%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP102^2 : 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 : n005.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 08:55:34 AM UTC 2026
% Result : Theorem 2.71s 1.62s
% Output : Refutation 2.71s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP102^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n005.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:26:44 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : ITP102^2 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.45 % Domain : Interactive Theorem Proving
% 0.19/0.45 % Problem : Sledgehammer ListInf problem prob_77__5408624_1
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English :
% 0.19/0.45
% 0.19/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.45 % Source : [Des21]
% 0.19/0.45 % Names : ListInf/prob_77__5408624_1 [Des21]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.00 v9.1.0, 0.33 v8.1.0, 0.50 v7.5.0
% 0.19/0.45 % Syntax : Number of formulae : 308 ( 58 unt; 34 typ; 0 def)
% 0.19/0.45 % Number of atoms : 925 ( 201 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 13 ( 3 avg)
% 0.19/0.45 % Number of connectives : 3253 ( 93 ~; 25 |; 45 &;2535 @)
% 0.19/0.45 % ( 0 <=>; 555 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 21 ( 8 avg)
% 0.19/0.45 % Number of types : 3 ( 2 usr)
% 0.19/0.45 % Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 33 ( 32 usr; 2 con; 0-4 aty)
% 0.19/0.45 % Number of variables : 992 ( 60 ^; 860 !; 46 ?; 992 :)
% 0.19/0.45 % ( 26 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.45 % SPC : TH1_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:15:42.791
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % Could-be-implicit typings (3)
% 0.19/0.45 thf(ty_t_List_Olist,type,
% 0.19/0.45 list: $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Nat_Onat,type,
% 0.19/0.45 nat: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_a,type,
% 0.19/0.45 a: $tType ).
% 0.19/0.45
% 0.19/0.45 % Explicit typings (31)
% 0.19/0.45 thf(sy_cl_HOL_Otype,type,
% 0.19/0.45 type:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Nat_Osize,type,
% 0.19/0.45 size:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.19/0.45 ord:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.45 order:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Ono__bot,type,
% 0.19/0.45 no_bot:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Ono__top,type,
% 0.19/0.45 no_top:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.19/0.45 linorder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.19/0.45 preorder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.19/0.45 wellorder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Odense__order,type,
% 0.19/0.45 dense_order:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.19/0.45 dense_linorder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.19/0.45 condit1656338222tinuum:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.19/0.45 condit1037483654norder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.19/0.45 minus_minus:
% 0.19/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List2_Olist__asc,type,
% 0.19/0.45 list_asc:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List2_Olist__desc,type,
% 0.19/0.45 list_desc:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List2_Olist__strict__asc,type,
% 0.19/0.45 list_strict_asc:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List2_Olist__strict__desc,type,
% 0.19/0.45 list_strict_desc:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
% 0.19/0.45 listIn521021761append:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List_Olinorder__class_Osorted,type,
% 0.19/0.45 linorder_sorted:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List_Olist__ex,type,
% 0.19/0.45 list_ex:
% 0.19/0.45 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List_Onth,type,
% 0.19/0.45 nth:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List_Orev,type,
% 0.19/0.45 rev:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_OSuc,type,
% 0.19/0.45 suc: nat > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.19/0.45 size_size:
% 0.19/0.45 !>[A: $tType] : ( A > nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.19/0.45 ord_less:
% 0.19/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.45 ord_less_eq:
% 0.19/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_f,type,
% 0.19/0.45 f: nat > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_g,type,
% 0.19/0.45 g: nat > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_xs,type,
% 0.19/0.45 xs: list @ a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_ys,type,
% 0.19/0.45 ys: list @ a ).
% 0.19/0.45
% 0.19/0.45 % Relevant facts (255)
% 0.19/0.45 thf(fact_0_i__append__eq__i__append__conv,axiom,
% 0.19/0.45 ! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
% 0.19/0.45 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.19/0.45 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.19/0.45 => ( ( ( listIn521021761append @ A @ Xs @ F )
% 0.19/0.45 = ( listIn521021761append @ A @ Ys @ G ) )
% 0.19/0.45 = ( ( Xs = Ys )
% 0.19/0.45 & ( F = G ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % i_append_eq_i_append_conv
% 0.19/0.45 thf(fact_1_i__append__nth1,axiom,
% 0.19/0.45 ! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
% 0.19/0.45 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.45 => ( ( listIn521021761append @ A @ Xs @ F @ N )
% 0.19/0.45 = ( nth @ A @ Xs @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % i_append_nth1
% 0.19/0.45 thf(fact_2_nth__equalityI,axiom,
% 0.19/0.45 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.19/0.45 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.19/0.45 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.19/0.45 => ( ! [I: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.45 => ( ( nth @ A @ Xs @ I )
% 0.19/0.45 = ( nth @ A @ Ys @ I ) ) )
% 0.19/0.45 => ( Xs = Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nth_equalityI
% 0.19/0.45 thf(fact_3_Skolem__list__nth,axiom,
% 0.19/0.45 ! [A: $tType,K: nat,P: nat > A > $o] :
% 0.19/0.45 ( ( ! [I2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I2 @ K )
% 0.19/0.45 => ? [X: A] : ( P @ I2 @ X ) ) )
% 0.19/0.45 = ( ? [Xs2: list @ A] :
% 0.19/0.45 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.19/0.45 = K )
% 0.19/0.45 & ! [I2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I2 @ K )
% 0.19/0.45 => ( P @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Skolem_list_nth
% 0.19/0.45 thf(fact_4_list__eq__iff__nth__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ^ [Y: list @ A,Z: list @ A] : ( Y = Z ) )
% 0.19/0.45 = ( ^ [Xs2: list @ A,Ys2: list @ A] :
% 0.19/0.45 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.19/0.45 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.19/0.45 & ! [I2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.45 => ( ( nth @ A @ Xs2 @ I2 )
% 0.19/0.45 = ( nth @ A @ Ys2 @ I2 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % list_eq_iff_nth_eq
% 0.19/0.45 thf(fact_5_order__refl,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( preorder @ A )
% 0.19/0.45 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_refl
% 0.19/0.45 thf(fact_6_list__desc__trans__le,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( preorder @ A )
% 0.19/0.45 => ( ( list_desc @ A )
% 0.19/0.45 = ( ^ [Xs2: list @ A] :
% 0.19/0.45 ! [J: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.45 => ! [I2: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.19/0.45 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % list_desc_trans_le
% 0.19/0.45 thf(fact_7_length__induct,axiom,
% 0.19/0.45 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.19/0.45 ( ! [Xs3: list @ A] :
% 0.19/0.45 ( ! [Ys3: list @ A] :
% 0.19/0.45 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.19/0.45 => ( P @ Ys3 ) )
% 0.19/0.45 => ( P @ Xs3 ) )
% 0.19/0.45 => ( P @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % length_induct
% 0.19/0.45 thf(fact_8_nat__less__le,axiom,
% 0.19/0.45 ( ( ord_less @ nat )
% 0.19/0.45 = ( ^ [M: nat,N2: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.19/0.45 & ( M != N2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_less_le
% 0.19/0.45 thf(fact_9_less__imp__le__nat,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.45 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_imp_le_nat
% 0.19/0.45 thf(fact_10_le__eq__less__or__eq,axiom,
% 0.19/0.45 ( ( ord_less_eq @ nat )
% 0.19/0.45 = ( ^ [M: nat,N2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M @ N2 )
% 0.19/0.45 | ( M = N2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_eq_less_or_eq
% 0.19/0.45 thf(fact_11_less__or__eq__imp__le,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.45 | ( M2 = N ) )
% 0.19/0.45 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_or_eq_imp_le
% 0.19/0.45 thf(fact_12_le__neq__implies__less,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.45 => ( ( M2 != N )
% 0.19/0.45 => ( ord_less @ nat @ M2 @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_neq_implies_less
% 0.19/0.45 thf(fact_13_less__mono__imp__le__mono,axiom,
% 0.19/0.45 ! [F: nat > nat,I3: nat,J2: nat] :
% 0.19/0.45 ( ! [I: nat,J3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I @ J3 )
% 0.19/0.45 => ( ord_less @ nat @ ( F @ I ) @ ( F @ J3 ) ) )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ I3 @ J2 )
% 0.19/0.45 => ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_mono_imp_le_mono
% 0.19/0.45 thf(fact_14_dual__order_Oantisym,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.45 => ! [B: A,A2: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ B @ A2 )
% 0.19/0.45 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.45 => ( A2 = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.antisym
% 0.19/0.45 thf(fact_15_dual__order_Oeq__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.45 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.19/0.45 = ( ^ [A3: A,B2: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.19/0.45 & ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.eq_iff
% 0.19/0.45 thf(fact_16_dual__order_Otrans,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.45 => ! [B: A,A2: A,C: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ B @ A2 )
% 0.19/0.45 => ( ( ord_less_eq @ A @ C @ B )
% 0.19/0.45 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.trans
% 0.19/0.45 thf(fact_17_linorder__wlog,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [P: A > A > $o,A2: A,B: A] :
% 0.19/0.45 ( ! [A4: A,B3: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.45 => ( P @ A4 @ B3 ) )
% 0.19/0.45 => ( ! [A4: A,B3: A] :
% 0.19/0.45 ( ( P @ B3 @ A4 )
% 0.19/0.45 => ( P @ A4 @ B3 ) )
% 0.19/0.45 => ( P @ A2 @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % linorder_wlog
