1.58 = ( minus_minus @ nat @ N @ K ) )
% 2.59/1.58 => ( ( ord_less @ nat @ K @ N )
% 2.59/1.58 => ( M = N ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_12_nat__diff__right__cancel__less,axiom,
% 2.59/1.58 ! [N: nat,K: nat,M: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) )
% 2.59/1.58 => ( ord_less @ nat @ N @ M ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_13_length__induct,axiom,
% 2.59/1.58 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.59/1.58 ( ! [Xs2: list @ A] :
% 2.59/1.58 ( ! [Ys2: list @ A] :
% 2.59/1.58 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.59/1.58 => ( P @ Ys2 ) )
% 2.59/1.58 => ( P @ Xs2 ) )
% 2.59/1.58 => ( P @ Xs ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_14_measure__induct__rule,axiom,
% 2.59/1.58 ! [B: $tType,A: $tType] :
% 2.59/1.58 ( ( wellorder @ B )
% 2.59/1.58 => ! [F: A > B,P: A > $o,A3: A] :
% 2.59/1.58 ( ! [X2: A] :
% 2.59/1.58 ( ! [Y: A] :
% 2.59/1.58 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 2.59/1.58 => ( P @ Y ) )
% 2.59/1.58 => ( P @ X2 ) )
% 2.59/1.58 => ( P @ A3 ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_15_measure__induct,axiom,
% 2.59/1.58 ! [B: $tType,A: $tType] :
% 2.59/1.58 ( ( wellorder @ B )
% 2.59/1.58 => ! [F: A > B,P: A > $o,A3: A] :
% 2.59/1.58 ( ! [X2: A] :
% 2.59/1.58 ( ! [Y: A] :
% 2.59/1.58 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 2.59/1.58 => ( P @ Y ) )
% 2.59/1.58 => ( P @ X2 ) )
% 2.59/1.58 => ( P @ A3 ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_16_fun__diff__def,axiom,
% 2.59/1.58 ! [B: $tType,A: $tType] :
% 2.59/1.58 ( ( minus @ B )
% 2.59/1.58 => ( ( minus_minus @ ( A > B ) )
% 2.59/1.58 = ( ^ [A2: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A2 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_17_infinite__descent__measure,axiom,
% 2.59/1.58 ! [A: $tType,P: A > $o,V: A > nat,X3: A] :
% 2.59/1.58 ( ! [X2: A] :
% 2.59/1.58 ( ~ ( P @ X2 )
% 2.59/1.58 => ? [Y: A] :
% 2.59/1.58 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 2.59/1.58 & ~ ( P @ Y ) ) )
% 2.59/1.58 => ( P @ X3 ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_18_linorder__neqE__nat,axiom,
% 2.59/1.58 ! [X3: nat,Y2: nat] :
% 2.59/1.58 ( ( X3 != Y2 )
% 2.59/1.58 => ( ~ ( ord_less @ nat @ X3 @ Y2 )
% 2.59/1.58 => ( ord_less @ nat @ Y2 @ X3 ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_19_infinite__descent,axiom,
% 2.59/1.58 ! [P: nat > $o,N: nat] :
% 2.59/1.58 ( ! [N2: nat] :
% 2.59/1.58 ( ~ ( P @ N2 )
% 2.59/1.58 => ? [M2: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ M2 @ N2 )
% 2.59/1.58 & ~ ( P @ M2 ) ) )
% 2.59/1.58 => ( P @ N ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_20_nat__less__induct,axiom,
% 2.59/1.58 ! [P: nat > $o,N: nat] :
% 2.59/1.58 ( ! [N2: nat] :
% 2.59/1.58 ( ! [M2: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ M2 @ N2 )
% 2.59/1.58 => ( P @ M2 ) )
% 2.59/1.58 => ( P @ N2 ) )
% 2.59/1.58 => ( P @ N ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_21_less__irrefl__nat,axiom,
% 2.59/1.58 ! [N: nat] :
% 2.59/1.58 ~ ( ord_less @ nat @ N @ N ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_22_less__not__refl3,axiom,
% 2.59/1.58 ! [S: nat,T: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ S @ T )
% 2.59/1.58 => ( S != T ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_23_less__not__refl2,axiom,
% 2.59/1.58 ! [N: nat,M: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ N @ M )
% 2.59/1.58 => ( M != N ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_24_less__not__refl,axiom,
% 2.59/1.58 ! [N: nat] :
% 2.59/1.58 ~ ( ord_less @ nat @ N @ N ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_25_nat__neq__iff,axiom,
% 2.59/1.58 ! [M: nat,N: nat] :
% 2.59/1.58 ( ( M != N )
% 2.59/1.58 = ( ( ord_less @ nat @ M @ N )
% 2.59/1.58 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_26_size__neq__size__imp__neq,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( size @ A )
% 2.59/1.58 => ! [X3: A,Y2: A] :
% 2.59/1.58 ( ( ( size_size @ A @ X3 )
% 2.59/1.58 != ( size_size @ A @ Y2 ) )
% 2.59/1.58 => ( X3 != Y2 ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_27_neq__if__length__neq,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.59/1.58 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.59/1.58 != ( size_size @ ( list @ A ) @ Ys ) )
% 2.59/1.58 => ( Xs != Ys ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_28_Ex__list__of__length,axiom,
% 2.59/1.58 ! [A: $tType,N: nat] :
% 2.59/1.58 ? [Xs2: list @ A] :
% 2.59/1.58 ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.59/1.58 = N ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_29_eq__imp__diff__eq,axiom,
% 2.59/1.58 ! [M: nat,N: nat,K: nat] :
% 2.59/1.58 ( ( M = N )
% 2.59/1.58 => ( ( minus_minus @ nat @ M @ K )
% 2.59/1.58 = ( minus_minus @ nat @ N @ K ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_30_diff__commute,axiom,
% 2.59/1.58 ! [I: nat,J: nat,K: nat] :
% 2.59/1.58 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.59/1.58 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_31_diff__strict__right__mono,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( ordered_ab_group_add @ A )
% 2.59/1.58 => ! [A3: A,B3: A,C: A] :
% 2.59/1.58 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.58 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_32_diff__strict__left__mono,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( ordered_ab_group_add @ A )
% 2.59/1.58 => ! [B3: A,A3: A,C: A] :
% 2.59/1.58 ( ( ord_less @ A @ B3 @ A3 )
% 2.59/1.58 => ( ord_less @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_33_diff__eq__diff__less,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( ordered_ab_group_add @ A )
% 2.59/1.58 => ! [A3: A,B3: A,C: A,D: A] :
% 2.59/1.58 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.59/1.58 = ( minus_minus @ A @ C @ D ) )
% 2.59/1.58 => ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.58 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_34_diff__strict__mono,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( ordered_ab_group_add @ A )
% 2.59/1.58 => ! [A3: A,B3: A,D: A,C: A] :
% 2.59/1.58 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.58 => ( ( ord_less @ A @ D @ C )
% 2.59/1.58 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_35_i__append__i__drop__eq1,axiom,
% 2.59/1.58 ! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.58 => ( ( listIn1569887318i_drop @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
% 2.59/1.58 = ( listIn521021761append @ A @ ( drop @ A @ N @ Xs ) @ F ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_36_i__append__i__drop__eq2,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 2.59/1.58 => ( ( listIn1569887318i_drop @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
% 2.59/1.58 = ( listIn1569887318i_drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ F ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_37_last__drop,axiom,
% 2.59/1.58 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.59/1.58 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.58 => ( ( last @ A @ ( drop @ A @ N @ Xs ) )
% 2.59/1.58 = ( last @ A @ Xs ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_38_i__append__nth2,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 2.59/1.58 => ( ( listIn521021761append @ A @ Xs @ F @ N )
% 2.59/1.58 = ( F @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_39_i__append__nth,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( listIn521021761append @ A )
% 2.59/1.58 = ( ^ [Xs3: list @ A,F2: nat > A,N3: nat] : ( if @ A @ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ ( nth @ A @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_40_i__append__def,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( listIn521021761append @ A )
% 2.59/1.58 = ( ^ [Xs3: list @ A,F2: nat > A,N3: nat] : ( if @ A @ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ ( nth @ A @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_41_drop__append,axiom,
% 2.59/1.58 ! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
% 2.59/1.58 ( ( drop @ A @ N @ ( append @ A @ Xs @ Ys ) )
% 2.59/1.58 = ( append @ A @ ( drop @ A @ N @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_42_append_Oassoc,axiom,
% 2.59/1.58 ! [A: $tType,A3: list @ A,B3: list @ A,C: list @ A] :
% 2.59/1.58 ( ( append @ A @ ( append @ A @ A3 @ B3 ) @ C )
% 2.59/1.58 = ( append @ A @ A3 @ ( append @ A @ B3 @ C ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_43_append__assoc,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 2.59/1.58 ( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
% 2.59/1.58 = ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_44_append__same__eq,axiom,
% 2.59/1.58 ! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
% 2.59/1.58 ( ( ( append @ A @ Ys @ Xs )
% 2.59/1.58 = ( append @ A @ Zs @ Xs ) )
% 2.59/1.58 = ( Ys = Zs ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_45_ext,axiom,
% 2.59/1.58 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.59/1.58 ( ! [X2: A] :
% 2.59/1.58 ( ( F @ X2 )
% 2.59/1.58 = ( G @ X2 ) )
% 2.59/1.58 => ( F = G ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_46_same__append__eq,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 2.59/1.58 ( ( ( append @ A @ Xs @ Ys )
% 2.59/1.58 = ( append @ A @ Xs @ Zs ) )
% 2.59/1.58 = ( Ys = Zs ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_47_diff__diff__cancel,axiom,
% 2.59/1.58 ! [I: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ I @ N )
% 2.59/1.58 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 2.59/1.58 = I ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_48_append__eq__append__conv,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
% 2.59/1.58 ( ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.59/1.58 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.59/1.58 | ( ( size_size @ ( list @ A ) @ Us )
% 2.59/1.58 = ( size_size @ ( list @ A ) @ Vs ) ) )
% 2.59/1.58 => ( ( ( append @ A @ Xs @ Us )
% 2.59/1.58 = ( append @ A @ Ys @ Vs ) )
% 2.59/1.58 = ( ( Xs = Ys )
% 2.59/1.58 & ( Us = Vs ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_49_i__append__assoc,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A] :
% 2.59/1.58 ( ( listIn521021761append @ A @ Xs @ ( listIn521021761append @ A @ Ys @ F ) )
% 2.59/1.58 = ( listIn521021761append @ A @ ( append @ A @ Xs @ Ys ) @ F ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_50_i__append__nth1,axiom,