% 0.19/0.45 thf(fact_18_dual__order_Orefl,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.45 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.refl
% 0.19/0.45 thf(fact_19_order__trans,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( preorder @ A )
% 0.19/0.45 => ! [X2: A,Y2: A,Z2: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.45 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.19/0.45 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_trans
% 0.19/0.45 thf(fact_20_order__class_Oorder_Oantisym,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.45 => ! [A2: A,B: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.45 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.19/0.45 => ( A2 = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_class.order.antisym
% 0.19/0.45 thf(fact_21_ord__le__eq__trans,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ord @ A )
% 0.19/0.45 => ! [A2: A,B: A,C: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.45 => ( ( B = C )
% 0.19/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ord_le_eq_trans
% 0.19/0.45 thf(fact_22_ord__eq__le__trans,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ord @ A )
% 0.19/0.45 => ! [A2: A,B: A,C: A] :
% 0.19/0.45 ( ( A2 = B )
% 0.19/0.45 => ( ( ord_less_eq @ A @ B @ C )
% 0.19/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ord_eq_le_trans
% 0.19/0.45 thf(fact_23_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.45 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.19/0.45 = ( ^ [A3: A,B2: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.45 & ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_class.order.eq_iff
% 0.19/0.45 thf(fact_24_antisym__conv,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.45 => ! [Y2: A,X2: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.19/0.45 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.45 = ( X2 = Y2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % antisym_conv
% 0.19/0.45 thf(fact_25_le__cases3,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [X2: A,Y2: A,Z2: A] :
% 0.19/0.45 ( ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.45 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.19/0.45 => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.19/0.45 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.19/0.45 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.19/0.45 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.19/0.45 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.19/0.45 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
% 0.19/0.45 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.19/0.45 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 0.19/0.45 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.19/0.45 => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_cases3
% 0.19/0.45 thf(fact_26_order_Otrans,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B @ C )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_27_le__cases,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_28_eq__refl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( X2 = Y2 )
% 0.19/0.46 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_29_linear,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.46 | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_30_antisym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.19/0.46 => ( X2 = Y2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_31_eq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.19/0.46 = ( ^ [X3: A,Y3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.46 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_32_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: $tType,B4: $tType] :
% 0.19/0.46 ( ( ( ord @ B4 )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A2: A,B: A,F: A > B4,C: B4] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X4: A,Y4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq @ B4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_33_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: $tType,B4: $tType] :
% 0.19/0.46 ( ( ( ord @ B4 )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 0.19/0.46 ( ( A2
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq @ B4 @ B @ C )
% 0.19/0.46 => ( ! [X4: B4,Y4: B4] :
% 0.19/0.46 ( ( ord_less_eq @ B4 @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_34_order__subst2,axiom,
% 0.19/0.46 ! [A: $tType,C2: $tType] :
% 0.19/0.46 ( ( ( order @ C2 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.46 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X4: A,Y4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_35_order__subst1,axiom,
% 0.19/0.46 ! [A: $tType,B4: $tType] :
% 0.19/0.46 ( ( ( order @ B4 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq @ B4 @ B @ C )
% 0.19/0.46 => ( ! [X4: B4,Y4: B4] :
% 0.19/0.46 ( ( ord_less_eq @ B4 @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_36_le__fun__def,axiom,
% 0.19/0.46 ! [B4: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B4 )
% 0.19/0.46 => ( ( ord_less_eq @ ( A > B4 ) )
% 0.19/0.46 = ( ^ [F2: A > B4,G2: A > B4] :
% 0.19/0.46 ! [X3: A] : ( ord_less_eq @ B4 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_fun_def
% 0.19/0.46 thf(fact_37_le__funI,axiom,
% 0.19/0.46 ! [B4: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B4 )
% 0.19/0.46 => ! [F: A > B4,G: A > B4] :
% 0.19/0.46 ( ! [X4: A] : ( ord_less_eq @ B4 @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.19/0.46 => ( ord_less_eq @ ( A > B4 ) @ F @ G ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funI
% 0.19/0.46 thf(fact_38_le__funE,axiom,
% 0.19/0.46 ! [B4: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B4 )
% 0.19/0.46 => ! [F: A > B4,G: A > B4,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ ( A > B4 ) @ F @ G )
% 0.19/0.46 => ( ord_less_eq @ B4 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funE
% 0.19/0.46 thf(fact_39_le__funD,axiom,
% 0.19/0.46 ! [B4: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B4 )
% 0.19/0.46 => ! [F: A > B4,G: A > B4,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ ( A > B4 ) @ F @ G )
% 0.19/0.46 => ( ord_less_eq @ B4 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funD
% 0.19/0.46 thf(fact_40_dual__order_Ostrict__implies__not__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B: A,A2: A] :
% 0.19/0.46 ( ( ord_less @ A @ B @ A2 )
% 0.19/0.46 => ( A2 != B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_implies_not_eq
% 0.19/0.46 thf(fact_41_order_Ostrict__implies__not__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( A2 != B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_implies_not_eq
% 0.19/0.46 thf(fact_42_not__less__iff__gr__or__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 0.19/0.46 = ( ( ord_less @ A @ Y2 @ X2 )
% 0.19/0.46 | ( X2 = Y2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_less_iff_gr_or_eq
% 0.19/0.46 thf(fact_43_dual__order_Ostrict__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B: A,A2: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ B @ A2 )
% 0.19/0.46 => ( ( ord_less @ A @ C @ B )
% 0.19/0.46 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_trans
% 0.19/0.46 thf(fact_44_linorder__less__wlog,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > A > $o,A2: A,B: A] :
% 0.19/0.46 ( ! [A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( ! [A4: A] : ( P @ A4 @ A4 )
% 0.19/0.46 => ( ! [A4: A,B3: A] :
% 0.19/0.46 ( ( P @ B3 @ A4 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( P @ A2 @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_less_wlog
% 0.19/0.46 thf(fact_45_ext,axiom,
% 0.19/0.46 ! [B4: $tType,A: $tType,F: A > B4,G: A > B4] :
% 0.19/0.46 ( ! [X4: A] :
% 0.19/0.46 ( ( F @ X4 )
% 0.19/0.46 = ( G @ X4 ) )
% 0.19/0.46 => ( F = G ) ) ).
% 0.19/0.46
% 0.19/0.46 % ext
% 0.19/0.46 thf(fact_46_exists__least__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( wellorder @ A )
% 0.19/0.46 => ( ( ^ [P2: A > $o] :
% 0.19/0.46 ? [X: A] : ( P2 @ X ) )
% 0.19/0.46 = ( ^ [P3: A > $o] :
% 0.19/0.46 ? [N2: A] :
% 0.19/0.46 ( ( P3 @ N2 )
% 0.19/0.46 & ! [M: A] :
% 0.19/0.46 ( ( ord_less @ A @ M @ N2 )
% 0.19/0.46 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % exists_least_iff
% 0.19/0.46 thf(fact_47_less__imp__not__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_less
% 0.19/0.46 thf(fact_48_order_Ostrict__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( ( ord_less @ A @ B @ C )
% 0.19/0.46 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_trans
% 0.19/0.46 thf(fact_49_dual__order_Oirrefl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.irrefl
% 0.19/0.46 thf(fact_50_linorder__cases,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( X2 != Y2 )
% 0.19/0.46 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_cases
% 0.19/0.46 thf(fact_51_less__imp__triv,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A,P: $o] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( ord_less @ A @ Y2 @ X2 )
% 0.19/0.46 => P ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_triv
% 0.19/0.46 thf(fact_52_less__imp__not__eq2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( Y2 != X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_eq2
% 0.19/0.46 thf(fact_53_antisym__conv3,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [Y2: A,X2: A] :
% 0.19/0.46 ( ~ ( ord_less @ A @ Y2 @ X2 )
% 0.19/0.46 => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 0.19/0.46 = ( X2 = Y2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv3
% 0.19/0.46 thf(fact_54_less__induct,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( wellorder @ A )
% 0.19/0.46 => ! [P: A > $o,A2: A] :
% 0.19/0.46 ( ! [X4: A] :
% 0.19/0.46 ( ! [Y5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Y5 @ X4 )
% 0.19/0.46 => ( P @ Y5 ) )
% 0.19/0.46 => ( P @ X4 ) )
% 0.19/0.46 => ( P @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_induct
% 0.19/0.46 thf(fact_55_less__not__sym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_sym
% 0.19/0.46 thf(fact_56_less__imp__not__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( X2 != Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_eq
% 0.19/0.46 thf(fact_57_dual__order_Oasym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B: A,A2: A] :
% 0.19/0.46 ( ( ord_less @ A @ B @ A2 )
% 0.19/0.46 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.asym
% 0.19/0.46 thf(fact_58_ord__less__eq__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ord @ A )
% 0.19/0.46 => ! [A2: A,B: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_trans
% 0.19/0.46 thf(fact_59_ord__eq__less__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ord @ A )
% 0.19/0.46 => ! [A2: A,B: A,C: A] :
% 0.19/0.46 ( ( A2 = B )
% 0.19/0.46 => ( ( ord_less @ A @ B @ C )
% 0.19/0.46 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_trans
% 0.19/0.46 thf(fact_60_less__irrefl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A] :
% 0.19/0.46 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_irrefl
% 0.19/0.46 thf(fact_61_less__linear,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 | ( X2 = Y2 )
% 0.19/0.46 | ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_linear
% 0.19/0.46 thf(fact_62_less__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A,Z2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.19/0.46 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_trans
% 0.19/0.46 thf(fact_63_less__asym_H,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [A2: A,B: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym'
% 0.19/0.46 thf(fact_64_less__asym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym
% 0.19/0.46 thf(fact_65_less__imp__neq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( X2 != Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_neq
% 0.19/0.46 thf(fact_66_dense,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_order @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ? [Z3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Z3 )
% 0.19/0.46 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense
% 0.19/0.46 thf(fact_67_order_Oasym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.asym
% 0.19/0.46 thf(fact_68_neq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( X2 != Y2 )
% 0.19/0.46 = ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 | ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq_iff
% 0.19/0.46 thf(fact_69_neqE,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( X2 != Y2 )
% 0.19/0.46 => ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neqE
% 0.19/0.46 thf(fact_70_gt__ex,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( no_top @ A )
% 0.19/0.46 => ! [X2: A] :
% 0.19/0.46 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 0.19/0.46
% 0.19/0.46 % gt_ex
% 0.19/0.46 thf(fact_71_lt__ex,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( no_bot @ A )
% 0.19/0.46 => ! [X2: A] :
% 0.19/0.46 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % lt_ex
% 0.19/0.46 thf(fact_72_order__less__subst2,axiom,
% 0.19/0.46 ! [A: $tType,C2: $tType] :
% 0.19/0.46 ( ( ( order @ C2 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X4: A,Y4: A] :
% 0.19/0.46 ( ( ord_less @ A @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst2
% 0.19/0.46 thf(fact_73_order__less__subst1,axiom,
% 0.19/0.46 ! [A: $tType,B4: $tType] :
% 0.19/0.46 ( ( ( order @ B4 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less @ B4 @ B @ C )
% 0.19/0.46 => ( ! [X4: B4,Y4: B4] :
% 0.19/0.46 ( ( ord_less @ B4 @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst1
% 0.19/0.46 thf(fact_74_ord__less__eq__subst,axiom,
% 0.19/0.46 ! [A: $tType,B4: $tType] :
% 0.19/0.46 ( ( ( ord @ B4 )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A2: A,B: A,F: A > B4,C: B4] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X4: A,Y4: A] :
% 0.19/0.46 ( ( ord_less @ A @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less @ B4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_subst
% 0.19/0.46 thf(fact_75_ord__eq__less__subst,axiom,
% 0.19/0.46 ! [A: $tType,B4: $tType] :
% 0.19/0.46 ( ( ( ord @ B4 )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 0.19/0.46 ( ( A2
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less @ B4 @ B @ C )
% 0.19/0.46 => ( ! [X4: B4,Y4: B4] :
% 0.19/0.46 ( ( ord_less @ B4 @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_subst
% 0.19/0.46 thf(fact_76_measure__induct__rule,axiom,
% 0.19/0.46 ! [B4: $tType,A: $tType] :
% 0.19/0.46 ( ( wellorder @ B4 )
% 0.19/0.46 => ! [F: A > B4,P: A > $o,A2: A] :
% 0.19/0.46 ( ! [X4: A] :
% 0.19/0.46 ( ! [Y5: A] :
% 0.19/0.46 ( ( ord_less @ B4 @ ( F @ Y5 ) @ ( F @ X4 ) )
% 0.19/0.46 => ( P @ Y5 ) )
% 0.19/0.46 => ( P @ X4 ) )
% 0.19/0.46 => ( P @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % measure_induct_rule
% 0.19/0.46 thf(fact_77_measure__induct,axiom,
% 0.19/0.46 ! [B4: $tType,A: $tType] :
% 0.19/0.46 ( ( wellorder @ B4 )
% 0.19/0.46 => ! [F: A > B4,P: A > $o,A2: A] :
% 0.19/0.46 ( ! [X4: A] :
% 0.19/0.46 ( ! [Y5: A] :
% 0.19/0.46 ( ( ord_less @ B4 @ ( F @ Y5 ) @ ( F @ X4 ) )
% 0.19/0.46 => ( P @ Y5 ) )
% 0.19/0.46 => ( P @ X4 ) )
% 0.19/0.46 => ( P @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % measure_induct
% 0.19/0.46 thf(fact_78_infinite__descent__measure,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 0.19/0.46 ( ! [X4: A] :
% 0.19/0.46 ( ~ ( P @ X4 )
% 0.19/0.46 => ? [Y5: A] :
% 0.19/0.46 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X4 ) )
% 0.19/0.46 & ~ ( P @ Y5 ) ) )
% 0.19/0.46 => ( P @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % infinite_descent_measure
% 0.19/0.46 thf(fact_79_linorder__neqE__nat,axiom,
% 0.19/0.46 ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( X2 != Y2 )
% 0.19/0.46 => ( ~ ( ord_less @ nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_neqE_nat
% 0.19/0.46 thf(fact_80_infinite__descent,axiom,
% 0.19/0.46 ! [P: nat > $o,N: nat] :
% 0.19/0.46 ( ! [N3: nat] :
% 0.19/0.46 ( ~ ( P @ N3 )
% 0.19/0.46 => ? [M3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M3 @ N3 )
% 0.19/0.46 & ~ ( P @ M3 ) ) )
% 0.19/0.46 => ( P @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % infinite_descent
% 0.19/0.46 thf(fact_81_nat__less__induct,axiom,
% 0.19/0.46 ! [P: nat > $o,N: nat] :
% 0.19/0.46 ( ! [N3: nat] :
% 0.19/0.46 ( ! [M3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M3 @ N3 )
% 0.19/0.46 => ( P @ M3 ) )
% 0.19/0.46 => ( P @ N3 ) )
% 0.19/0.46 => ( P @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_less_induct
% 0.19/0.46 thf(fact_82_less__irrefl__nat,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % less_irrefl_nat
% 0.19/0.46 thf(fact_83_less__not__refl3,axiom,
% 0.19/0.46 ! [S: nat,T: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ S @ T )
% 0.19/0.46 => ( S != T ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl3
% 0.19/0.46 thf(fact_84_less__not__refl2,axiom,
% 0.19/0.46 ! [N: nat,M2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ N @ M2 )
% 0.19/0.46 => ( M2 != N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl2
% 0.19/0.46 thf(fact_85_less__not__refl,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl
% 0.19/0.46 thf(fact_86_nat__neq__iff,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( M2 != N )
% 0.19/0.46 = ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.46 | ( ord_less @ nat @ N @ M2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_neq_iff
% 0.19/0.46 thf(fact_87_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.46 ! [P: nat > $o,K: nat,B: nat] :
% 0.19/0.46 ( ( P @ K )
% 0.19/0.46 => ( ! [Y4: nat] :
% 0.19/0.46 ( ( P @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ nat @ Y4 @ B ) )
% 0.19/0.46 => ? [X4: nat] :
% 0.19/0.46 ( ( P @ X4 )
% 0.19/0.46 & ! [Y5: nat] :
% 0.19/0.46 ( ( P @ Y5 )
% 0.19/0.46 => ( ord_less_eq @ nat @ Y5 @ X4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.ex_has_greatest_nat
% 0.19/0.46 thf(fact_88_nat__le__linear,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 | ( ord_less_eq @ nat @ N @ M2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_le_linear
% 0.19/0.46 thf(fact_89_le__antisym,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ N @ M2 )
% 0.19/0.46 => ( M2 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_antisym
% 0.19/0.46 thf(fact_90_eq__imp__le,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( M2 = N )
% 0.19/0.46 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_imp_le
% 0.19/0.46 thf(fact_91_le__trans,axiom,
% 0.19/0.46 ! [I3: nat,J2: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I3 @ J2 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ J2 @ K )
% 0.19/0.46 => ( ord_less_eq @ nat @ I3 @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_trans
% 0.19/0.46 thf(fact_92_le__refl,axiom,
% 0.19/0.46 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % le_refl
% 0.19/0.46 thf(fact_93_size__neq__size__imp__neq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( size @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ( size_size @ A @ X2 )
% 0.19/0.46 != ( size_size @ A @ Y2 ) )
% 0.19/0.46 => ( X2 != Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % size_neq_size_imp_neq
% 0.19/0.46 thf(fact_94_neq__if__length__neq,axiom,
% 0.19/0.46 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.19/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.19/0.46 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.19/0.46 => ( Xs != Ys ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq_if_length_neq
% 0.19/0.46 thf(fact_95_Ex__list__of__length,axiom,
% 0.19/0.46 ! [A: $tType,N: nat] :
% 0.19/0.46 ? [Xs3: list @ A] :
% 0.19/0.46 ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.19/0.46 = N ) ).
% 0.19/0.46
% 0.19/0.46 % Ex_list_of_length
% 0.19/0.46 thf(fact_96_list__desc__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ( ( list_desc @ A )
% 0.19/0.46 = ( ^ [Xs2: list @ A] :
% 0.19/0.46 ! [J: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_desc_trans
% 0.19/0.46 thf(fact_97_order_Onot__eq__order__implies__strict,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A] :
% 0.19/0.46 ( ( A2 != B )
% 0.19/0.46 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.46 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.not_eq_order_implies_strict
% 0.19/0.46 thf(fact_98_dual__order_Ostrict__implies__order,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B: A,A2: A] :
% 0.19/0.46 ( ( ord_less @ A @ B @ A2 )
% 0.19/0.46 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_implies_order
% 0.19/0.46 thf(fact_99_dual__order_Ostrict__iff__order,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [B2: A,A3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.19/0.46 & ( A3 != B2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_iff_order
% 0.19/0.46 thf(fact_100_dual__order_Oorder__iff__strict,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less_eq @ A )
% 0.19/0.46 = ( ^ [B2: A,A3: A] :
% 0.19/0.46 ( ( ord_less @ A @ B2 @ A3 )
% 0.19/0.46 | ( A3 = B2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.order_iff_strict
% 0.19/0.46 thf(fact_101_order_Ostrict__implies__order,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_implies_order
% 0.19/0.46 thf(fact_102_dense__le__bounded,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A,Z2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ! [W: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ W )
% 0.19/0.46 => ( ( ord_less @ A @ W @ Y2 )
% 0.19/0.46 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense_le_bounded
% 0.19/0.46 thf(fact_103_dense__ge__bounded,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_linorder @ A )
% 0.19/0.46 => ! [Z2: A,X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X2 )