% 2.59/1.58 ! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
% 2.59/1.58 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.58 => ( ( listIn521021761append @ A @ Xs @ F @ N )
% 2.59/1.58 = ( nth @ A @ Xs @ N ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_51_le__refl,axiom,
% 2.59/1.58 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_52_le__trans,axiom,
% 2.59/1.58 ! [I: nat,J: nat,K: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ J @ K )
% 2.59/1.58 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_53_eq__imp__le,axiom,
% 2.59/1.58 ! [M: nat,N: nat] :
% 2.59/1.58 ( ( M = N )
% 2.59/1.58 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_54_le__antisym,axiom,
% 2.59/1.58 ! [M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ N @ M )
% 2.59/1.58 => ( M = N ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_55_nat__le__linear,axiom,
% 2.59/1.58 ! [M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.58 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_56_append__eq__appendI,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
% 2.59/1.58 ( ( ( append @ A @ Xs @ Xs1 )
% 2.59/1.58 = Zs )
% 2.59/1.58 => ( ( Ys
% 2.59/1.58 = ( append @ A @ Xs1 @ Us ) )
% 2.59/1.58 => ( ( append @ A @ Xs @ Ys )
% 2.59/1.58 = ( append @ A @ Zs @ Us ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_57_Nat_Oex__has__greatest__nat,axiom,
% 2.59/1.58 ! [P: nat > $o,K: nat,B3: nat] :
% 2.59/1.58 ( ( P @ K )
% 2.59/1.58 => ( ! [Y3: nat] :
% 2.59/1.58 ( ( P @ Y3 )
% 2.59/1.58 => ( ord_less_eq @ nat @ Y3 @ B3 ) )
% 2.59/1.58 => ? [X2: nat] :
% 2.59/1.58 ( ( P @ X2 )
% 2.59/1.58 & ! [Y: nat] :
% 2.59/1.58 ( ( P @ Y )
% 2.59/1.58 => ( ord_less_eq @ nat @ Y @ X2 ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_58_append__eq__append__conv2,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
% 2.59/1.58 ( ( ( append @ A @ Xs @ Ys )
% 2.59/1.58 = ( append @ A @ Zs @ Ts ) )
% 2.59/1.58 = ( ? [Us2: list @ A] :
% 2.59/1.58 ( ( ( Xs
% 2.59/1.58 = ( append @ A @ Zs @ Us2 ) )
% 2.59/1.58 & ( ( append @ A @ Us2 @ Ys )
% 2.59/1.58 = Ts ) )
% 2.59/1.58 | ( ( ( append @ A @ Xs @ Us2 )
% 2.59/1.58 = Zs )
% 2.59/1.58 & ( Ys
% 2.59/1.58 = ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_59_diff__mono,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( ordered_ab_group_add @ A )
% 2.59/1.58 => ! [A3: A,B3: A,D: A,C: A] :
% 2.59/1.58 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.58 => ( ( ord_less_eq @ A @ D @ C )
% 2.59/1.58 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_60_diff__left__mono,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( ordered_ab_group_add @ A )
% 2.59/1.58 => ! [B3: A,A3: A,C: A] :
% 2.59/1.58 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.59/1.58 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_61_diff__right__mono,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( ordered_ab_group_add @ A )
% 2.59/1.58 => ! [A3: A,B3: A,C: A] :
% 2.59/1.58 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.58 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_62_diff__eq__diff__less__eq,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( ordered_ab_group_add @ A )
% 2.59/1.58 => ! [A3: A,B3: A,C: A,D: A] :
% 2.59/1.58 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.59/1.58 = ( minus_minus @ A @ C @ D ) )
% 2.59/1.58 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.58 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_63_i__append__eq__i__append__conv2__aux,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,F: nat > A,Ys: list @ A,G: nat > A] :
% 2.59/1.58 ( ( ( listIn521021761append @ A @ Xs @ F )
% 2.59/1.58 = ( listIn521021761append @ A @ Ys @ G ) )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
% 2.59/1.58 => ? [Zs2: list @ A] :
% 2.59/1.58 ( ( ( append @ A @ Xs @ Zs2 )
% 2.59/1.58 = Ys )
% 2.59/1.58 & ( F
% 2.59/1.58 = ( listIn521021761append @ A @ Zs2 @ G ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_64_ge__less__neq__conv,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( linorder @ A )
% 2.59/1.58 => ( ( ord_less_eq @ A )
% 2.59/1.58 = ( ^ [A4: A,N3: A] :
% 2.59/1.58 ! [X: A] :
% 2.59/1.58 ( ( ord_less @ A @ X @ A4 )
% 2.59/1.58 => ( N3 != X ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_65_less__ge__neq__conv,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( linorder @ A )
% 2.59/1.58 => ( ( ord_less @ A )
% 2.59/1.58 = ( ^ [N3: A,A4: A] :
% 2.59/1.58 ! [X: A] :
% 2.59/1.58 ( ( ord_less_eq @ A @ A4 @ X )
% 2.59/1.58 => ( N3 != X ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_66_greater__le__neq__conv,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( linorder @ A )
% 2.59/1.58 => ( ( ord_less @ A )
% 2.59/1.58 = ( ^ [A4: A,N3: A] :
% 2.59/1.58 ! [X: A] :
% 2.59/1.58 ( ( ord_less_eq @ A @ X @ A4 )
% 2.59/1.58 => ( N3 != X ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_67_le__greater__neq__conv,axiom,
% 2.59/1.58 ! [A: $tType] :
% 2.59/1.58 ( ( linorder @ A )
% 2.59/1.58 => ( ( ord_less_eq @ A )
% 2.59/1.58 = ( ^ [N3: A,A4: A] :
% 2.59/1.58 ! [X: A] :
% 2.59/1.58 ( ( ord_less @ A @ A4 @ X )
% 2.59/1.58 => ( N3 != X ) ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_68_nat__less__le,axiom,
% 2.59/1.58 ( ( ord_less @ nat )
% 2.59/1.58 = ( ^ [M3: nat,N3: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 2.59/1.58 & ( M3 != N3 ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_69_less__imp__le__nat,axiom,
% 2.59/1.58 ! [M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ M @ N )
% 2.59/1.58 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_70_le__eq__less__or__eq,axiom,
% 2.59/1.58 ( ( ord_less_eq @ nat )
% 2.59/1.58 = ( ^ [M3: nat,N3: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ M3 @ N3 )
% 2.59/1.58 | ( M3 = N3 ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_71_less__or__eq__imp__le,axiom,
% 2.59/1.58 ! [M: nat,N: nat] :
% 2.59/1.58 ( ( ( ord_less @ nat @ M @ N )
% 2.59/1.58 | ( M = N ) )
% 2.59/1.58 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_72_le__neq__implies__less,axiom,
% 2.59/1.58 ! [M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.58 => ( ( M != N )
% 2.59/1.58 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_73_less__mono__imp__le__mono,axiom,
% 2.59/1.58 ! [F: nat > nat,I: nat,J: nat] :
% 2.59/1.58 ( ! [I2: nat,J2: nat] :
% 2.59/1.58 ( ( ord_less @ nat @ I2 @ J2 )
% 2.59/1.58 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.58 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_74_eq__diff__iff,axiom,
% 2.59/1.58 ! [K: nat,M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ K @ M )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.58 => ( ( ( minus_minus @ nat @ M @ K )
% 2.59/1.58 = ( minus_minus @ nat @ N @ K ) )
% 2.59/1.58 = ( M = N ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_75_le__diff__iff,axiom,
% 2.59/1.58 ! [K: nat,M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ K @ M )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.59/1.58 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_76_Nat_Odiff__diff__eq,axiom,
% 2.59/1.58 ! [K: nat,M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ K @ M )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.58 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.59/1.58 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_77_diff__le__mono,axiom,
% 2.59/1.58 ! [M: nat,N: nat,L: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.58 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_78_diff__le__self,axiom,
% 2.59/1.58 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_79_le__diff__iff_H,axiom,
% 2.59/1.58 ! [A3: nat,C: nat,B3: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ A3 @ C )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ B3 @ C )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A3 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 2.59/1.58 = ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_80_diff__le__mono2,axiom,
% 2.59/1.58 ! [M: nat,N: nat,L: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.58 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_81_le__diff__swap,axiom,
% 2.59/1.58 ! [I: nat,K: nat,J: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ I @ K )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ J @ K )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ J ) @ I )
% 2.59/1.58 = ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ I ) @ J ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_82_le__diff__imp__le,axiom,
% 2.59/1.58 ! [I: nat,J: nat,M: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
% 2.59/1.58 => ( ord_less_eq @ nat @ I @ J ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_83_le__imp__diff__le,axiom,
% 2.59/1.58 ! [J: nat,K: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ J @ K )
% 2.59/1.58 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_84_eq__diff__left__iff,axiom,
% 2.59/1.58 ! [M: nat,K: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ M @ K )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ N @ K )
% 2.59/1.58 => ( ( ( minus_minus @ nat @ K @ M )
% 2.59/1.58 = ( minus_minus @ nat @ K @ N ) )
% 2.59/1.58 = ( M = N ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_85_le__diff__le__imp__le,axiom,
% 2.59/1.58 ! [I: nat,J: nat,M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ N @ M )
% 2.59/1.58 => ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ N ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_86_nat__diff__left__cancel__le2,axiom,
% 2.59/1.58 ! [K: nat,M: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ N @ K )
% 2.59/1.58 => ( ord_less_eq @ nat @ N @ M ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_87_nat__diff__right__cancel__le2,axiom,
% 2.59/1.58 ! [M: nat,K: nat,N: nat] :
% 2.59/1.58 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.59/1.58 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.58 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_88_i__append__eq__i__appendI,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,Xs4: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
% 2.59/1.58 ( ( ( append @ A @ Xs @ Xs4 )
% 2.59/1.58 = Ys )
% 2.59/1.58 => ( ( F
% 2.59/1.58 = ( listIn521021761append @ A @ Xs4 @ G ) )
% 2.59/1.58 => ( ( listIn521021761append @ A @ Xs @ F )
% 2.59/1.58 = ( listIn521021761append @ A @ Ys @ G ) ) ) ) ).