% 0.19/0.46 => ( ! [W: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ W )
% 0.19/0.46 => ( ( ord_less @ A @ W @ X2 )
% 0.19/0.46 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense_ge_bounded
% 0.19/0.46 thf(fact_104_dual__order_Ostrict__trans2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B: A,A2: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ B @ A2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ C @ B )
% 0.19/0.46 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_trans2
% 0.19/0.46 thf(fact_105_dual__order_Ostrict__trans1,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B: A,A2: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B @ A2 )
% 0.19/0.46 => ( ( ord_less @ A @ C @ B )
% 0.19/0.46 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_trans1
% 0.19/0.46 thf(fact_106_order_Ostrict__iff__order,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [A3: A,B2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.46 & ( A3 != B2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_iff_order
% 0.19/0.46 thf(fact_107_order_Oorder__iff__strict,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less_eq @ A )
% 0.19/0.46 = ( ^ [A3: A,B2: A] :
% 0.19/0.46 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.46 | ( A3 = B2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.order_iff_strict
% 0.19/0.46 thf(fact_108_order_Ostrict__trans2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B @ C )
% 0.19/0.46 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_trans2
% 0.19/0.46 thf(fact_109_order_Ostrict__trans1,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.46 => ( ( ord_less @ A @ B @ C )
% 0.19/0.46 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_trans1
% 0.19/0.46 thf(fact_110_not__le__imp__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [Y2: A,X2: A] :
% 0.19/0.46 ( ~ ( ord_less_eq @ A @ Y2 @ X2 )
% 0.19/0.46 => ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_le_imp_less
% 0.19/0.46 thf(fact_111_less__le__not__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [X3: A,Y3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.46 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_le_not_le
% 0.19/0.46 thf(fact_112_le__imp__less__or__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 | ( X2 = Y2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_imp_less_or_eq
% 0.19/0.46 thf(fact_113_le__less__linear,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.46 | ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_less_linear
% 0.19/0.46 thf(fact_114_dense__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_linorder @ A )
% 0.19/0.46 => ! [Y2: A,Z2: A] :
% 0.19/0.46 ( ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ X4 @ Y2 )
% 0.19/0.46 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.19/0.46 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense_le
% 0.19/0.46 thf(fact_115_dense__ge,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_linorder @ A )
% 0.19/0.46 => ! [Z2: A,Y2: A] :
% 0.19/0.46 ( ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X4 )
% 0.19/0.46 => ( ord_less_eq @ A @ Y2 @ X4 ) )
% 0.19/0.46 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense_ge
% 0.19/0.46 thf(fact_116_less__le__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A,Z2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.19/0.46 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_le_trans
% 0.19/0.46 thf(fact_117_le__less__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A,Z2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.19/0.46 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_less_trans
% 0.19/0.46 thf(fact_118_less__imp__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_le
% 0.19/0.46 thf(fact_119_antisym__conv2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 0.19/0.46 = ( X2 = Y2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv2
% 0.19/0.46 thf(fact_120_antisym__conv1,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.46 = ( X2 = Y2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv1
% 0.19/0.46 thf(fact_121_le__neq__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.46 => ( ( A2 != B )
% 0.19/0.46 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_neq_trans
% 0.19/0.46 thf(fact_122_not__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 0.19/0.46 = ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_less
% 0.19/0.46 thf(fact_123_not__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ( ~ ( ord_less_eq @ A @ X2 @ Y2 ) )
% 0.19/0.46 = ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_le
% 0.19/0.46 thf(fact_124_order__less__le__subst2,axiom,
% 0.19/0.46 ! [A: $tType,C2: $tType] :
% 0.19/0.46 ( ( ( order @ C2 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X4: A,Y4: A] :
% 0.19/0.46 ( ( ord_less @ A @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_le_subst2
% 0.19/0.46 thf(fact_125_order__less__le__subst1,axiom,
% 0.19/0.46 ! [A: $tType,B4: $tType] :
% 0.19/0.46 ( ( ( order @ B4 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq @ B4 @ B @ C )
% 0.19/0.46 => ( ! [X4: B4,Y4: B4] :
% 0.19/0.46 ( ( ord_less_eq @ B4 @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_le_subst1
% 0.19/0.46 thf(fact_126_order__le__less__subst2,axiom,
% 0.19/0.46 ! [A: $tType,C2: $tType] :
% 0.19/0.46 ( ( ( order @ C2 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.46 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X4: A,Y4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_le_less_subst2
% 0.19/0.46 thf(fact_127_order__le__less__subst1,axiom,
% 0.19/0.46 ! [A: $tType,B4: $tType] :
% 0.19/0.46 ( ( ( order @ B4 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less @ B4 @ B @ C )
% 0.19/0.46 => ( ! [X4: B4,Y4: B4] :
% 0.19/0.46 ( ( ord_less @ B4 @ X4 @ Y4 )
% 0.19/0.46 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_le_less_subst1
% 0.19/0.46 thf(fact_128_less__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [X3: A,Y3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.46 & ( X3 != Y3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_le
% 0.19/0.46 thf(fact_129_le__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less_eq @ A )
% 0.19/0.46 = ( ^ [X3: A,Y3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X3 @ Y3 )
% 0.19/0.46 | ( X3 = Y3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_less
% 0.19/0.46 thf(fact_130_leI,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y2: A] :
% 0.19/0.46 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % leI
% 0.19/0.46 thf(fact_131_leD,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [Y2: A,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.19/0.46 => ~ ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % leD
% 0.19/0.46 thf(fact_132_list__strict__asc__trans__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [Xs: list @ A] :
% 0.19/0.46 ( ( list_strict_asc @ A @ Xs )
% 0.19/0.46 => ! [J4: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.46 => ! [I4: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I4 @ J4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_strict_asc_trans_le
% 0.19/0.46 thf(fact_133_list__asc__trans__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ( ( list_asc @ A )
% 0.19/0.46 = ( ^ [Xs2: list @ A] :
% 0.19/0.46 ! [J: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ! [I2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_asc_trans_le
% 0.19/0.46 thf(fact_134_list__strict__desc__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ( ( list_strict_desc @ A )
% 0.19/0.46 = ( ^ [Xs2: list @ A] :
% 0.19/0.46 ! [J: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.19/0.46 => ( ord_less @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_strict_desc_trans
% 0.19/0.46 thf(fact_135_list__strict__asc__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ( ( list_strict_asc @ A )
% 0.19/0.46 = ( ^ [Xs2: list @ A] :
% 0.19/0.46 ! [J: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.19/0.46 => ( ord_less @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_strict_asc_trans
% 0.19/0.46 thf(fact_136_list__asc__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ( ( list_asc @ A )
% 0.19/0.46 = ( ^ [Xs2: list @ A] :
% 0.19/0.46 ! [J: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_asc_trans
% 0.19/0.46 thf(fact_137_Lattices__Big_Oex__has__greatest__nat,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o,K: A,F: A > nat,B: nat] :
% 0.19/0.46 ( ( P @ K )
% 0.19/0.46 => ( ! [Y4: A] :
% 0.19/0.46 ( ( P @ Y4 )
% 0.19/0.46 => ( ord_less @ nat @ ( F @ Y4 ) @ B ) )
% 0.19/0.46 => ? [X4: A] :
% 0.19/0.46 ( ( P @ X4 )
% 0.19/0.46 & ! [Y5: A] :
% 0.19/0.46 ( ( P @ Y5 )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( F @ Y5 ) @ ( F @ X4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lattices_Big.ex_has_greatest_nat
% 0.19/0.46 thf(fact_138_nat__descend__induct,axiom,
% 0.19/0.46 ! [N: nat,P: nat > $o,M2: nat] :
% 0.19/0.46 ( ! [K2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ N @ K2 )
% 0.19/0.46 => ( P @ K2 ) )
% 0.19/0.46 => ( ! [K2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.19/0.46 => ( ! [I4: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ K2 @ I4 )
% 0.19/0.46 => ( P @ I4 ) )
% 0.19/0.46 => ( P @ K2 ) ) )
% 0.19/0.46 => ( P @ M2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_descend_induct
% 0.19/0.46 thf(fact_139_complete__interval,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( condit1037483654norder @ A )
% 0.19/0.46 => ! [A2: A,B: A,P: A > $o] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.19/0.46 => ( ( P @ A2 )
% 0.19/0.46 => ( ~ ( P @ B )
% 0.19/0.46 => ? [C3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.19/0.46 & ( ord_less_eq @ A @ C3 @ B )
% 0.19/0.46 & ! [X5: A] :
% 0.19/0.46 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 0.19/0.46 & ( ord_less @ A @ X5 @ C3 ) )
% 0.19/0.46 => ( P @ X5 ) )
% 0.19/0.46 & ! [D: A] :
% 0.19/0.46 ( ! [X4: A] :
% 0.19/0.46 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 0.19/0.46 & ( ord_less @ A @ X4 @ D ) )
% 0.19/0.46 => ( P @ X4 ) )
% 0.19/0.46 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % complete_interval
% 0.19/0.46 thf(fact_140_le__greater__neq__conv,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ( ( ord_less_eq @ A )
% 0.19/0.46 = ( ^ [N2: A,A3: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ A3 @ X3 )
% 0.19/0.46 => ( N2 != X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_greater_neq_conv
% 0.19/0.46 thf(fact_141_greater__le__neq__conv,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [A3: A,N2: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X3 @ A3 )
% 0.19/0.46 => ( N2 != X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % greater_le_neq_conv
% 0.19/0.46 thf(fact_142_less__ge__neq__conv,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [N2: A,A3: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A3 @ X3 )
% 0.19/0.46 => ( N2 != X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_ge_neq_conv
% 0.19/0.46 thf(fact_143_less__fun__def,axiom,
% 0.19/0.46 ! [B4: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B4 )
% 0.19/0.46 => ( ( ord_less @ ( A > B4 ) )
% 0.19/0.46 = ( ^ [F2: A > B4,G2: A > B4] :
% 0.19/0.46 ( ( ord_less_eq @ ( A > B4 ) @ F2 @ G2 )
% 0.19/0.46 & ~ ( ord_less_eq @ ( A > B4 ) @ G2 @ F2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_fun_def
% 0.19/0.46 thf(fact_144_list__strict__asc__imp__list__asc,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [Xs: list @ A] :
% 0.19/0.46 ( ( list_strict_asc @ A @ Xs )
% 0.19/0.46 => ( list_asc @ A @ Xs ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_strict_asc_imp_list_asc
% 0.19/0.46 thf(fact_145_ex__gt__or__lt,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( condit1656338222tinuum @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ? [B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B3 )
% 0.19/0.46 | ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ex_gt_or_lt
% 0.19/0.46 thf(fact_146_ex__has__least__nat,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.19/0.46 ( ( P @ K )
% 0.19/0.46 => ? [X4: A] :
% 0.19/0.46 ( ( P @ X4 )
% 0.19/0.46 & ! [Y5: A] :
% 0.19/0.46 ( ( P @ Y5 )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( M2 @ X4 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ex_has_least_nat
% 0.19/0.46 thf(fact_147_list__strict__desc__imp__list__desc,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [Xs: list @ A] :
% 0.19/0.46 ( ( list_strict_desc @ A @ Xs )
% 0.19/0.46 => ( list_desc @ A @ Xs ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_strict_desc_imp_list_desc
% 0.19/0.46 thf(fact_148_ge__less__neq__conv,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ( ( ord_less_eq @ A )
% 0.19/0.46 = ( ^ [A3: A,N2: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X3 @ A3 )
% 0.19/0.46 => ( N2 != X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ge_less_neq_conv
% 0.19/0.46 thf(fact_149_minf_I8_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(8)
% 0.19/0.46 thf(fact_150_minf_I6_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z3 )
% 0.19/0.46 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(6)
% 0.19/0.46 thf(fact_151_pinf_I8_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z3 @ X5 )
% 0.19/0.46 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(8)
% 0.19/0.46 thf(fact_152_pinf_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.19/0.46 ( ? [Z4: A] :
% 0.19/0.46 ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z4 @ X4 )
% 0.19/0.46 => ( ( P @ X4 )
% 0.19/0.46 = ( P4 @ X4 ) ) )
% 0.19/0.46 => ( ? [Z4: A] :
% 0.19/0.46 ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z4 @ X4 )
% 0.19/0.46 => ( ( Q @ X4 )
% 0.19/0.46 = ( Q2 @ X4 ) ) )
% 0.19/0.46 => ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z3 @ X5 )
% 0.19/0.46 => ( ( ( P @ X5 )
% 0.19/0.46 & ( Q @ X5 ) )
% 0.19/0.46 = ( ( P4 @ X5 )
% 0.19/0.46 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(1)
% 0.19/0.46 thf(fact_153_pinf_I2_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.19/0.46 ( ? [Z4: A] :
% 0.19/0.46 ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z4 @ X4 )
% 0.19/0.46 => ( ( P @ X4 )
% 0.19/0.46 = ( P4 @ X4 ) ) )
% 0.19/0.46 => ( ? [Z4: A] :
% 0.19/0.46 ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z4 @ X4 )
% 0.19/0.46 => ( ( Q @ X4 )
% 0.19/0.46 = ( Q2 @ X4 ) ) )
% 0.19/0.46 => ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z3 @ X5 )
% 0.19/0.46 => ( ( ( P @ X5 )
% 0.19/0.46 | ( Q @ X5 ) )
% 0.19/0.46 = ( ( P4 @ X5 )
% 0.19/0.46 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(2)
% 0.19/0.46 thf(fact_154_pinf_I3_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z3 @ X5 )
% 0.19/0.46 => ( X5 != T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(3)
% 0.19/0.46 thf(fact_155_pinf_I4_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z3 @ X5 )
% 0.19/0.46 => ( X5 != T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(4)
% 0.19/0.46 thf(fact_156_pinf_I5_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z3 @ X5 )
% 0.19/0.46 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(5)
% 0.19/0.46 thf(fact_157_pinf_I7_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z3 @ X5 )
% 0.19/0.46 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(7)
% 0.19/0.46 thf(fact_158_pinf_I11_J,axiom,
% 0.19/0.46 ! [C2: $tType,D2: $tType] :
% 0.19/0.46 ( ( ord @ C2 )
% 0.19/0.46 => ! [F3: D2] :
% 0.19/0.46 ? [Z3: C2] :
% 0.19/0.46 ! [X5: C2] :
% 0.19/0.46 ( ( ord_less @ C2 @ Z3 @ X5 )
% 0.19/0.46 => ( F3 = F3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(11)
% 0.19/0.46 thf(fact_159_minf_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.19/0.46 ( ? [Z4: A] :
% 0.19/0.46 ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ X4 @ Z4 )
% 0.19/0.46 => ( ( P @ X4 )
% 0.19/0.46 = ( P4 @ X4 ) ) )
% 0.19/0.46 => ( ? [Z4: A] :
% 0.19/0.46 ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ X4 @ Z4 )
% 0.19/0.46 => ( ( Q @ X4 )
% 0.19/0.46 = ( Q2 @ X4 ) ) )
% 0.19/0.46 => ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z3 )
% 0.19/0.46 => ( ( ( P @ X5 )
% 0.19/0.46 & ( Q @ X5 ) )
% 0.19/0.46 = ( ( P4 @ X5 )
% 0.19/0.46 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(1)
% 0.19/0.46 thf(fact_160_minf_I2_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.19/0.46 ( ? [Z4: A] :
% 0.19/0.46 ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ X4 @ Z4 )
% 0.19/0.46 => ( ( P @ X4 )
% 0.19/0.46 = ( P4 @ X4 ) ) )
% 0.19/0.46 => ( ? [Z4: A] :
% 0.19/0.46 ! [X4: A] :
% 0.19/0.46 ( ( ord_less @ A @ X4 @ Z4 )
% 0.19/0.46 => ( ( Q @ X4 )
% 0.19/0.46 = ( Q2 @ X4 ) ) )
% 0.19/0.46 => ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z3 )
% 0.19/0.46 => ( ( ( P @ X5 )
% 0.19/0.46 | ( Q @ X5 ) )
% 0.19/0.46 = ( ( P4 @ X5 )
% 0.19/0.46 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(2)
% 0.19/0.46 thf(fact_161_minf_I3_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z3 )
% 0.19/0.46 => ( X5 != T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(3)
% 0.19/0.46 thf(fact_162_minf_I4_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z3 )
% 0.19/0.46 => ( X5 != T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(4)
% 0.19/0.46 thf(fact_163_minf_I5_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z3 )
% 0.19/0.46 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(5)
% 0.19/0.46 thf(fact_164_minf_I7_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z3 )
% 0.19/0.46 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(7)
% 0.19/0.46 thf(fact_165_minf_I11_J,axiom,
% 0.19/0.46 ! [C2: $tType,D2: $tType] :
% 0.19/0.46 ( ( ord @ C2 )
% 0.19/0.46 => ! [F3: D2] :
% 0.19/0.46 ? [Z3: C2] :
% 0.19/0.46 ! [X5: C2] :
% 0.19/0.46 ( ( ord_less @ C2 @ X5 @ Z3 )
% 0.19/0.46 => ( F3 = F3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(11)
% 0.19/0.46 thf(fact_166_pinf_I6_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z3: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z3 @ X5 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(6)
% 0.19/0.46 thf(fact_167_verit__comp__simplify1_I3_J,axiom,
% 0.19/0.46 ! [B4: $tType] :
% 0.19/0.46 ( ( linorder @ B4 )
% 0.19/0.46 => ! [B5: B4,A5: B4] :
% 0.19/0.46 ( ( ~ ( ord_less_eq @ B4 @ B5 @ A5 ) )
% 0.19/0.46 = ( ord_less @ B4 @ A5 @ B5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_comp_simplify1(3)
% 0.19/0.46 thf(fact_168_sorted__iff__nth__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ( ( linorder_sorted @ A )
% 0.19/0.46 = ( ^ [Xs2: list @ A] :
% 0.19/0.46 ! [I2: nat,J: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.19/0.46 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % sorted_iff_nth_mono
% 0.19/0.46 thf(fact_169_sorted__nth__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [Xs: list @ A,I3: nat,J2: nat] :
% 0.19/0.46 ( ( linorder_sorted @ A @ Xs )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ I3 @ J2 )
% 0.19/0.46 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % sorted_nth_mono
% 0.19/0.46 thf(fact_170_list__ord__le__sorted__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ( ( list_asc @ A )
% 0.19/0.46 = ( linorder_sorted @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_ord_le_sorted_eq
% 0.19/0.46 thf(fact_171_verit__la__disequality,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [A2: A,B: A] :
% 0.19/0.46 ( ( A2 = B )
% 0.19/0.46 | ~ ( ord_less_eq @ A @ A2 @ B )
% 0.19/0.46 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_la_disequality
% 0.19/0.46 thf(fact_172_verit__comp__simplify1_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_comp_simplify1(1)
% 0.19/0.46 thf(fact_173_sorted__iff__nth__mono__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ( ( linorder_sorted @ A )
% 0.19/0.46 = ( ^ [Xs2: list @ A] :
% 0.19/0.46 ! [I2: nat,J: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.19/0.46 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % sorted_iff_nth_mono_less
% 0.19/0.46 thf(fact_174_sorted__rev__nth__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [Xs: list @ A,I3: nat,J2: nat] :
% 0.19/0.46 ( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ I3 @ J2 )
% 0.19/0.46 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J2 ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % sorted_rev_nth_mono
% 0.19/0.46 thf(fact_175_sorted__rev__iff__nth__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [Xs: list @ A] :
% 0.19/0.46 ( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
% 0.19/0.46 = ( ! [I2: nat,J: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.19/0.46 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % sorted_rev_iff_nth_mono
% 0.19/0.46 thf(fact_176_sorted__iff__nth__Suc,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ( ( linorder_sorted @ A )
% 0.19/0.46 = ( ^ [Xs2: list @ A] :
% 0.19/0.46 ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % sorted_iff_nth_Suc
% 0.19/0.46 thf(fact_177_old_Onat_Oinject,axiom,
% 0.19/0.46 ! [Nat: nat,Nat2: nat] :
% 0.19/0.46 ( ( ( suc @ Nat )
% 0.19/0.46 = ( suc @ Nat2 ) )
% 0.19/0.46 = ( Nat = Nat2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % old.nat.inject
% 0.19/0.46 thf(fact_178_nat_Oinject,axiom,
% 0.19/0.46 ! [X22: nat,Y22: nat] :
% 0.19/0.46 ( ( ( suc @ X22 )
% 0.19/0.46 = ( suc @ Y22 ) )
% 0.19/0.46 = ( X22 = Y22 ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat.inject
% 0.19/0.46 thf(fact_179_rev__is__rev__conv,axiom,
% 0.19/0.46 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.19/0.46 ( ( ( rev @ A @ Xs )
% 0.19/0.46 = ( rev @ A @ Ys ) )
% 0.19/0.46 = ( Xs = Ys ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_is_rev_conv
% 0.19/0.46 thf(fact_180_rev__rev__ident,axiom,
% 0.19/0.46 ! [A: $tType,Xs: list @ A] :
% 0.19/0.46 ( ( rev @ A @ ( rev @ A @ Xs ) )
% 0.19/0.46 = Xs ) ).