% 2.59/1.58
% 2.59/1.58 thf(fact_89_i__append__eq__i__append__conv2,axiom,
% 2.59/1.58 ! [A: $tType,Xs: list @ A,F: nat > A,Ys: list @ A,G: nat > A] :
% 2.59/1.58 ( ( ( listIn521021761append @ A @ Xs @ F )
% 2.59/1.58 = ( listIn521021761append @ A @ Ys @ G ) )
% 2.59/1.58 = ( ? [Zs3: list @ A] :
% 2.59/1.58 ( ( ( Xs
% 2.59/1.58 = ( append @ A @ Ys @ Zs3 ) )
% 2.59/1.58 & ( ( listIn521021761append @ A @ Zs3 @ F )
% 2.59/1.58 = G ) )
% 2.59/1.58 | ( ( ( append @ A @ Xs @ Zs3 )
% 2.59/1.59 = Ys )
% 2.59/1.59 & ( F
% 2.59/1.59 = ( listIn521021761append @ A @ Zs3 @ G ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_90_nth__append,axiom,
% 2.59/1.59 ! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
% 2.59/1.59 ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.59 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
% 2.59/1.59 = ( nth @ A @ Xs @ N ) ) )
% 2.59/1.59 & ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.59 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
% 2.59/1.59 = ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_91_nth__equalityI,axiom,
% 2.59/1.59 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.59/1.59 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.59/1.59 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.59/1.59 => ( ! [I2: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.59 => ( ( nth @ A @ Xs @ I2 )
% 2.59/1.59 = ( nth @ A @ Ys @ I2 ) ) )
% 2.59/1.59 => ( Xs = Ys ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_92_Skolem__list__nth,axiom,
% 2.59/1.59 ! [A: $tType,K: nat,P: nat > A > $o] :
% 2.59/1.59 ( ( ! [I3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I3 @ K )
% 2.59/1.59 => ? [X4: A] : ( P @ I3 @ X4 ) ) )
% 2.59/1.59 = ( ? [Xs3: list @ A] :
% 2.59/1.59 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.59/1.59 = K )
% 2.59/1.59 & ! [I3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I3 @ K )
% 2.59/1.59 => ( P @ I3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_93_list__eq__iff__nth__eq,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ^ [Y4: list @ A,Z: list @ A] : ( Y4 = Z ) )
% 2.59/1.59 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 2.59/1.59 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.59/1.59 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 2.59/1.59 & ! [I3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.59/1.59 => ( ( nth @ A @ Xs3 @ I3 )
% 2.59/1.59 = ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_94_less__diff__iff,axiom,
% 2.59/1.59 ! [K: nat,M: nat,N: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ K @ M )
% 2.59/1.59 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.59 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.59/1.59 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_95_diff__less__mono,axiom,
% 2.59/1.59 ! [A3: nat,B3: nat,C: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less_eq @ nat @ C @ A3 )
% 2.59/1.59 => ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_96_less__diff__le__imp__less,axiom,
% 2.59/1.59 ! [I: nat,J: nat,M: nat,N: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
% 2.59/1.59 => ( ( ord_less_eq @ nat @ N @ M )
% 2.59/1.59 => ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ N ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_97_nat__diff__left__cancel__le1,axiom,
% 2.59/1.59 ! [K: nat,M: nat,N: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
% 2.59/1.59 => ( ( ord_less @ nat @ M @ K )
% 2.59/1.59 => ( ord_less_eq @ nat @ N @ M ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_98_nat__diff__right__cancel__le1,axiom,
% 2.59/1.59 ! [M: nat,K: nat,N: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.59/1.59 => ( ( ord_less @ nat @ K @ M )
% 2.59/1.59 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_99_i__drop__nth__sub,axiom,
% 2.59/1.59 ! [A: $tType,N: nat,X3: nat,S: nat > A] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ N @ X3 )
% 2.59/1.59 => ( ( listIn1569887318i_drop @ A @ N @ S @ ( minus_minus @ nat @ X3 @ N ) )
% 2.59/1.59 = ( S @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_100_diff__eq__diff__eq,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( group_add @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A,D: A] :
% 2.59/1.59 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.59/1.59 = ( minus_minus @ A @ C @ D ) )
% 2.59/1.59 => ( ( A3 = B3 )
% 2.59/1.59 = ( C = D ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_101_diff__right__commute,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( cancel146912293up_add @ A )
% 2.59/1.59 => ! [A3: A,C: A,B3: A] :
% 2.59/1.59 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B3 )
% 2.59/1.59 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_102_nth__append2,axiom,
% 2.59/1.59 ! [A: $tType,Xs: list @ A,N: nat,Ys: list @ A] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 2.59/1.59 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
% 2.59/1.59 = ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_103_nth__append1,axiom,
% 2.59/1.59 ! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
% 2.59/1.59 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.59 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
% 2.59/1.59 = ( nth @ A @ Xs @ N ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_104_order__refl,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_105_list__desc__trans__le,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ( ( list_desc @ A )
% 2.59/1.59 = ( ^ [Xs3: list @ A] :
% 2.59/1.59 ! [J3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.59/1.59 => ! [I3: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ I3 @ J3 )
% 2.59/1.59 => ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_106_nat__descend__induct,axiom,
% 2.59/1.59 ! [N: nat,P: nat > $o,M: nat] :
% 2.59/1.59 ( ! [K2: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ N @ K2 )
% 2.59/1.59 => ( P @ K2 ) )
% 2.59/1.59 => ( ! [K2: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ K2 @ N )
% 2.59/1.59 => ( ! [I4: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ K2 @ I4 )
% 2.59/1.59 => ( P @ I4 ) )
% 2.59/1.59 => ( P @ K2 ) ) )
% 2.59/1.59 => ( P @ M ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_107_Lattices__Big_Oex__has__greatest__nat,axiom,
% 2.59/1.59 ! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
% 2.59/1.59 ( ( P @ K )
% 2.59/1.59 => ( ! [Y3: A] :
% 2.59/1.59 ( ( P @ Y3 )
% 2.59/1.59 => ( ord_less @ nat @ ( F @ Y3 ) @ B3 ) )
% 2.59/1.59 => ? [X2: A] :
% 2.59/1.59 ( ( P @ X2 )
% 2.59/1.59 & ! [Y: A] :
% 2.59/1.59 ( ( P @ Y )
% 2.59/1.59 => ( ord_less_eq @ nat @ ( F @ Y ) @ ( F @ X2 ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_108_list__asc__trans__le,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ( ( list_asc @ A )
% 2.59/1.59 = ( ^ [Xs3: list @ A] :
% 2.59/1.59 ! [J3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.59/1.59 => ! [I3: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ I3 @ J3 )
% 2.59/1.59 => ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_109_list__strict__asc__trans__le,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [Xs: list @ A] :
% 2.59/1.59 ( ( list_strict_asc @ A @ Xs )
% 2.59/1.59 => ! [J4: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.59 => ! [I4: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ I4 @ J4 )
% 2.59/1.59 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_110_list__strict__asc__imp__list__asc,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [Xs: list @ A] :
% 2.59/1.59 ( ( list_strict_asc @ A @ Xs )
% 2.59/1.59 => ( list_asc @ A @ Xs ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_111_le__funD,axiom,
% 2.59/1.59 ! [B: $tType,A: $tType] :
% 2.59/1.59 ( ( ord @ B )
% 2.59/1.59 => ! [F: A > B,G: A > B,X3: A] :
% 2.59/1.59 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.59/1.59 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_112_le__funE,axiom,
% 2.59/1.59 ! [B: $tType,A: $tType] :
% 2.59/1.59 ( ( ord @ B )
% 2.59/1.59 => ! [F: A > B,G: A > B,X3: A] :
% 2.59/1.59 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.59/1.59 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_113_le__funI,axiom,
% 2.59/1.59 ! [B: $tType,A: $tType] :
% 2.59/1.59 ( ( ord @ B )
% 2.59/1.59 => ! [F: A > B,G: A > B] :
% 2.59/1.59 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 2.59/1.59 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_114_le__fun__def,axiom,
% 2.59/1.59 ! [B: $tType,A: $tType] :
% 2.59/1.59 ( ( ord @ B )
% 2.59/1.59 => ( ( ord_less_eq @ ( A > B ) )
% 2.59/1.59 = ( ^ [F2: A > B,G2: A > B] :
% 2.59/1.59 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_115_order__subst1,axiom,
% 2.59/1.59 ! [A: $tType,B: $tType] :
% 2.59/1.59 ( ( ( order @ B )
% 2.59/1.59 & ( order @ A ) )
% 2.59/1.59 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.59/1.59 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.59/1.59 => ( ! [X2: B,Y3: B] :
% 2.59/1.59 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_116_order__subst2,axiom,
% 2.59/1.59 ! [A: $tType,C2: $tType] :
% 2.59/1.59 ( ( ( order @ C2 )
% 2.59/1.59 & ( order @ A ) )
% 2.59/1.59 => ! [A3: A,B3: A,F: A > C2,C: C2] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 2.59/1.59 => ( ! [X2: A,Y3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less_eq @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_117_ord__eq__le__subst,axiom,
% 2.59/1.59 ! [A: $tType,B: $tType] :
% 2.59/1.59 ( ( ( ord @ B )
% 2.59/1.59 & ( ord @ A ) )
% 2.59/1.59 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.59/1.59 ( ( A3
% 2.59/1.59 = ( F @ B3 ) )
% 2.59/1.59 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.59/1.59 => ( ! [X2: B,Y3: B] :
% 2.59/1.59 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_118_ord__le__eq__subst,axiom,
% 2.59/1.59 ! [A: $tType,B: $tType] :
% 2.59/1.59 ( ( ( ord @ B )
% 2.59/1.59 & ( ord @ A ) )
% 2.59/1.59 => ! [A3: A,B3: A,F: A > B,C: B] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ( F @ B3 )
% 2.59/1.59 = C )
% 2.59/1.59 => ( ! [X2: A,Y3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_119_eq__iff,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.59/1.59 = ( ^ [X: A,Y5: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X @ Y5 )
% 2.59/1.59 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_120_antisym,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.59/1.59 => ( X3 = Y2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_121_linear,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 | ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_122_eq__refl,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( X3 = Y2 )
% 2.59/1.59 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_123_le__cases,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ~ ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_124_order_Otrans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.59/1.59 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_125_le__cases3,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A,Z2: A] :
% 2.59/1.59 ( ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.59/1.59 => ( ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.59/1.59 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.59/1.59 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 2.59/1.59 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.59/1.59 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.59/1.59 => ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
% 2.59/1.59 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.59/1.59 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 2.59/1.59 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 2.59/1.59 => ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_126_antisym__conv,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [Y2: A,X3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 = ( X3 = Y2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_127_order__class_Oorder_Oeq__iff,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.59/1.59 = ( ^ [A4: A,B4: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.59/1.59 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_128_ord__eq__le__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ord @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( A3 = B3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.59/1.59 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_129_ord__le__eq__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ord @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( B3 = C )
% 2.59/1.59 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_130_order__class_Oorder_Oantisym,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.59/1.59 => ( A3 = B3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_131_order__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A,Z2: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.59/1.59 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_132_dual__order_Orefl,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_133_linorder__wlog,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [P: A > A > $o,A3: A,B3: A] :
% 2.59/1.59 ( ! [A5: A,B5: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.59/1.59 => ( P @ A5 @ B5 ) )
% 2.59/1.59 => ( ! [A5: A,B5: A] :
% 2.59/1.59 ( ( P @ B5 @ A5 )
% 2.59/1.59 => ( P @ A5 @ B5 ) )
% 2.59/1.59 => ( P @ A3 @ B3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_134_dual__order_Otrans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [B3: A,A3: A,C: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.59/1.59 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_135_dual__order_Oeq__iff,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.59/1.59 = ( ^ [A4: A,B4: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.59/1.59 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_136_dual__order_Oantisym,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [B3: A,A3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( A3 = B3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_137_dual__order_Ostrict__implies__not__eq,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [B3: A,A3: A] :
% 2.59/1.59 ( ( ord_less @ A @ B3 @ A3 )
% 2.59/1.59 => ( A3 != B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_138_order_Ostrict__implies__not__eq,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( A3 != B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_139_not__less__iff__gr__or__eq,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.59/1.59 = ( ( ord_less @ A @ Y2 @ X3 )
% 2.59/1.59 | ( X3 = Y2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_140_dual__order_Ostrict__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [B3: A,A3: A,C: A] :
% 2.59/1.59 ( ( ord_less @ A @ B3 @ A3 )
% 2.59/1.59 => ( ( ord_less @ A @ C @ B3 )
% 2.59/1.59 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_141_linorder__less__wlog,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [P: A > A > $o,A3: A,B3: A] :
% 2.59/1.59 ( ! [A5: A,B5: A] :
% 2.59/1.59 ( ( ord_less @ A @ A5 @ B5 )
% 2.59/1.59 => ( P @ A5 @ B5 ) )
% 2.59/1.59 => ( ! [A5: A] : ( P @ A5 @ A5 )
% 2.59/1.59 => ( ! [A5: A,B5: A] :