% 0.19/0.46
% 0.19/0.46 % rev_rev_ident
% 0.19/0.46 thf(fact_181_Suc__less__eq,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
% 0.19/0.46 = ( ord_less @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_less_eq
% 0.19/0.46 thf(fact_182_Suc__mono,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.46 => ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_mono
% 0.19/0.46 thf(fact_183_lessI,axiom,
% 0.19/0.46 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % lessI
% 0.19/0.46 thf(fact_184_Suc__le__mono,axiom,
% 0.19/0.46 ! [N: nat,M2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M2 ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ N @ M2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_le_mono
% 0.19/0.46 thf(fact_185_length__rev,axiom,
% 0.19/0.46 ! [A: $tType,Xs: list @ A] :
% 0.19/0.46 ( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
% 0.19/0.46 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.19/0.46
% 0.19/0.46 % length_rev
% 0.19/0.46 thf(fact_186_Suc__leD,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_leD
% 0.19/0.46 thf(fact_187_le__SucE,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.46 => ( ~ ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( M2
% 0.19/0.46 = ( suc @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_SucE
% 0.19/0.46 thf(fact_188_le__SucI,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ M2 @ ( suc @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_SucI
% 0.19/0.46 thf(fact_189_Suc__le__D,axiom,
% 0.19/0.46 ! [N: nat,M4: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
% 0.19/0.46 => ? [M5: nat] :
% 0.19/0.46 ( M4
% 0.19/0.46 = ( suc @ M5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_le_D
% 0.19/0.46 thf(fact_190_le__Suc__eq,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.46 = ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 | ( M2
% 0.19/0.46 = ( suc @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_Suc_eq
% 0.19/0.46 thf(fact_191_Suc__n__not__le__n,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_n_not_le_n
% 0.19/0.46 thf(fact_192_not__less__eq__eq,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ~ ( ord_less_eq @ nat @ M2 @ N ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ ( suc @ N ) @ M2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_less_eq_eq
% 0.19/0.46 thf(fact_193_full__nat__induct,axiom,
% 0.19/0.46 ! [P: nat > $o,N: nat] :
% 0.19/0.46 ( ! [N3: nat] :
% 0.19/0.46 ( ! [M3: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
% 0.19/0.46 => ( P @ M3 ) )
% 0.19/0.46 => ( P @ N3 ) )
% 0.19/0.46 => ( P @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % full_nat_induct
% 0.19/0.46 thf(fact_194_nat__induct__at__least,axiom,
% 0.19/0.46 ! [M2: nat,N: nat,P: nat > $o] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( ( P @ M2 )
% 0.19/0.46 => ( ! [N3: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N3 )
% 0.19/0.46 => ( ( P @ N3 )
% 0.19/0.46 => ( P @ ( suc @ N3 ) ) ) )
% 0.19/0.46 => ( P @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_induct_at_least
% 0.19/0.46 thf(fact_195_transitive__stepwise__le,axiom,
% 0.19/0.46 ! [M2: nat,N: nat,R: nat > nat > $o] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( ! [X4: nat] : ( R @ X4 @ X4 )
% 0.19/0.46 => ( ! [X4: nat,Y4: nat,Z3: nat] :
% 0.19/0.46 ( ( R @ X4 @ Y4 )
% 0.19/0.46 => ( ( R @ Y4 @ Z3 )
% 0.19/0.46 => ( R @ X4 @ Z3 ) ) )
% 0.19/0.46 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 0.19/0.46 => ( R @ M2 @ N ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % transitive_stepwise_le
% 0.19/0.46 thf(fact_196_n__not__Suc__n,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( N
% 0.19/0.46 != ( suc @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % n_not_Suc_n
% 0.19/0.46 thf(fact_197_Suc__inject,axiom,
% 0.19/0.46 ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ( suc @ X2 )
% 0.19/0.46 = ( suc @ Y2 ) )
% 0.19/0.46 => ( X2 = Y2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_inject
% 0.19/0.46 thf(fact_198_rev__swap,axiom,
% 0.19/0.46 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.19/0.46 ( ( ( rev @ A @ Xs )
% 0.19/0.46 = Ys )
% 0.19/0.46 = ( Xs
% 0.19/0.46 = ( rev @ A @ Ys ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_swap
% 0.19/0.46 thf(fact_199_not__less__less__Suc__eq,axiom,
% 0.19/0.46 ! [N: nat,M2: nat] :
% 0.19/0.46 ( ~ ( ord_less @ nat @ N @ M2 )
% 0.19/0.46 => ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
% 0.19/0.46 = ( N = M2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_less_less_Suc_eq
% 0.19/0.46 thf(fact_200_strict__inc__induct,axiom,
% 0.19/0.46 ! [I3: nat,J2: nat,P: nat > $o] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ J2 )
% 0.19/0.46 => ( ! [I: nat] :
% 0.19/0.46 ( ( J2
% 0.19/0.46 = ( suc @ I ) )
% 0.19/0.46 => ( P @ I ) )
% 0.19/0.46 => ( ! [I: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I @ J2 )
% 0.19/0.46 => ( ( P @ ( suc @ I ) )
% 0.19/0.46 => ( P @ I ) ) )
% 0.19/0.46 => ( P @ I3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % strict_inc_induct
% 0.19/0.46 thf(fact_201_less__Suc__induct,axiom,
% 0.19/0.46 ! [I3: nat,J2: nat,P: nat > nat > $o] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ J2 )
% 0.19/0.46 => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
% 0.19/0.46 => ( ! [I: nat,J3: nat,K2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I @ J3 )
% 0.19/0.46 => ( ( ord_less @ nat @ J3 @ K2 )
% 0.19/0.46 => ( ( P @ I @ J3 )
% 0.19/0.46 => ( ( P @ J3 @ K2 )
% 0.19/0.46 => ( P @ I @ K2 ) ) ) ) )
% 0.19/0.46 => ( P @ I3 @ J2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_Suc_induct
% 0.19/0.46 thf(fact_202_less__trans__Suc,axiom,
% 0.19/0.46 ! [I3: nat,J2: nat,K: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ J2 )
% 0.19/0.46 => ( ( ord_less @ nat @ J2 @ K )
% 0.19/0.46 => ( ord_less @ nat @ ( suc @ I3 ) @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_trans_Suc
% 0.19/0.46 thf(fact_203_Suc__less__SucD,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
% 0.19/0.46 => ( ord_less @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_less_SucD
% 0.19/0.46 thf(fact_204_less__antisym,axiom,
% 0.19/0.46 ! [N: nat,M2: nat] :
% 0.19/0.46 ( ~ ( ord_less @ nat @ N @ M2 )
% 0.19/0.46 => ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
% 0.19/0.46 => ( M2 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_antisym
% 0.19/0.46 thf(fact_205_Suc__less__eq2,axiom,
% 0.19/0.46 ! [N: nat,M2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 0.19/0.46 = ( ? [M6: nat] :
% 0.19/0.46 ( ( M2
% 0.19/0.46 = ( suc @ M6 ) )
% 0.19/0.46 & ( ord_less @ nat @ N @ M6 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_less_eq2
% 0.19/0.46 thf(fact_206_All__less__Suc,axiom,
% 0.19/0.46 ! [N: nat,P: nat > $o] :
% 0.19/0.46 ( ( ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 0.19/0.46 => ( P @ I2 ) ) )
% 0.19/0.46 = ( ( P @ N )
% 0.19/0.46 & ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ N )
% 0.19/0.46 => ( P @ I2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % All_less_Suc
% 0.19/0.46 thf(fact_207_not__less__eq,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ~ ( ord_less @ nat @ M2 @ N ) )
% 0.19/0.46 = ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_less_eq
% 0.19/0.46 thf(fact_208_less__Suc__eq,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.46 = ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.46 | ( M2 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_Suc_eq
% 0.19/0.46 thf(fact_209_Ex__less__Suc,axiom,
% 0.19/0.46 ! [N: nat,P: nat > $o] :
% 0.19/0.46 ( ( ? [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 0.19/0.46 & ( P @ I2 ) ) )
% 0.19/0.46 = ( ( P @ N )
% 0.19/0.46 | ? [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ N )
% 0.19/0.46 & ( P @ I2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Ex_less_Suc
% 0.19/0.46 thf(fact_210_less__SucI,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.46 => ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_SucI
% 0.19/0.46 thf(fact_211_less__SucE,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.46 => ( ~ ( ord_less @ nat @ M2 @ N )
% 0.19/0.46 => ( M2 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_SucE
% 0.19/0.46 thf(fact_212_Suc__lessI,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.46 => ( ( ( suc @ M2 )
% 0.19/0.46 != N )
% 0.19/0.46 => ( ord_less @ nat @ ( suc @ M2 ) @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_lessI
% 0.19/0.46 thf(fact_213_Suc__lessE,axiom,
% 0.19/0.46 ! [I3: nat,K: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( suc @ I3 ) @ K )
% 0.19/0.46 => ~ ! [J3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ J3 )
% 0.19/0.46 => ( K
% 0.19/0.46 != ( suc @ J3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_lessE
% 0.19/0.46 thf(fact_214_Suc__lessD,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( suc @ M2 ) @ N )
% 0.19/0.46 => ( ord_less @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_lessD
% 0.19/0.46 thf(fact_215_Nat_OlessE,axiom,
% 0.19/0.46 ! [I3: nat,K: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ K )
% 0.19/0.46 => ( ( K
% 0.19/0.46 != ( suc @ I3 ) )
% 0.19/0.46 => ~ ! [J3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ J3 )
% 0.19/0.46 => ( K
% 0.19/0.46 != ( suc @ J3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.lessE
% 0.19/0.46 thf(fact_216_lift__Suc__mono__less__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [F: nat > A,N: nat,M2: nat] :
% 0.19/0.46 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.19/0.46 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M2 ) )
% 0.19/0.46 = ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lift_Suc_mono_less_iff
% 0.19/0.46 thf(fact_217_lift__Suc__mono__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [F: nat > A,N: nat,N4: nat] :
% 0.19/0.46 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.19/0.46 => ( ( ord_less @ nat @ N @ N4 )
% 0.19/0.46 => ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lift_Suc_mono_less
% 0.19/0.46 thf(fact_218_lift__Suc__antimono__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [F: nat > A,N: nat,N4: nat] :
% 0.19/0.46 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ N @ N4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lift_Suc_antimono_le
% 0.19/0.46 thf(fact_219_lift__Suc__mono__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [F: nat > A,N: nat,N4: nat] :
% 0.19/0.46 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ N @ N4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lift_Suc_mono_le
% 0.19/0.46 thf(fact_220_le__imp__less__Suc,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_imp_less_Suc
% 0.19/0.46 thf(fact_221_less__eq__Suc__le,axiom,
% 0.19/0.46 ( ( ord_less @ nat )
% 0.19/0.46 = ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eq_Suc_le
% 0.19/0.46 thf(fact_222_less__Suc__eq__le,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_Suc_eq_le
% 0.19/0.46 thf(fact_223_le__less__Suc__eq,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
% 0.19/0.46 = ( N = M2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_less_Suc_eq
% 0.19/0.46 thf(fact_224_Suc__le__lessD,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
% 0.19/0.46 => ( ord_less @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_le_lessD
% 0.19/0.46 thf(fact_225_inc__induct,axiom,
% 0.19/0.46 ! [I3: nat,J2: nat,P: nat > $o] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I3 @ J2 )
% 0.19/0.46 => ( ( P @ J2 )
% 0.19/0.46 => ( ! [N3: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I3 @ N3 )
% 0.19/0.46 => ( ( ord_less @ nat @ N3 @ J2 )
% 0.19/0.46 => ( ( P @ ( suc @ N3 ) )
% 0.19/0.46 => ( P @ N3 ) ) ) )
% 0.19/0.46 => ( P @ I3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % inc_induct
% 0.19/0.46 thf(fact_226_dec__induct,axiom,
% 0.19/0.46 ! [I3: nat,J2: nat,P: nat > $o] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I3 @ J2 )
% 0.19/0.46 => ( ( P @ I3 )
% 0.19/0.46 => ( ! [N3: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I3 @ N3 )
% 0.19/0.46 => ( ( ord_less @ nat @ N3 @ J2 )
% 0.19/0.46 => ( ( P @ N3 )
% 0.19/0.46 => ( P @ ( suc @ N3 ) ) ) ) )
% 0.19/0.46 => ( P @ J2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dec_induct
% 0.19/0.46 thf(fact_227_Suc__le__eq,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
% 0.19/0.46 = ( ord_less @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_le_eq
% 0.19/0.46 thf(fact_228_Suc__leI,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( suc @ M2 ) @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_leI
% 0.19/0.46 thf(fact_229_sorted__rev__iff__nth__Suc,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [Xs: list @ A] :
% 0.19/0.46 ( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
% 0.19/0.46 = ( ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I2 ) ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % sorted_rev_iff_nth_Suc
% 0.19/0.46 thf(fact_230_nat__induct_H,axiom,
% 0.19/0.46 ! [P: nat > $o,N0: nat,N: nat] :
% 0.19/0.46 ( ( P @ N0 )
% 0.19/0.46 => ( ! [N3: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ N0 @ N3 )
% 0.19/0.46 => ( ( P @ N3 )
% 0.19/0.46 => ( P @ ( suc @ N3 ) ) ) )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ N0 @ N )
% 0.19/0.46 => ( P @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_induct'
% 0.19/0.46 thf(fact_231_rev__nth,axiom,
% 0.19/0.46 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.19/0.46 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.46 => ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
% 0.19/0.46 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_nth
% 0.19/0.46 thf(fact_232_list__ex__length,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( list_ex @ A )
% 0.19/0.46 = ( ^ [P3: A > $o,Xs2: list @ A] :
% 0.19/0.46 ? [N2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 & ( P3 @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_ex_length
% 0.19/0.46 thf(fact_233_Suc__diff__diff,axiom,
% 0.19/0.46 ! [M2: nat,N: nat,K: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
% 0.19/0.46 = ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_diff_diff
% 0.19/0.46 thf(fact_234_diff__Suc__Suc,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
% 0.19/0.46 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_Suc_Suc
% 0.19/0.46 thf(fact_235_diff__diff__cancel,axiom,
% 0.19/0.46 ! [I3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I3 @ N )
% 0.19/0.46 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I3 ) )
% 0.19/0.46 = I3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_cancel
% 0.19/0.46 thf(fact_236_list__ex__rev,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 0.19/0.46 ( ( list_ex @ A @ P @ ( rev @ A @ Xs ) )
% 0.19/0.46 = ( list_ex @ A @ P @ Xs ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_ex_rev
% 0.19/0.46 thf(fact_237_i__append__nth2,axiom,
% 0.19/0.46 ! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 0.19/0.46 => ( ( listIn521021761append @ A @ Xs @ F @ N )
% 0.19/0.46 = ( F @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % i_append_nth2
% 0.19/0.46 thf(fact_238_zero__induct__lemma,axiom,
% 0.19/0.46 ! [P: nat > $o,K: nat,I3: nat] :
% 0.19/0.46 ( ( P @ K )
% 0.19/0.46 => ( ! [N3: nat] :
% 0.19/0.46 ( ( P @ ( suc @ N3 ) )
% 0.19/0.46 => ( P @ N3 ) )
% 0.19/0.46 => ( P @ ( minus_minus @ nat @ K @ I3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_induct_lemma
% 0.19/0.46 thf(fact_239_less__diff__imp__less,axiom,
% 0.19/0.46 ! [I3: nat,J2: nat,M2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ ( minus_minus @ nat @ J2 @ M2 ) )
% 0.19/0.46 => ( ord_less @ nat @ I3 @ J2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_diff_imp_less
% 0.19/0.46 thf(fact_240_nat__diff__left__cancel__eq1,axiom,
% 0.19/0.46 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.46 ( ( ( minus_minus @ nat @ K @ M2 )
% 0.19/0.46 = ( minus_minus @ nat @ K @ N ) )
% 0.19/0.46 => ( ( ord_less @ nat @ M2 @ K )
% 0.19/0.46 => ( M2 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_diff_left_cancel_eq1
% 0.19/0.46 thf(fact_241_nat__diff__left__cancel__eq2,axiom,
% 0.19/0.46 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.46 ( ( ( minus_minus @ nat @ K @ M2 )
% 0.19/0.46 = ( minus_minus @ nat @ K @ N ) )
% 0.19/0.46 => ( ( ord_less @ nat @ N @ K )
% 0.19/0.46 => ( M2 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_diff_left_cancel_eq2
% 0.19/0.46 thf(fact_242_nat__diff__left__cancel__less,axiom,
% 0.19/0.46 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( minus_minus @ nat @ K @ M2 ) @ ( minus_minus @ nat @ K @ N ) )
% 0.19/0.46 => ( ord_less @ nat @ N @ M2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_diff_left_cancel_less
% 0.19/0.46 thf(fact_243_nat__diff__right__cancel__eq1,axiom,
% 0.19/0.46 ! [M2: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ( minus_minus @ nat @ M2 @ K )
% 0.19/0.46 = ( minus_minus @ nat @ N @ K ) )
% 0.19/0.46 => ( ( ord_less @ nat @ K @ M2 )
% 0.19/0.46 => ( M2 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_diff_right_cancel_eq1
% 0.19/0.46 thf(fact_244_nat__diff__right__cancel__eq2,axiom,
% 0.19/0.46 ! [M2: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ( minus_minus @ nat @ M2 @ K )
% 0.19/0.46 = ( minus_minus @ nat @ N @ K ) )
% 0.19/0.46 => ( ( ord_less @ nat @ K @ N )
% 0.19/0.46 => ( M2 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_diff_right_cancel_eq2
% 0.19/0.46 thf(fact_245_nat__diff__right__cancel__less,axiom,
% 0.19/0.46 ! [N: nat,K: nat,M2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M2 @ K ) )
% 0.19/0.46 => ( ord_less @ nat @ N @ M2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_diff_right_cancel_less
% 0.19/0.46 thf(fact_246_diff__less__mono2,axiom,
% 0.19/0.46 ! [M2: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.46 => ( ( ord_less @ nat @ M2 @ L )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less_mono2
% 0.19/0.46 thf(fact_247_less__imp__diff__less,axiom,
% 0.19/0.46 ! [J2: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J2 @ K )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ N ) @ K ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_diff_less
% 0.19/0.46 thf(fact_248_diff__commute,axiom,
% 0.19/0.46 ! [I3: nat,J2: nat,K: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J2 ) @ K )
% 0.19/0.46 = ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ K ) @ J2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_commute
% 0.19/0.46 thf(fact_249_diff__le__mono2,axiom,
% 0.19/0.46 ! [M2: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_mono2
% 0.19/0.46 thf(fact_250_le__diff__iff_H,axiom,
% 0.19/0.46 ! [A2: nat,C: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ A2 @ C )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ B @ C )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_iff'
% 0.19/0.46 thf(fact_251_diff__le__self,axiom,
% 0.19/0.46 ! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_self
% 0.19/0.46 thf(fact_252_diff__le__mono,axiom,
% 0.19/0.46 ! [M2: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_mono
% 0.19/0.46 thf(fact_253_Nat_Odiff__diff__eq,axiom,
% 0.19/0.46 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ M2 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.19/0.46 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.19/0.46 = ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_diff_eq
% 0.19/0.46 thf(fact_254_le__diff__iff,axiom,
% 0.19/0.46 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ M2 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_iff
% 0.19/0.46
% 0.19/0.46 % Type constructors (16)
% 0.19/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.19/0.46 ! [A6: $tType,A7: $tType] :
% 0.19/0.46 ( ( preorder @ A7 )
% 0.19/0.46 => ( preorder @ ( A6 > A7 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.19/0.46 ! [A6: $tType,A7: $tType] :
% 0.19/0.46 ( ( order @ A7 )
% 0.19/0.46 => ( order @ ( A6 > A7 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.19/0.46 ! [A6: $tType,A7: $tType] :
% 0.19/0.46 ( ( ord @ A7 )
% 0.19/0.46 => ( ord @ ( A6 > A7 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.19/0.46 condit1037483654norder @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.19/0.46 wellorder @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.19/0.46 preorder @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.19/0.46 linorder @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.19/0.46 no_top @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.19/0.46 order @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.19/0.46 ord @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.19/0.46 size @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 0.19/0.46 preorder @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
% 0.19/0.46 linorder @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
% 0.19/0.46 order @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
% 0.19/0.46 ord @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_List_Olist___Nat_Osize_8,axiom,
% 0.19/0.46 ! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Conjectures (3)
% 0.19/0.46 thf(conj_0,hypothesis,
% 0.19/0.46 ! [X5: nat] :
% 0.19/0.46 ( ( listIn521021761append @ a @ xs @ f @ X5 )
% 0.19/0.46 = ( listIn521021761append @ a @ ys @ g @ X5 ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_1,hypothesis,
% 0.19/0.46 ord_less_eq @ nat @ ( size_size @ ( list @ a ) @ xs ) @ ( size_size @ ( list @ a ) @ ys ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_2,conjecture,
% 0.19/0.46 ! [I: nat] :
% 0.19/0.46 ( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ a ) @ xs ) )
% 0.19/0.46 | ( ( nth @ a @ xs @ I )
% 0.19/0.46 = ( nth @ a @ ys @ I ) ) ) ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 2.42/1.49 ---- Embedded in THF ---
% 2.42/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.42/1.49 %%% Generated on Tue Feb 24 08:26:45 EST 2026
% 2.42/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.42/1.49 %%% Logic specification used:
% 2.42/1.49 %%% thf(spec, logic, $$dhol).