% 2.59/1.59 ( ( P @ B5 @ A5 )
% 2.59/1.59 => ( P @ A5 @ B5 ) )
% 2.59/1.59 => ( P @ A3 @ B3 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_142_exists__least__iff,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( wellorder @ A )
% 2.59/1.59 => ( ( ^ [P2: A > $o] :
% 2.59/1.59 ? [X4: A] : ( P2 @ X4 ) )
% 2.59/1.59 = ( ^ [P3: A > $o] :
% 2.59/1.59 ? [N3: A] :
% 2.59/1.59 ( ( P3 @ N3 )
% 2.59/1.59 & ! [M3: A] :
% 2.59/1.59 ( ( ord_less @ A @ M3 @ N3 )
% 2.59/1.59 => ~ ( P3 @ M3 ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_143_less__imp__not__less,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_144_order_Ostrict__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less @ A @ B3 @ C )
% 2.59/1.59 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_145_dual__order_Oirrefl,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A] :
% 2.59/1.59 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_146_linorder__cases,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( X3 != Y2 )
% 2.59/1.59 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_147_less__imp__triv,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A,P: $o] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ord_less @ A @ Y2 @ X3 )
% 2.59/1.59 => P ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_148_less__imp__not__eq2,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( Y2 != X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_149_antisym__conv3,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [Y2: A,X3: A] :
% 2.59/1.59 ( ~ ( ord_less @ A @ Y2 @ X3 )
% 2.59/1.59 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.59/1.59 = ( X3 = Y2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_150_less__induct,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( wellorder @ A )
% 2.59/1.59 => ! [P: A > $o,A3: A] :
% 2.59/1.59 ( ! [X2: A] :
% 2.59/1.59 ( ! [Y: A] :
% 2.59/1.59 ( ( ord_less @ A @ Y @ X2 )
% 2.59/1.59 => ( P @ Y ) )
% 2.59/1.59 => ( P @ X2 ) )
% 2.59/1.59 => ( P @ A3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_151_less__not__sym,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_152_less__imp__not__eq,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( X3 != Y2 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_153_dual__order_Oasym,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [B3: A,A3: A] :
% 2.59/1.59 ( ( ord_less @ A @ B3 @ A3 )
% 2.59/1.59 => ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_154_ord__less__eq__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ord @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( B3 = C )
% 2.59/1.59 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_155_ord__eq__less__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ord @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( A3 = B3 )
% 2.59/1.59 => ( ( ord_less @ A @ B3 @ C )
% 2.59/1.59 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_156_less__irrefl,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A] :
% 2.59/1.59 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_157_less__linear,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 | ( X3 = Y2 )
% 2.59/1.59 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_158_less__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A,Z2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.59/1.59 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_159_less__asym_H,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_160_less__asym,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_161_less__imp__neq,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( X3 != Y2 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_162_dense,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( dense_order @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ? [Z3: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Z3 )
% 2.59/1.59 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_163_order_Oasym,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_164_neq__iff,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( X3 != Y2 )
% 2.59/1.59 = ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 | ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_165_neqE,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( X3 != Y2 )
% 2.59/1.59 => ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_166_gt__ex,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( no_top @ A )
% 2.59/1.59 => ! [X3: A] :
% 2.59/1.59 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_167_lt__ex,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( no_bot @ A )
% 2.59/1.59 => ! [X3: A] :
% 2.59/1.59 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_168_order__less__subst2,axiom,
% 2.59/1.59 ! [A: $tType,C2: $tType] :
% 2.59/1.59 ( ( ( order @ C2 )
% 2.59/1.59 & ( order @ A ) )
% 2.59/1.59 => ! [A3: A,B3: A,F: A > C2,C: C2] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 2.59/1.59 => ( ! [X2: A,Y3: A] :
% 2.59/1.59 ( ( ord_less @ A @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_169_order__less__subst1,axiom,
% 2.59/1.59 ! [A: $tType,B: $tType] :
% 2.59/1.59 ( ( ( order @ B )
% 2.59/1.59 & ( order @ A ) )
% 2.59/1.59 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 2.59/1.59 => ( ( ord_less @ B @ B3 @ C )
% 2.59/1.59 => ( ! [X2: B,Y3: B] :
% 2.59/1.59 ( ( ord_less @ B @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_170_ord__less__eq__subst,axiom,
% 2.59/1.59 ! [A: $tType,B: $tType] :
% 2.59/1.59 ( ( ( ord @ B )
% 2.59/1.59 & ( ord @ A ) )
% 2.59/1.59 => ! [A3: A,B3: A,F: A > B,C: B] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ( F @ B3 )
% 2.59/1.59 = C )
% 2.59/1.59 => ( ! [X2: A,Y3: A] :
% 2.59/1.59 ( ( ord_less @ A @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_171_ord__eq__less__subst,axiom,
% 2.59/1.59 ! [A: $tType,B: $tType] :
% 2.59/1.59 ( ( ( ord @ B )
% 2.59/1.59 & ( ord @ A ) )
% 2.59/1.59 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.59/1.59 ( ( A3
% 2.59/1.59 = ( F @ B3 ) )
% 2.59/1.59 => ( ( ord_less @ B @ B3 @ C )
% 2.59/1.59 => ( ! [X2: B,Y3: B] :
% 2.59/1.59 ( ( ord_less @ B @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_172_ex__has__least__nat,axiom,
% 2.59/1.59 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 2.59/1.59 ( ( P @ K )
% 2.59/1.59 => ? [X2: A] :
% 2.59/1.59 ( ( P @ X2 )
% 2.59/1.59 & ! [Y: A] :
% 2.59/1.59 ( ( P @ Y )
% 2.59/1.59 => ( ord_less_eq @ nat @ ( M @ X2 ) @ ( M @ Y ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_173_list__drop__eq__conv,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ^ [Y4: list @ A,Z: list @ A] : ( Y4 = Z ) )
% 2.59/1.59 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 2.59/1.59 ! [N3: nat] :
% 2.59/1.59 ( ( drop @ A @ N3 @ Xs3 )
% 2.59/1.59 = ( drop @ A @ N3 @ Ys3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_174_list__asc__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ( ( list_asc @ A )
% 2.59/1.59 = ( ^ [Xs3: list @ A] :
% 2.59/1.59 ! [J3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.59/1.59 => ! [I3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I3 @ J3 )
% 2.59/1.59 => ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_175_list__strict__asc__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ( ( list_strict_asc @ A )
% 2.59/1.59 = ( ^ [Xs3: list @ A] :
% 2.59/1.59 ! [J3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.59/1.59 => ! [I3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I3 @ J3 )
% 2.59/1.59 => ( ord_less @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_176_list__desc__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ( ( list_desc @ A )
% 2.59/1.59 = ( ^ [Xs3: list @ A] :
% 2.59/1.59 ! [J3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.59/1.59 => ! [I3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I3 @ J3 )
% 2.59/1.59 => ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_177_order_Onot__eq__order__implies__strict,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( A3 != B3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_178_dual__order_Ostrict__implies__order,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [B3: A,A3: A] :
% 2.59/1.59 ( ( ord_less @ A @ B3 @ A3 )
% 2.59/1.59 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_179_dual__order_Ostrict__iff__order,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ord_less @ A )
% 2.59/1.59 = ( ^ [B4: A,A4: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.59/1.59 & ( A4 != B4 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_180_dual__order_Oorder__iff__strict,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ord_less_eq @ A )
% 2.59/1.59 = ( ^ [B4: A,A4: A] :
% 2.59/1.59 ( ( ord_less @ A @ B4 @ A4 )
% 2.59/1.59 | ( A4 = B4 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_181_order_Ostrict__implies__order,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_182_dense__le__bounded,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( dense_linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A,Z2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ! [W: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ W )
% 2.59/1.59 => ( ( ord_less @ A @ W @ Y2 )
% 2.59/1.59 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.59/1.59 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_183_dense__ge__bounded,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( dense_linorder @ A )
% 2.59/1.59 => ! [Z2: A,X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z2 @ X3 )
% 2.59/1.59 => ( ! [W: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z2 @ W )
% 2.59/1.59 => ( ( ord_less @ A @ W @ X3 )
% 2.59/1.59 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 2.59/1.59 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_184_dual__order_Ostrict__trans2,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [B3: A,A3: A,C: A] :
% 2.59/1.59 ( ( ord_less @ A @ B3 @ A3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.59/1.59 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_185_dual__order_Ostrict__trans1,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [B3: A,A3: A,C: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.59/1.59 => ( ( ord_less @ A @ C @ B3 )
% 2.59/1.59 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_186_order_Ostrict__iff__order,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ord_less @ A )
% 2.59/1.59 = ( ^ [A4: A,B4: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.59/1.59 & ( A4 != B4 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_187_order_Oorder__iff__strict,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ord_less_eq @ A )
% 2.59/1.59 = ( ^ [A4: A,B4: A] :
% 2.59/1.59 ( ( ord_less @ A @ A4 @ B4 )
% 2.59/1.59 | ( A4 = B4 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_188_order_Ostrict__trans2,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.59/1.59 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_189_order_Ostrict__trans1,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less @ A @ B3 @ C )
% 2.59/1.59 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_190_not__le__imp__less,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [Y2: A,X3: A] :
% 2.59/1.59 ( ~ ( ord_less_eq @ A @ Y2 @ X3 )
% 2.59/1.59 => ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_191_less__le__not__le,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ( ( ord_less @ A )
% 2.59/1.59 = ( ^ [X: A,Y5: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X @ Y5 )
% 2.59/1.59 & ~ ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_192_le__imp__less__or__eq,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 | ( X3 = Y2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_193_le__less__linear,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_194_dense__le,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( dense_linorder @ A )
% 2.59/1.59 => ! [Y2: A,Z2: A] :
% 2.59/1.59 ( ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X2 @ Y2 )
% 2.59/1.59 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.59/1.59 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_195_dense__ge,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( dense_linorder @ A )
% 2.59/1.59 => ! [Z2: A,Y2: A] :
% 2.59/1.59 ( ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z2 @ X2 )
% 2.59/1.59 => ( ord_less_eq @ A @ Y2 @ X2 ) )
% 2.59/1.59 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_196_less__le__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A,Z2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.59/1.59 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_197_le__less__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A,Z2: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.59/1.59 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_198_less__imp__le,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_199_antisym__conv2,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.59/1.59 = ( X3 = Y2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_200_antisym__conv1,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.59/1.59 = ( X3 = Y2 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_201_le__neq__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( A3 != B3 )
% 2.59/1.59 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_202_not__less,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.59/1.59 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_203_not__le,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
% 2.59/1.59 = ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_204_order__less__le__subst2,axiom,
% 2.59/1.59 ! [A: $tType,C2: $tType] :
% 2.59/1.59 ( ( ( order @ C2 )
% 2.59/1.59 & ( order @ A ) )
% 2.59/1.59 => ! [A3: A,B3: A,F: A > C2,C: C2] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 2.59/1.59 => ( ! [X2: A,Y3: A] :
% 2.59/1.59 ( ( ord_less @ A @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_205_order__less__le__subst1,axiom,
% 2.59/1.59 ! [A: $tType,B: $tType] :
% 2.59/1.59 ( ( ( order @ B )
% 2.59/1.59 & ( order @ A ) )
% 2.59/1.59 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 2.59/1.59 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.59/1.59 => ( ! [X2: B,Y3: B] :
% 2.59/1.59 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_206_order__le__less__subst2,axiom,
% 2.59/1.59 ! [A: $tType,C2: $tType] :
% 2.59/1.59 ( ( ( order @ C2 )
% 2.59/1.59 & ( order @ A ) )
% 2.59/1.59 => ! [A3: A,B3: A,F: A > C2,C: C2] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 2.59/1.59 => ( ! [X2: A,Y3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_207_order__le__less__subst1,axiom,
% 2.59/1.59 ! [A: $tType,B: $tType] :
% 2.59/1.59 ( ( ( order @ B )
% 2.59/1.59 & ( order @ A ) )
% 2.59/1.59 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.59/1.59 => ( ( ord_less @ B @ B3 @ C )
% 2.59/1.59 => ( ! [X2: B,Y3: B] :
% 2.59/1.59 ( ( ord_less @ B @ X2 @ Y3 )
% 2.59/1.59 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.59/1.59 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_208_less__le,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ord_less @ A )
% 2.59/1.59 = ( ^ [X: A,Y5: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ X @ Y5 )
% 2.59/1.59 & ( X != Y5 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_209_le__less,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ( ( ord_less_eq @ A )
% 2.59/1.59 = ( ^ [X: A,Y5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X @ Y5 )
% 2.59/1.59 | ( X = Y5 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_210_leI,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [X3: A,Y2: A] :
% 2.59/1.59 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.59/1.59 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_211_leD,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [Y2: A,X3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.59/1.59 => ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_212_list__strict__desc__trans,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ( ( list_strict_desc @ A )
% 2.59/1.59 = ( ^ [Xs3: list @ A] :
% 2.59/1.59 ! [J3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.59/1.59 => ! [I3: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ I3 @ J3 )
% 2.59/1.59 => ( ord_less @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_213_complete__interval,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( condit1037483654norder @ A )
% 2.59/1.59 => ! [A3: A,B3: A,P: A > $o] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B3 )
% 2.59/1.59 => ( ( P @ A3 )
% 2.59/1.59 => ( ~ ( P @ B3 )
% 2.59/1.59 => ? [C3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ A3 @ C3 )
% 2.59/1.59 & ( ord_less_eq @ A @ C3 @ B3 )
% 2.59/1.59 & ! [X5: A] :
% 2.59/1.59 ( ( ( ord_less_eq @ A @ A3 @ X5 )
% 2.59/1.59 & ( ord_less @ A @ X5 @ C3 ) )
% 2.59/1.59 => ( P @ X5 ) )
% 2.59/1.59 & ! [D2: A] :
% 2.59/1.59 ( ! [X2: A] :
% 2.59/1.59 ( ( ( ord_less_eq @ A @ A3 @ X2 )
% 2.59/1.59 & ( ord_less @ A @ X2 @ D2 ) )
% 2.59/1.59 => ( P @ X2 ) )
% 2.59/1.59 => ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_214_verit__comp__simplify1_I3_J,axiom,
% 2.59/1.59 ! [B: $tType] :
% 2.59/1.59 ( ( linorder @ B )
% 2.59/1.59 => ! [B6: B,A6: B] :
% 2.59/1.59 ( ( ~ ( ord_less_eq @ B @ B6 @ A6 ) )
% 2.59/1.59 = ( ord_less @ B @ A6 @ B6 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_215_pinf_I6_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z3 @ X5 )
% 2.59/1.59 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_216_pinf_I8_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z3 @ X5 )
% 2.59/1.59 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_217_less__fun__def,axiom,
% 2.59/1.59 ! [B: $tType,A: $tType] :
% 2.59/1.59 ( ( ord @ B )
% 2.59/1.59 => ( ( ord_less @ ( A > B ) )
% 2.59/1.59 = ( ^ [F2: A > B,G2: A > B] :
% 2.59/1.59 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.59/1.59 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_218_verit__la__disequality,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( A3 = B3 )
% 2.59/1.59 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 2.59/1.59 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_219_minf_I11_J,axiom,
% 2.59/1.59 ! [C2: $tType,D3: $tType] :
% 2.59/1.59 ( ( ord @ C2 )
% 2.59/1.59 => ! [F3: D3] :
% 2.59/1.59 ? [Z3: C2] :
% 2.59/1.59 ! [X5: C2] :
% 2.59/1.59 ( ( ord_less @ C2 @ X5 @ Z3 )
% 2.59/1.59 => ( F3 = F3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_220_minf_I7_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X5 @ Z3 )
% 2.59/1.59 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_221_minf_I5_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X5 @ Z3 )
% 2.59/1.59 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_222_minf_I4_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X5 @ Z3 )
% 2.59/1.59 => ( X5 != T ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_223_minf_I3_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X5 @ Z3 )
% 2.59/1.59 => ( X5 != T ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_224_minf_I2_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.59/1.59 ( ? [Z4: A] :
% 2.59/1.59 ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X2 @ Z4 )
% 2.59/1.59 => ( ( P @ X2 )
% 2.59/1.59 = ( P4 @ X2 ) ) )
% 2.59/1.59 => ( ? [Z4: A] :
% 2.59/1.59 ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X2 @ Z4 )
% 2.59/1.59 => ( ( Q @ X2 )
% 2.59/1.59 = ( Q2 @ X2 ) ) )
% 2.59/1.59 => ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X5 @ Z3 )
% 2.59/1.59 => ( ( ( P @ X5 )
% 2.59/1.59 | ( Q @ X5 ) )
% 2.59/1.59 = ( ( P4 @ X5 )
% 2.59/1.59 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_225_minf_I1_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.59/1.59 ( ? [Z4: A] :
% 2.59/1.59 ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X2 @ Z4 )
% 2.59/1.59 => ( ( P @ X2 )
% 2.59/1.59 = ( P4 @ X2 ) ) )
% 2.59/1.59 => ( ? [Z4: A] :
% 2.59/1.59 ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ X2 @ Z4 )
% 2.59/1.59 => ( ( Q @ X2 )
% 2.59/1.59 = ( Q2 @ X2 ) ) )
% 2.59/1.59 => ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X5 @ Z3 )
% 2.59/1.59 => ( ( ( P @ X5 )
% 2.59/1.59 & ( Q @ X5 ) )
% 2.59/1.59 = ( ( P4 @ X5 )
% 2.59/1.59 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_226_pinf_I11_J,axiom,
% 2.59/1.59 ! [C2: $tType,D3: $tType] :
% 2.59/1.59 ( ( ord @ C2 )
% 2.59/1.59 => ! [F3: D3] :
% 2.59/1.59 ? [Z3: C2] :
% 2.59/1.59 ! [X5: C2] :
% 2.59/1.59 ( ( ord_less @ C2 @ Z3 @ X5 )
% 2.59/1.59 => ( F3 = F3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_227_pinf_I7_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z3 @ X5 )
% 2.59/1.59 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_228_pinf_I5_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z3 @ X5 )
% 2.59/1.59 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_229_pinf_I4_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z3 @ X5 )
% 2.59/1.59 => ( X5 != T ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_230_pinf_I3_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z3 @ X5 )
% 2.59/1.59 => ( X5 != T ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_231_pinf_I2_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.59/1.59 ( ? [Z4: A] :
% 2.59/1.59 ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z4 @ X2 )
% 2.59/1.59 => ( ( P @ X2 )
% 2.59/1.59 = ( P4 @ X2 ) ) )
% 2.59/1.59 => ( ? [Z4: A] :
% 2.59/1.59 ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z4 @ X2 )
% 2.59/1.59 => ( ( Q @ X2 )
% 2.59/1.59 = ( Q2 @ X2 ) ) )
% 2.59/1.59 => ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z3 @ X5 )
% 2.59/1.59 => ( ( ( P @ X5 )
% 2.59/1.59 | ( Q @ X5 ) )
% 2.59/1.59 = ( ( P4 @ X5 )
% 2.59/1.59 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_232_pinf_I1_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.59/1.59 ( ? [Z4: A] :
% 2.59/1.59 ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z4 @ X2 )
% 2.59/1.59 => ( ( P @ X2 )
% 2.59/1.59 = ( P4 @ X2 ) ) )
% 2.59/1.59 => ( ? [Z4: A] :
% 2.59/1.59 ! [X2: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z4 @ X2 )
% 2.59/1.59 => ( ( Q @ X2 )
% 2.59/1.59 = ( Q2 @ X2 ) ) )
% 2.59/1.59 => ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ Z3 @ X5 )
% 2.59/1.59 => ( ( ( P @ X5 )
% 2.59/1.59 & ( Q @ X5 ) )
% 2.59/1.59 = ( ( P4 @ X5 )
% 2.59/1.59 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_233_verit__comp__simplify1_I1_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( order @ A )
% 2.59/1.59 => ! [A3: A] :
% 2.59/1.59 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_234_ex__gt__or__lt,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( condit1656338222tinuum @ A )
% 2.59/1.59 => ! [A3: A] :
% 2.59/1.59 ? [B5: A] :
% 2.59/1.59 ( ( ord_less @ A @ A3 @ B5 )
% 2.59/1.59 | ( ord_less @ A @ B5 @ A3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_235_list__strict__desc__imp__list__desc,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( preorder @ A )
% 2.59/1.59 => ! [Xs: list @ A] :
% 2.59/1.59 ( ( list_strict_desc @ A @ Xs )
% 2.59/1.59 => ( list_desc @ A @ Xs ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_236_minf_I8_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X5 @ Z3 )
% 2.59/1.59 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_237_minf_I6_J,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( linorder @ A )
% 2.59/1.59 => ! [T: A] :
% 2.59/1.59 ? [Z3: A] :
% 2.59/1.59 ! [X5: A] :
% 2.59/1.59 ( ( ord_less @ A @ X5 @ Z3 )
% 2.59/1.59 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_238_i__append__i__take__eq2,axiom,
% 2.59/1.59 ! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 2.59/1.59 => ( ( listIn1033672622i_take @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
% 2.59/1.59 = ( append @ A @ Xs @ ( listIn1033672622i_take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ F ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_239_nth__drop,axiom,
% 2.59/1.59 ! [A: $tType,N: nat,Xs: list @ A,I: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.59 => ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
% 2.59/1.59 = ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_240_add__right__cancel,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( cancel_semigroup_add @ A )
% 2.59/1.59 => ! [B3: A,A3: A,C: A] :
% 2.59/1.59 ( ( ( plus_plus @ A @ B3 @ A3 )
% 2.59/1.59 = ( plus_plus @ A @ C @ A3 ) )
% 2.59/1.59 = ( B3 = C ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_241_add__left__cancel,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( cancel_semigroup_add @ A )
% 2.59/1.59 => ! [A3: A,B3: A,C: A] :
% 2.59/1.59 ( ( ( plus_plus @ A @ A3 @ B3 )
% 2.59/1.59 = ( plus_plus @ A @ A3 @ C ) )
% 2.59/1.59 = ( B3 = C ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_242_add__le__cancel__right,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ordere236663937imp_le @ A )
% 2.59/1.59 => ! [A3: A,C: A,B3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.59 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_243_add__le__cancel__left,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ordere236663937imp_le @ A )
% 2.59/1.59 => ! [C: A,A3: A,B3: A] :
% 2.59/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
% 2.59/1.59 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_244_add__less__cancel__left,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ordere236663937imp_le @ A )
% 2.59/1.59 => ! [C: A,A3: A,B3: A] :
% 2.59/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
% 2.59/1.59 = ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_245_add__less__cancel__right,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( ordere236663937imp_le @ A )
% 2.59/1.59 => ! [A3: A,C: A,B3: A] :
% 2.59/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.59 = ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_246_add__diff__cancel,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( group_add @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
% 2.59/1.59 = A3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_247_diff__add__cancel,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( group_add @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
% 2.59/1.59 = A3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_248_add__diff__cancel__left,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( cancel146912293up_add @ A )
% 2.59/1.59 => ! [C: A,A3: A,B3: A] :
% 2.59/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
% 2.59/1.59 = ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_249_add__diff__cancel__left_H,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( cancel146912293up_add @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ A3 )
% 2.59/1.59 = B3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_250_add__diff__cancel__right,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( cancel146912293up_add @ A )
% 2.59/1.59 => ! [A3: A,C: A,B3: A] :
% 2.59/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.59 = ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_251_add__diff__cancel__right_H,axiom,
% 2.59/1.59 ! [A: $tType] :
% 2.59/1.59 ( ( cancel146912293up_add @ A )
% 2.59/1.59 => ! [A3: A,B3: A] :
% 2.59/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
% 2.59/1.59 = A3 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_252_nat__add__left__cancel__less,axiom,
% 2.59/1.59 ! [K: nat,M: nat,N: nat] :
% 2.59/1.59 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.59/1.59 = ( ord_less @ nat @ M @ N ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_253_nat__add__left__cancel__le,axiom,
% 2.59/1.59 ! [K: nat,M: nat,N: nat] :
% 2.59/1.59 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.59/1.59 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_254_diff__diff__left,axiom,
% 2.59/1.59 ! [I: nat,J: nat,K: nat] :
% 2.59/1.59 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.59/1.59 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(fact_255_drop__drop,axiom,
% 2.59/1.59 ! [A: $tType,N: nat,M: nat,Xs: list @ A] :
% 2.59/1.59 ( ( drop @ A @ N @ ( drop @ A @ M @ Xs ) )
% 2.59/1.59 = ( drop @ A @ ( plus_plus @ nat @ N @ M ) @ Xs ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.59/1.59 ! [A7: $tType,A8: $tType] :
% 2.59/1.59 ( ( preorder @ A8 )
% 2.59/1.59 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.59/1.59 ! [A7: $tType,A8: $tType] :
% 2.59/1.59 ( ( order @ A8 )
% 2.59/1.59 => ( order @ ( A7 > A8 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_fun___Orderings_Oord,axiom,
% 2.59/1.59 ! [A7: $tType,A8: $tType] :
% 2.59/1.59 ( ( ord @ A8 )
% 2.59/1.59 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_fun___Groups_Ominus,axiom,
% 2.59/1.59 ! [A7: $tType,A8: $tType] :
% 2.59/1.59 ( ( minus @ A8 )
% 2.59/1.59 => ( minus @ ( A7 > A8 ) ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.59/1.59 condit1037483654norder @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.59/1.59 ordere236663937imp_le @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.59/1.59 cancel146912293up_add @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.59/1.59 cancel_semigroup_add @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.59/1.59 wellorder @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.59/1.59 preorder @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.59/1.59 linorder @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.59/1.59 no_top @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.59/1.59 order @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.59/1.59 ord @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
% 2.59/1.59 minus @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 2.59/1.59 size @ nat ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
% 2.59/1.59 preorder @ $o ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
% 2.59/1.59 linorder @ $o ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
% 2.59/1.59 order @ $o ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 2.59/1.59 ord @ $o ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
% 2.59/1.59 minus @ $o ).
% 2.59/1.59
% 2.59/1.59 thf(tcon_List_Olist___Nat_Osize_10,axiom,
% 2.59/1.59 ! [A7: $tType] : ( size @ ( list @ A7 ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(help_If_3_1_T,axiom,
% 2.59/1.59 ! [P: $o] :
% 2.59/1.59 ( ( P = $true )
% 2.59/1.59 | ( P = $false ) ) ).
% 2.59/1.59
% 2.59/1.59 thf(help_If_2_1_T,axiom,
% 2.59/1.59 ! [A: $tType,X3: A,Y2: A] :
% 2.59/1.59 ( ( if @ A @ $false @ X3 @ Y2 )
% 2.59/1.59 = Y2 ) ).
% 2.59/1.59
% 2.59/1.59 thf(help_If_1_1_T,axiom,
% 2.59/1.59 ! [A: $tType,X3: A,Y2: A] :
% 2.59/1.59 ( ( if @ A @ $true @ X3 @ Y2 )
% 2.59/1.59 = X3 ) ).
% 2.59/1.59
% 2.59/1.59 thf(conj_0,conjecture,
% 2.59/1.59 ( ( ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
% 2.59/1.59 => ( ( listIn1569887318i_drop @ a @ n @ ( listIn521021761append @ a @ xs @ f ) )
% 2.59/1.59 = ( listIn521021761append @ a @ ( drop @ a @ n @ xs ) @ f ) ) )
% 2.59/1.59 & ( ~ ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
% 2.59/1.59 => ( ( listIn1569887318i_drop @ a @ n @ ( listIn521021761append @ a @ xs @ f ) )
% 2.59/1.59 = ( listIn1569887318i_drop @ a @ ( minus_minus @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) ) @ f ) ) ) ) ).
% 2.59/1.59 ------------------------
% 2.69/1.61 % (11638)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/183Mi)
% 2.69/1.62 % (11641)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/2Mi)
% 2.69/1.62 % (11640)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/27Mi)
% 2.69/1.62 % (11642)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/2Mi)
% 2.69/1.62 % (11641)Instruction limit reached!
% 2.69/1.62 % (11641)------------------------------
% 2.69/1.62 % (11641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.62 % (11641)Termination reason: Unknown
% 2.69/1.62 % (11641)Termination phase: shuffling
% 2.69/1.62
% 2.69/1.62 % (11641)Memory used [KB]: 1407
% 2.69/1.62 % (11641)Time elapsed: 0.003 s
% 2.69/1.62 % (11641)Instructions burned: 3 (million)
% 2.69/1.62 % (11641)------------------------------
% 2.69/1.62 % (11641)------------------------------
% 2.69/1.62 % (11643)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/275Mi)
% 2.69/1.62 % (11644)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/18Mi)
% 2.69/1.62 % (11639)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/4Mi)
% 2.69/1.62 % (11642)Instruction limit reached!
% 2.69/1.62 % (11642)------------------------------
% 2.69/1.62 % (11642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.62 % (11642)Termination reason: Unknown
% 2.69/1.62 % (11642)Termination phase: shuffling
% 2.69/1.62
% 2.69/1.62 % (11642)Memory used [KB]: 1407
% 2.69/1.62 % (11642)Time elapsed: 0.003 s
% 2.69/1.62 % (11642)Instructions burned: 2 (million)
% 2.69/1.62 % (11642)------------------------------
% 2.69/1.62 % (11642)------------------------------
% 2.69/1.63 % (11639)Instruction limit reached!
% 2.69/1.63 % (11639)------------------------------
% 2.69/1.63 % (11639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.63 % (11639)Termination reason: Unknown
% 2.69/1.63 % (11639)Termination phase: shuffling
% 2.69/1.63
% 2.69/1.63 % (11639)Memory used [KB]: 1407
% 2.69/1.63 % (11639)Time elapsed: 0.004 s
% 2.69/1.63 % (11639)Instructions burned: 4 (million)
% 2.69/1.63 % (11639)------------------------------
% 2.69/1.63 % (11639)------------------------------
% 2.69/1.63 % (11644)Instruction limit reached!
% 2.69/1.63 % (11644)------------------------------
% 2.69/1.63 % (11644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.63 % (11644)Termination reason: Unknown
% 2.69/1.63 % (11644)Termination phase: shuffling
% 2.69/1.63
% 2.69/1.63 % (11644)Memory used [KB]: 1791
% 2.69/1.63 % (11644)Time elapsed: 0.012 s
% 2.69/1.63 % (11644)Instructions burned: 18 (million)
% 2.69/1.63 % (11644)------------------------------
% 2.69/1.63 % (11644)------------------------------
% 2.69/1.64 % (11640)Instruction limit reached!
% 2.69/1.64 % (11640)------------------------------
% 2.69/1.64 % (11640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.64 % (11640)Termination reason: Unknown
% 2.69/1.64 % (11640)Termination phase: Property scanning
% 2.69/1.64
% 2.69/1.64 % (11640)Memory used [KB]: 1791
% 2.69/1.64 % (11640)Time elapsed: 0.017 s
% 2.69/1.64 % (11640)Instructions burned: 28 (million)
% 2.69/1.64 % (11640)------------------------------
% 2.69/1.64 % (11640)------------------------------
% 2.69/1.64 % (11650)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/37Mi)
% 2.69/1.64 % (11649)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/3Mi)
% 2.69/1.64 % (11649)Instruction limit reached!
% 2.69/1.64 % (11649)------------------------------
% 2.69/1.64 % (11649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.64 % (11649)Termination reason: Unknown
% 2.69/1.64 % (11649)Termination phase: shuffling
% 2.69/1.64
% 2.69/1.64 % (11649)Memory used [KB]: 1407
% 2.69/1.64 % (11649)Time elapsed: 0.003 s
% 2.69/1.64 % (11649)Instructions burned: 3 (million)
% 2.69/1.64 % (11649)------------------------------
% 2.69/1.64 % (11649)------------------------------
% 2.69/1.64 % (11651)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/15Mi)
% 2.69/1.65 % (11651)Instruction limit reached!
% 2.69/1.65 % (11651)------------------------------
% 2.69/1.65 % (11651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.65 % (11651)Termination reason: Unknown
% 2.69/1.65 % (11651)Termination phase: shuffling
% 2.69/1.65
% 2.69/1.65 % (11651)Memory used [KB]: 1663
% 2.69/1.65 % (11651)Time elapsed: 0.007 s
% 2.69/1.65 % (11651)Instructions burned: 15 (million)
% 2.69/1.65 % (11651)------------------------------
% 2.69/1.65 % (11651)------------------------------
% 2.69/1.65 % (11653)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/1041Mi)
% 2.69/1.65 % (11652)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/3Mi)
% 2.69/1.65 % (11652)Instruction limit reached!
% 2.69/1.65 % (11652)------------------------------
% 2.69/1.65 % (11652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.65 % (11652)Termination reason: Unknown
% 2.69/1.65 % (11652)Termination phase: shuffling
% 2.69/1.65
% 2.69/1.65 % (11652)Memory used [KB]: 1407
% 2.69/1.65 % (11652)Time elapsed: 0.004 s
% 2.69/1.65 % (11652)Instructions burned: 4 (million)
% 2.69/1.65 % (11652)------------------------------
% 2.69/1.65 % (11652)------------------------------
% 2.69/1.66 % (11650)Instruction limit reached!
% 2.69/1.66 % (11650)------------------------------
% 2.69/1.66 % (11650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.66 % (11650)Termination reason: Unknown
% 2.69/1.66 % (11650)Termination phase: shuffling
% 2.69/1.66
% 2.69/1.66 % (11650)Memory used [KB]: 2174
% 2.69/1.66 % (11650)Time elapsed: 0.019 s
% 2.69/1.66 % (11650)Instructions burned: 38 (million)
% 2.69/1.66 % (11650)------------------------------
% 2.69/1.66 % (11650)------------------------------
% 2.69/1.66 % (11657)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/16Mi)
% 2.69/1.66 % (11655)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/7Mi)
% 2.69/1.66 % (11655)Instruction limit reached!
% 2.69/1.66 % (11655)------------------------------
% 2.69/1.66 % (11655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.66 % (11655)Termination reason: Unknown
% 2.69/1.66 % (11655)Termination phase: shuffling
% 2.69/1.66
% 2.69/1.66 % (11655)Memory used [KB]: 1535
% 2.69/1.66 % (11655)Time elapsed: 0.006 s
% 2.69/1.66 % (11655)Instructions burned: 7 (million)
% 2.69/1.66 % (11655)------------------------------
% 2.69/1.66 % (11655)------------------------------
% 2.69/1.66 % (11657)Instruction limit reached!
% 2.69/1.66 % (11657)------------------------------
% 2.69/1.66 % (11657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.66 % (11657)Termination reason: Unknown
% 2.69/1.66 % (11657)Termination phase: shuffling
% 2.69/1.66
% 2.69/1.66 % (11657)Memory used [KB]: 1663
% 2.69/1.66 % (11657)Time elapsed: 0.008 s
% 2.69/1.66 % (11657)Instructions burned: 17 (million)
% 2.69/1.66 % (11657)------------------------------
% 2.69/1.66 % (11657)------------------------------
% 2.69/1.67 % (11658)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/3Mi)
% 2.69/1.67 % (11658)Instruction limit reached!
% 2.69/1.67 % (11658)------------------------------
% 2.69/1.67 % (11658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.67 % (11658)Termination reason: Unknown
% 2.69/1.67 % (11658)Termination phase: shuffling
% 2.69/1.67
% 2.69/1.67 % (11658)Memory used [KB]: 1407
% 2.69/1.67 % (11658)Time elapsed: 0.003 s
% 2.69/1.67 % (11658)Instructions burned: 3 (million)
% 2.69/1.67 % (11658)------------------------------
% 2.69/1.67 % (11658)------------------------------
% 2.69/1.67 % (11659)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/3Mi)
% 2.69/1.67 % (11659)Instruction limit reached!
% 2.69/1.67 % (11659)------------------------------
% 2.69/1.67 % (11659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.67 % (11659)Termination reason: Unknown
% 2.69/1.67 % (11659)Termination phase: shuffling
% 2.69/1.67
% 2.69/1.67 % (11659)Memory used [KB]: 1407
% 2.69/1.67 % (11659)Time elapsed: 0.003 s
% 2.69/1.68 % (11659)Instructions burned: 3 (million)
% 2.69/1.68 % (11659)------------------------------
% 2.69/1.68 % (11659)------------------------------
% 2.69/1.68 % (11643)First to succeed.
% 2.69/1.68 % (11661)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11420 for (2999ds/3Mi)
% 2.69/1.68 % (11638)Instruction limit reached!
% 2.69/1.68 % (11638)------------------------------
% 2.69/1.68 % (11638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.68 % (11638)Termination reason: Unknown
% 2.69/1.68 % (11638)Termination phase: Saturation
% 2.69/1.68
% 2.69/1.68 % (11638)Memory used [KB]: 7164
% 2.69/1.68 % (11638)Time elapsed: 0.066 s
% 2.69/1.68 % (11638)Instructions burned: 183 (million)
% 2.69/1.68 % (11638)------------------------------
% 2.69/1.68 % (11638)------------------------------
% 2.69/1.68 % (11643)Refutation found. Thanks to Tanya!
% 2.69/1.68 % SZS status Theorem for DTF2THF_11420
% 2.69/1.68 % SZS output start Proof for DTF2THF_11420
% 2.69/1.68 thf(type_def_6, type, list: ($tType) > $tType).
% 2.69/1.68 thf(type_def_7, type, nat: $tType).
% 2.69/1.68 thf(type_def_8, type, a: $tType).
% 2.69/1.68 thf(func_def_0, type, list: ($tType) > $tType).
% 2.69/1.68 thf(func_def_1, type, nat: $tType).
% 2.69/1.68 thf(func_def_2, type, a: $tType).
% 2.69/1.68 thf(func_def_3, type, type: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_4, type, size: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_5, type, minus: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_6, type, ord: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_7, type, order: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_8, type, group_add: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_9, type, no_bot: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_10, type, no_top: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_13, type, wellorder: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_14, type, dense_order: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_15, type, dense_linorder: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_16, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_17, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_18, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_19, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_20, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_21, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.69/1.68 thf(func_def_22, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.69/1.68 thf(func_def_23, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.69/1.68 thf(func_def_24, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.69/1.68 thf(func_def_25, type, list_asc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.69/1.68 thf(func_def_26, type, list_desc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.69/1.68 thf(func_def_27, type, list_strict_asc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.69/1.68 thf(func_def_28, type, list_strict_desc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.69/1.68 thf(func_def_29, type, listIn521021761append: !>[X0: $tType]:(list @ X0 > (nat > X0) > nat > X0)).
% 2.69/1.68 thf(func_def_30, type, listIn1569887318i_drop: !>[X0: $tType]:(nat > (nat > X0) > nat > X0)).
% 2.69/1.68 thf(func_def_31, type, listIn1033672622i_take: !>[X0: $tType]:(nat > (nat > X0) > list @ X0)).
% 2.69/1.68 thf(func_def_32, type, append: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.69/1.68 thf(func_def_33, type, drop: !>[X0: $tType]:(nat > list @ X0 > list @ X0)).
% 2.69/1.68 thf(func_def_34, type, last: !>[X0: $tType]:(list @ X0 > X0)).
% 2.69/1.68 thf(func_def_35, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 2.69/1.68 thf(func_def_36, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.69/1.68 thf(func_def_37, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.69/1.68 thf(func_def_38, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.69/1.68 thf(func_def_39, type, f: nat > a).
% 2.69/1.68 thf(func_def_40, type, n: nat).
% 2.69/1.68 thf(func_def_41, type, xs: list @ a).
% 2.69/1.68 thf(func_def_56, type, sK0: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.69/1.68 thf(func_def_57, type, sK1: !>[X0: $tType]:(nat > (X0 > nat) > (X0 > $o) > X0)).
% 2.69/1.68 thf(func_def_58, type, sK2: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.69/1.68 thf(func_def_59, type, sK3: !>[X0: $tType]:((nat > X0) > list @ X0 > (nat > X0) > list @ X0 > list @ X0)).
% 2.69/1.68 thf(func_def_60, type, sK4: !>[X0: $tType]:(list @ X0 > list @ X0 > nat)).
% 2.69/1.68 thf(func_def_61, type, sK5: (nat > $o) > nat > nat).
% 2.69/1.68 thf(func_def_62, type, sK6: (nat > $o) > nat > nat).
% 2.69/1.68 thf(func_def_63, type, sK7: (nat > nat) > nat).
% 2.69/1.68 thf(func_def_64, type, sK8: (nat > nat) > nat).
% 2.69/1.68 thf(func_def_65, type, sK9: (nat > $o) > nat).
% 2.69/1.68 thf(func_def_66, type, sK10: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.69/1.68 thf(func_def_67, type, sK11: (nat > $o) > nat).
% 2.69/1.68 thf(func_def_68, type, sK12: (nat > $o) > nat).
% 2.69/1.68 thf(func_def_69, type, sK13: nat > (nat > $o) > nat).
% 2.69/1.68 thf(func_def_70, type, sK14: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.69/1.68 thf(func_def_71, type, sK15: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0 > list @ X0 > list @ X0)).
% 2.69/1.68 thf(func_def_72, type, sK16: !>[X0: $tType]:((nat > X0) > list @ X0 > (nat > X0) > list @ X0 > list @ X0)).
% 2.69/1.68 thf(func_def_73, type, sK17: !>[X0: $tType]:(nat > (nat > X0 > $o) > X0)).
% 2.69/1.68 thf(func_def_74, type, sK18: !>[X0: $tType]:(list @ X0 > (nat > X0 > $o) > nat > nat)).
% 2.69/1.68 thf(func_def_75, type, sK19: !>[X0: $tType]:((nat > X0 > $o) > nat > list @ X0)).
% 2.69/1.68 thf(func_def_76, type, sK20: !>[X0: $tType]:((nat > X0 > $o) > nat > nat)).
% 2.69/1.68 thf(func_def_77, type, sK21: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.69/1.68 thf(func_def_78, type, sK22: !>[X0: $tType]:(nat > list @ X0)).
% 2.69/1.68 thf(func_def_80, type, ph24: !>[X0: $tType]:(X0)).
% 2.69/1.68 thf(func_def_81, type, sK25: nat).
% 2.69/1.68 thf(func_def_82, type, sK26: nat).
% 2.69/1.68 thf(f1277,plain,(
% 2.69/1.68 $false),
% 2.69/1.68 inference(avatar_sat_refutation,[status(thm)],[f1184,f1189,f1190,f1217,f1220,f1221,f1242,f1269,f1274])).
% 2.69/1.68 thf(f1274,plain,(
% 2.69/1.68 spl23_8 | spl23_4),
% 2.69/1.68 inference(avatar_split_clause,[status(thm)],[f1265,f1196,f1239])).
% 2.69/1.68 thf(f1239,definition,(
% 2.69/1.68 spl23_8 <=> ($true = (ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs)))),
% 2.69/1.68 introduced(definition,[new_symbols(naming,[spl23_8])],[avatar_definition])).
% 2.69/1.68 thf(f1196,definition,(
% 2.69/1.68 spl23_4 <=> ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) = $true)),
% 2.69/1.68 introduced(definition,[new_symbols(naming,[spl23_4])],[avatar_definition])).
% 2.69/1.68 thf(f1265,plain,(
% 2.69/1.68 ($true = (ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs))) | spl23_4),
% 2.69/1.68 inference(trivial_inequality_removal,[status(thm)],[f1260])).
% 2.69/1.68 thf(f1260,plain,(
% 2.69/1.68 ($true = (ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs))) | ($true != $true) | spl23_4),
% 2.69/1.68 inference(superposition,[status(thm)],[f1198,f1058])).
% 2.69/1.68 thf(f1058,plain,(
% 2.69/1.68 ( ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less_eq @ nat @ X1 @ X0) = $true)) )),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f288])).
% 2.69/1.68 thf(f288,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : (((ord_less_eq @ nat @ X1 @ X0) = $true) | ($true = (ord_less_eq @ nat @ X0 @ X1)))),
% 2.69/1.68 inference(fool_elimination,[status(thm)],[f287])).
% 2.69/1.68 thf(f287,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X1 @ X0) | (ord_less_eq @ nat @ X0 @ X1))),
% 2.69/1.68 inference(rectify,[status(thm)],[f56])).
% 2.69/1.68 thf(f56,axiom,(
% 2.69/1.68 ! [X10 : nat,X5 : nat] : ((ord_less_eq @ nat @ X5 @ X10) | (ord_less_eq @ nat @ X10 @ X5))),
% 2.69/1.68 file('/export/starexec/sandbox2/tmp/tmp.rtTF0N8tiU/DTF2THF_11420.p',fact_55_nat__le__linear)).
% 2.69/1.68 thf(f1198,plain,(
% 2.69/1.68 ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) != $true) | spl23_4),
% 2.69/1.68 inference(avatar_component_clause,[status(thm)],[f1196])).
% 2.69/1.68 thf(f1269,plain,(
% 2.69/1.68 spl23_4 | ~spl23_6),
% 2.69/1.68 inference(avatar_contradiction_clause,[status(thm)],[f1268])).
% 2.69/1.68 thf(f1268,plain,(
% 2.69/1.68 $false | (spl23_4 | ~spl23_6)),
% 2.69/1.68 inference(subsumption_resolution,[status(thm)],[f1258,f1155])).
% 2.69/1.68 thf(f1155,plain,(
% 2.69/1.68 ( ! [X1 : nat] : (($true = (ord_less_eq @ nat @ X1 @ X1))) )),
% 2.69/1.68 inference(equality_resolution,[status(thm)],[f1069])).
% 2.69/1.68 thf(f1069,plain,(
% 2.69/1.68 ( ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) | (X0 != X1)) )),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f850])).
% 2.69/1.68 thf(f850,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) | (((ord_less @ nat @ X0 @ X1) != $true) & (X0 != X1)))),
% 2.69/1.68 inference(ennf_transformation,[status(thm)],[f728])).
% 2.69/1.68 thf(f728,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : ((((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1)) => ($true = (ord_less_eq @ nat @ X0 @ X1)))),
% 2.69/1.68 inference(fool_elimination,[status(thm)],[f727])).
% 2.69/1.68 thf(f727,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) | (X0 = X1)) => (ord_less_eq @ nat @ X0 @ X1))),
% 2.69/1.68 inference(rectify,[status(thm)],[f72])).
% 2.69/1.68 thf(f72,axiom,(
% 2.69/1.68 ! [X10 : nat,X5 : nat] : (((ord_less @ nat @ X10 @ X5) | (X5 = X10)) => (ord_less_eq @ nat @ X10 @ X5))),
% 2.69/1.68 file('/export/starexec/sandbox2/tmp/tmp.rtTF0N8tiU/DTF2THF_11420.p',fact_71_less__or__eq__imp__le)).
% 2.69/1.68 thf(f1258,plain,(
% 2.69/1.68 ($true != (ord_less_eq @ nat @ n @ n)) | (spl23_4 | ~spl23_6)),
% 2.69/1.68 inference(superposition,[status(thm)],[f1198,f1231])).
% 2.69/1.68 thf(f1231,plain,(
% 2.69/1.68 (n = (size_size @ list @ a @ xs)) | ~spl23_6),
% 2.69/1.68 inference(avatar_component_clause,[status(thm)],[f1229])).
% 2.69/1.68 thf(f1229,definition,(
% 2.69/1.68 spl23_6 <=> (n = (size_size @ list @ a @ xs))),
% 2.69/1.68 introduced(definition,[new_symbols(naming,[spl23_6])],[avatar_definition])).
% 2.69/1.68 thf(f1242,plain,(
% 2.69/1.68 ~spl23_8 | spl23_6 | spl23_1),
% 2.69/1.68 inference(avatar_split_clause,[status(thm)],[f1225,f1177,f1229,f1239])).
% 2.69/1.68 thf(f1177,definition,(
% 2.69/1.68 spl23_1 <=> ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true)),
% 2.69/1.68 introduced(definition,[new_symbols(naming,[spl23_1])],[avatar_definition])).
% 2.69/1.68 thf(f1225,plain,(
% 2.69/1.68 ($true != (ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs))) | (n = (size_size @ list @ a @ xs)) | spl23_1),
% 2.69/1.68 inference(trivial_inequality_removal,[status(thm)],[f1224])).
% 2.69/1.68 thf(f1224,plain,(
% 2.69/1.68 ($true != (ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs))) | (n = (size_size @ list @ a @ xs)) | ($true != $true) | spl23_1),
% 2.69/1.68 inference(superposition,[status(thm)],[f1179,f1100])).
% 2.69/1.68 thf(f1100,plain,(
% 2.69/1.68 ( ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1) | ($true != (ord_less_eq @ nat @ X0 @ X1))) )),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f891])).
% 2.69/1.68 thf(f891,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : (($true != (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1))),
% 2.69/1.68 inference(flattening,[status(thm)],[f890])).
% 2.69/1.68 thf(f890,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : ((((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1)) | ($true != (ord_less_eq @ nat @ X0 @ X1)))),
% 2.69/1.68 inference(ennf_transformation,[status(thm)],[f787])).
% 2.69/1.68 thf(f787,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) => ((X0 != X1) => ((ord_less @ nat @ X0 @ X1) = $true)))),
% 2.69/1.68 inference(fool_elimination,[status(thm)],[f786])).
% 2.69/1.68 thf(f786,plain,(
% 2.69/1.68 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) => ((X0 != X1) => (ord_less @ nat @ X0 @ X1)))),
% 2.69/1.68 inference(rectify,[status(thm)],[f73])).
% 2.69/1.68 thf(f73,axiom,(
% 2.69/1.68 ! [X10 : nat,X5 : nat] : ((ord_less_eq @ nat @ X10 @ X5) => ((X5 != X10) => (ord_less @ nat @ X10 @ X5)))),
% 2.69/1.68 file('/export/starexec/sandbox2/tmp/tmp.rtTF0N8tiU/DTF2THF_11420.p',fact_72_le__neq__implies__less)).
% 2.69/1.68 thf(f1179,plain,(
% 2.69/1.68 ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true) | spl23_1),
% 2.69/1.68 inference(avatar_component_clause,[status(thm)],[f1177])).
% 2.69/1.68 thf(f1221,plain,(
% 2.69/1.68 ~spl23_4 | spl23_3),
% 2.69/1.68 inference(avatar_split_clause,[status(thm)],[f1207,f1186,f1196])).
% 2.69/1.68 thf(f1186,definition,(
% 2.69/1.68 spl23_3 <=> ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f))),
% 2.69/1.68 introduced(definition,[new_symbols(naming,[spl23_3])],[avatar_definition])).
% 2.69/1.68 thf(f1207,plain,(
% 2.69/1.68 ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) != $true) | spl23_3),
% 2.69/1.68 inference(trivial_inequality_removal,[status(thm)],[f1205])).
% 2.69/1.68 thf(f1205,plain,(
% 2.69/1.68 ((listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f) != (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f)) | ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) != $true) | spl23_3),
% 2.69/1.68 inference(superposition,[status(thm)],[f1188,f1050])).
% 2.69/1.68 thf(f1050,plain,(
% 2.69/1.68 ( ! [X0 : $tType,X2 : nat,X3 : nat > X0,X1 : list @ X0] : (((listIn1569887318i_drop @ X0 @ (minus_minus @ nat @ X2 @ (size_size @ list @ X0 @ X1)) @ X3) = (listIn1569887318i_drop @ X0 @ X2 @ (listIn521021761append @ X0 @ X1 @ X3))) | ($true != (ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X2))) )),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f861])).
% 2.69/1.68 thf(f861,plain,(
% 2.69/1.68 ! [X0 : $tType,X1 : list @ X0,X2 : nat,X3 : nat > X0] : (((listIn1569887318i_drop @ X0 @ (minus_minus @ nat @ X2 @ (size_size @ list @ X0 @ X1)) @ X3) = (listIn1569887318i_drop @ X0 @ X2 @ (listIn521021761append @ X0 @ X1 @ X3))) | ($true != (ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X2)))),
% 2.69/1.68 inference(ennf_transformation,[status(thm)],[f801])).
% 2.69/1.68 thf(f801,plain,(
% 2.69/1.68 ! [X0 : $tType,X1 : list @ X0,X2 : nat,X3 : nat > X0] : (($true = (ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X2)) => ((listIn1569887318i_drop @ X0 @ (minus_minus @ nat @ X2 @ (size_size @ list @ X0 @ X1)) @ X3) = (listIn1569887318i_drop @ X0 @ X2 @ (listIn521021761append @ X0 @ X1 @ X3))))),
% 2.69/1.68 inference(fool_elimination,[status(thm)],[f800])).
% 2.69/1.68 thf(f800,plain,(
% 2.69/1.68 ! [X0 : $tType,X1 : list @ X0,X2 : nat,X3 : nat > X0] : ((ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X2) => ((listIn1569887318i_drop @ X0 @ (minus_minus @ nat @ X2 @ (size_size @ list @ X0 @ X1)) @ X3) = (listIn1569887318i_drop @ X0 @ X2 @ (listIn521021761append @ X0 @ X1 @ X3))))),
% 2.69/1.68 inference(rectify,[status(thm)],[f37])).
% 2.69/1.68 thf(f37,axiom,(
% 2.69/1.68 ! [X0 : $tType,X1 : list @ X0,X5 : nat,X2 : nat > X0] : ((ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X5) => ((listIn1569887318i_drop @ X0 @ X5 @ (listIn521021761append @ X0 @ X1 @ X2)) = (listIn1569887318i_drop @ X0 @ (minus_minus @ nat @ X5 @ (size_size @ list @ X0 @ X1)) @ X2)))),
% 2.69/1.68 file('/export/starexec/sandbox2/tmp/tmp.rtTF0N8tiU/DTF2THF_11420.p',fact_36_i__append__i__drop__eq2)).
% 2.69/1.68 thf(f1188,plain,(
% 2.69/1.68 ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) != (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f)) | spl23_3),
% 2.69/1.68 inference(avatar_component_clause,[status(thm)],[f1186])).
% 2.69/1.68 thf(f1220,plain,(
% 2.69/1.68 spl23_4 | spl23_2),
% 2.69/1.68 inference(avatar_split_clause,[status(thm)],[f1212,f1181,f1196])).
% 2.69/1.68 thf(f1181,definition,(
% 2.69/1.68 spl23_2 <=> ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f))),
% 2.69/1.68 introduced(definition,[new_symbols(naming,[spl23_2])],[avatar_definition])).
% 2.69/1.68 thf(f1212,plain,(
% 2.69/1.68 ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) = $true) | spl23_2),
% 2.69/1.68 inference(trivial_inequality_removal,[status(thm)],[f1209])).
% 2.69/1.68 thf(f1209,plain,(
% 2.69/1.68 ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) = $true) | ($true != $true) | spl23_2),
% 2.69/1.68 inference(superposition,[status(thm)],[f1194,f1058])).
% 2.69/1.68 thf(f1194,plain,(
% 2.69/1.68 ($true != (ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs))) | spl23_2),
% 2.69/1.68 inference(trivial_inequality_removal,[status(thm)],[f1193])).
% 2.69/1.68 thf(f1193,plain,(
% 2.69/1.68 ((listIn521021761append @ a @ (drop @ a @ n @ xs) @ f) != (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f)) | ($true != (ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs))) | spl23_2),
% 2.69/1.68 inference(superposition,[status(thm)],[f1183,f1127])).
% 2.69/1.68 thf(f1127,plain,(
% 2.69/1.68 ( ! [X0 : $tType,X2 : list @ X0,X3 : nat > X0,X1 : nat] : (((listIn521021761append @ X0 @ (drop @ X0 @ X1 @ X2) @ X3) = (listIn1569887318i_drop @ X0 @ X1 @ (listIn521021761append @ X0 @ X2 @ X3))) | ($true != (ord_less_eq @ nat @ X1 @ (size_size @ list @ X0 @ X2)))) )),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f909])).
% 2.69/1.68 thf(f909,plain,(
% 2.69/1.68 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat > X0] : (($true != (ord_less_eq @ nat @ X1 @ (size_size @ list @ X0 @ X2))) | ((listIn521021761append @ X0 @ (drop @ X0 @ X1 @ X2) @ X3) = (listIn1569887318i_drop @ X0 @ X1 @ (listIn521021761append @ X0 @ X2 @ X3))))),
% 2.69/1.68 inference(ennf_transformation,[status(thm)],[f512])).
% 2.69/1.68 thf(f512,plain,(
% 2.69/1.68 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat > X0] : (($true = (ord_less_eq @ nat @ X1 @ (size_size @ list @ X0 @ X2))) => ((listIn521021761append @ X0 @ (drop @ X0 @ X1 @ X2) @ X3) = (listIn1569887318i_drop @ X0 @ X1 @ (listIn521021761append @ X0 @ X2 @ X3))))),
% 2.69/1.68 inference(fool_elimination,[status(thm)],[f511])).
% 2.69/1.68 thf(f511,plain,(
% 2.69/1.68 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat > X0] : ((ord_less_eq @ nat @ X1 @ (size_size @ list @ X0 @ X2)) => ((listIn521021761append @ X0 @ (drop @ X0 @ X1 @ X2) @ X3) = (listIn1569887318i_drop @ X0 @ X1 @ (listIn521021761append @ X0 @ X2 @ X3))))),
% 2.69/1.68 inference(rectify,[status(thm)],[f36])).
% 2.69/1.68 thf(f36,axiom,(
% 2.69/1.68 ! [X0 : $tType,X5 : nat,X1 : list @ X0,X2 : nat > X0] : ((ord_less_eq @ nat @ X5 @ (size_size @ list @ X0 @ X1)) => ((listIn1569887318i_drop @ X0 @ X5 @ (listIn521021761append @ X0 @ X1 @ X2)) = (listIn521021761append @ X0 @ (drop @ X0 @ X5 @ X1) @ X2)))),
% 2.69/1.68 file('/export/starexec/sandbox2/tmp/tmp.rtTF0N8tiU/DTF2THF_11420.p',fact_35_i__append__i__drop__eq1)).
% 2.69/1.68 thf(f1183,plain,(
% 2.69/1.68 ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) != (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f)) | spl23_2),
% 2.69/1.68 inference(avatar_component_clause,[status(thm)],[f1181])).
% 2.69/1.68 thf(f1217,plain,(
% 2.69/1.68 ~spl23_1 | spl23_2),
% 2.69/1.68 inference(avatar_split_clause,[status(thm)],[f1213,f1181,f1177])).
% 2.69/1.68 thf(f1213,plain,(
% 2.69/1.68 ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true) | spl23_2),
% 2.69/1.68 inference(trivial_inequality_removal,[status(thm)],[f1208])).
% 2.69/1.68 thf(f1208,plain,(
% 2.69/1.68 ($true != $true) | ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true) | spl23_2),
% 2.69/1.68 inference(superposition,[status(thm)],[f1194,f1070])).
% 2.69/1.68 thf(f1070,plain,(
% 2.69/1.68 ( ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less @ nat @ X0 @ X1) != $true)) )),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f850])).
% 2.69/1.68 thf(f1190,plain,(
% 2.69/1.68 spl23_1 | ~spl23_3),
% 2.69/1.68 inference(avatar_split_clause,[status(thm)],[f999,f1186,f1177])).
% 2.69/1.68 thf(f999,plain,(
% 2.69/1.68 ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) != (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f)) | ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true)),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f899])).
% 2.69/1.68 thf(f899,plain,(
% 2.69/1.68 (((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true) & ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) != (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f))) | (((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) != (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f)) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true))),
% 2.69/1.68 inference(ennf_transformation,[status(thm)],[f816])).
% 2.69/1.68 thf(f816,plain,(
% 2.69/1.68 ~((((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f))) & (((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f))))),
% 2.69/1.68 inference(flattening,[status(thm)],[f578])).
% 2.69/1.68 thf(f578,plain,(
% 2.69/1.68 ~((~((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f))) & (((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f))))),
% 2.69/1.68 inference(fool_elimination,[status(thm)],[f577])).
% 2.69/1.68 thf(f577,plain,(
% 2.69/1.68 ~((~(ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f))) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f))))),
% 2.69/1.68 inference(rectify,[status(thm)],[f283])).
% 2.69/1.68 thf(f283,negated_conjecture,(
% 2.69/1.68 ~((~(ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f))) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f))))),
% 2.69/1.68 inference(negated_conjecture,[status(cth)],[f282])).
% 2.69/1.68 thf(f282,conjecture,(
% 2.69/1.68 (~(ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f))) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) = (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f)))),
% 2.69/1.68 file('/export/starexec/sandbox2/tmp/tmp.rtTF0N8tiU/DTF2THF_11420.p',conj_0)).
% 2.69/1.68 thf(f1189,plain,(
% 2.69/1.68 ~spl23_2 | ~spl23_3),
% 2.69/1.68 inference(avatar_split_clause,[status(thm)],[f1000,f1186,f1181])).
% 2.69/1.68 thf(f1000,plain,(
% 2.69/1.68 ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) != (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f)) | ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) != (listIn1569887318i_drop @ a @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)) @ f))),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f899])).
% 2.69/1.68 thf(f1184,plain,(
% 2.69/1.68 ~spl23_1 | ~spl23_2),
% 2.69/1.68 inference(avatar_split_clause,[status(thm)],[f1002,f1181,f1177])).
% 2.69/1.68 thf(f1002,plain,(
% 2.69/1.68 ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true) | ((listIn1569887318i_drop @ a @ n @ (listIn521021761append @ a @ xs @ f)) != (listIn521021761append @ a @ (drop @ a @ n @ xs) @ f))),
% 2.69/1.68 inference(cnf_transformation,[status(thm)],[f899])).
% 2.69/1.68 % SZS output end Proof for DTF2THF_11420
% 2.69/1.68 % (11643)------------------------------
% 2.69/1.68 % (11643)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.68 % (11643)Termination reason: Refutation
% 2.69/1.68
% 2.69/1.68 % (11643)Memory used [KB]: 6908
% 2.69/1.68 % (11643)Time elapsed: 0.058 s
% 2.69/1.68 % (11643)Instructions burned: 99 (million)
% 2.69/1.68 % (11643)------------------------------
% 2.69/1.68 % (11643)------------------------------
% 2.69/1.68 % (11634)Success in time 0.098 s
%------------------------------------------------------------------------------