% 2.42/1.49
% 2.42/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2DTF11294.p
% 2.42/1.49 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.42/1.49 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.42/1.49 thf(ty_tf_a, type, a: $tType).
% 2.42/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 2.42/1.49 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.42/1.49 thf(sy_c_List2_Olist__asc, type, list_asc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.42/1.49 thf(sy_c_List2_Olist__desc, type, list_desc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.42/1.49 thf(sy_c_List2_Olist__strict__asc, type, list_strict_asc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.42/1.49 thf(sy_c_List2_Olist__strict__desc, type, list_strict_desc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.42/1.49 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append, type, listIn521021761append: (!> [A:$tType]: (((list @ A) > ((nat > A) > (nat > A)))))).
% 2.42/1.49 thf(sy_c_List_Olinorder__class_Osorted, type, linorder_sorted: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.42/1.49 thf(sy_c_List_Olist__ex, type, list_ex: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 2.42/1.49 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 2.42/1.49 thf(sy_c_List_Orev, type, rev: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.42/1.49 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.42/1.49 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.42/1.49 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.42/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.42/1.49 thf(sy_v_f, type, f: (nat > a)).
% 2.42/1.49 thf(sy_v_g, type, g: (nat > a)).
% 2.42/1.49 thf(sy_v_xs, type, xs: (list @ a)).
% 2.42/1.49 thf(sy_v_ys, type, ys: (list @ a)).
% 2.42/1.49 thf(fact_0_i__append__eq__i__append__conv, axiom, (! [A:$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))))))).
% 2.42/1.49 thf(fact_1_i__append__nth1, axiom, (! [A:$tType,N:nat,Xs:(list @ A),F:(nat > A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((((listIn521021761append @ A) @ Xs) @ F) @ N) = (((nth @ A) @ Xs) @ N)))))).
% 2.42/1.49 thf(fact_2_nth__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => ((! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I) = (((nth @ A) @ Ys) @ I))))) => (Xs = Ys)))))).
% 2.42/1.49 thf(fact_3_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => (? [X:A]: (((P @ I2) @ X)))))) = (? [Xs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = K) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => ((P @ I2) @ (((nth @ A) @ Xs2) @ I2)))))))))))).
% 2.42/1.49 thf(fact_4_list__eq__iff__nth__eq, axiom, (! [A:$tType]: (((^ [Y:(list @ A),Z:(list @ A)]: ((Y = Z))) = (^ [Xs2:(list @ A),Ys2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ A)) @ Ys2)) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs2)) => ((((nth @ A) @ Xs2) @ I2) = (((nth @ A) @ Ys2) @ I2)))))))))))).
% 2.42/1.49 thf(fact_5_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.42/1.49 thf(fact_6_list__desc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => ((list_desc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ J)) @ (((nth @ A) @ Xs2) @ I2))))))))))))))).
% 2.42/1.49 thf(fact_7_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs3:(list @ A)]: (((! [Ys3:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys3)) @ ((size_size @ (list @ A)) @ Xs3)) => (P @ Ys3)))) => (P @ Xs3)))) => (P @ Xs))))).
% 2.42/1.49 thf(fact_8_nat__less__le, axiom, ((ord_less @ nat) = (^ [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) & (M != N2)))))).
% 2.42/1.49 thf(fact_9_less__imp__le__nat, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_10_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) | (M = N2)))))).
% 2.42/1.49 thf(fact_11_less__or__eq__imp__le, axiom, (! [M2:nat,N:nat]: ((((((ord_less @ nat) @ M2) @ N) | (M2 = N)) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_12_le__neq__implies__less, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((M2 != N) => (((ord_less @ nat) @ M2) @ N)))))).
% 2.42/1.49 thf(fact_13_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I3:nat,J2:nat]: (((! [I:nat,J3:nat]: (((((ord_less @ nat) @ I) @ J3) => (((ord_less @ nat) @ (F @ I)) @ (F @ J3))))) => ((((ord_less_eq @ nat) @ I3) @ J2) => (((ord_less_eq @ nat) @ (F @ I3)) @ (F @ J2))))))).
% 2.42/1.49 thf(fact_14_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ A2) @ B) => (A2 = B))))))))).
% 2.42/1.49 thf(fact_15_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A3:A,B2:A]: (((((ord_less_eq @ A) @ B2) @ A3) & (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 2.42/1.49 thf(fact_16_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.42/1.49 thf(fact_17_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B:A]: (((! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((P @ A4) @ B3)))) => ((! [A4:A,B3:A]: ((((P @ B3) @ A4) => ((P @ A4) @ B3)))) => ((P @ A2) @ B))))))))).
% 2.42/1.49 thf(fact_18_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.42/1.49 thf(fact_19_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X2) @ Z2))))))))).
% 2.42/1.49 thf(fact_20_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ A2) => (A2 = B))))))))).
% 2.42/1.49 thf(fact_21_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((B = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.42/1.49 thf(fact_22_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.42/1.49 thf(fact_23_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) & (((ord_less_eq @ A) @ B2) @ A3))))))))).
% 2.42/1.49 thf(fact_24_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X2:A]: (((((ord_less_eq @ A) @ Y2) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y2) = (X2 = Y2))))))))).
% 2.42/1.49 thf(fact_25_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z2)))) => (((((ord_less_eq @ A) @ X2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X2)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y2))))))))))))))))).
% 2.42/1.49 thf(fact_26_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.42/1.49 thf(fact_27_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.42/1.49 thf(fact_28_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((X2 = Y2) => (((ord_less_eq @ A) @ X2) @ Y2)))))))).
% 2.42/1.49 thf(fact_29_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.42/1.49 thf(fact_30_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X2) => (X2 = Y2))))))))).
% 2.42/1.49 thf(fact_31_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 2.42/1.49 thf(fact_32_ord__le__eq__subst, axiom, (! [A:$tType,B4:$tType]: ((((ord @ B4) & (ord @ A)) => (! [A2:A,B:A,F:(A > B4),C:B4]: (((((ord_less_eq @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ B4) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ B4) @ (F @ A2)) @ C)))))))))).
% 2.42/1.49 thf(fact_33_ord__eq__le__subst, axiom, (! [A:$tType,B4:$tType]: ((((ord @ B4) & (ord @ A)) => (! [A2:A,F:(B4 > A),B:B4,C:B4]: (((A2 = (F @ B)) => ((((ord_less_eq @ B4) @ B) @ C) => ((! [X4:B4,Y4:B4]: (((((ord_less_eq @ B4) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.42/1.49 thf(fact_34_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C2) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.42/1.49 thf(fact_35_order__subst1, axiom, (! [A:$tType,B4:$tType]: ((((order @ B4) & (order @ A)) => (! [A2:A,F:(B4 > A),B:B4,C:B4]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B4) @ B) @ C) => ((! [X4:B4,Y4:B4]: (((((ord_less_eq @ B4) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.42/1.49 thf(fact_36_le__fun__def, axiom, (! [B4:$tType,A:$tType]: (((ord @ B4) => ((ord_less_eq @ (A > B4)) = (^ [F2:(A > B4),G2:(A > B4)]: ((! [X3:A]: ((((ord_less_eq @ B4) @ (F2 @ X3)) @ (G2 @ X3))))))))))).
% 2.42/1.49 thf(fact_37_le__funI, axiom, (! [B4:$tType,A:$tType]: (((ord @ B4) => (! [F:(A > B4),G:(A > B4)]: (((! [X4:A]: ((((ord_less_eq @ B4) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B4)) @ F) @ G)))))))).
% 2.42/1.49 thf(fact_38_le__funE, axiom, (! [B4:$tType,A:$tType]: (((ord @ B4) => (! [F:(A > B4),G:(A > B4),X2:A]: (((((ord_less_eq @ (A > B4)) @ F) @ G) => (((ord_less_eq @ B4) @ (F @ X2)) @ (G @ X2))))))))).
% 2.42/1.49 thf(fact_39_le__funD, axiom, (! [B4:$tType,A:$tType]: (((ord @ B4) => (! [F:(A > B4),G:(A > B4),X2:A]: (((((ord_less_eq @ (A > B4)) @ F) @ G) => (((ord_less_eq @ B4) @ (F @ X2)) @ (G @ X2))))))))).
% 2.42/1.49 thf(fact_40_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (A2 != B)))))))).
% 2.42/1.49 thf(fact_41_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (A2 != B)))))))).
% 2.42/1.49 thf(fact_42_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) = ((((ord_less @ A) @ Y2) @ X2) | (X2 = Y2))))))))).
% 2.42/1.49 thf(fact_43_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.42/1.49 thf(fact_44_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B:A]: (((! [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) => ((P @ A4) @ B3)))) => ((! [A4:A]: (((P @ A4) @ A4))) => ((! [A4:A,B3:A]: ((((P @ B3) @ A4) => ((P @ A4) @ B3)))) => ((P @ A2) @ B)))))))))).
% 2.42/1.49 thf(fact_45_ext, axiom, (! [B4:$tType,A:$tType,F:(A > B4),G:(A > B4)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.42/1.49 thf(fact_46_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X:A]: ((P2 @ X))))) = (^ [P3:(A > $o)]: ((? [N2:A]: (((P3 @ N2) & (! [M:A]: (((((ord_less @ A) @ M) @ N2) => (~ ((P3 @ M)))))))))))))))).
% 2.42/1.49 thf(fact_47_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X2)))))))))).
% 2.42/1.49 thf(fact_48_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.42/1.49 thf(fact_49_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.42/1.49 thf(fact_50_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) => ((X2 != Y2) => (((ord_less @ A) @ Y2) @ X2))))))))).
% 2.42/1.49 thf(fact_51_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,P:$o]: (((((ord_less @ A) @ X2) @ Y2) => ((((ord_less @ A) @ Y2) @ X2) => P)))))))).
% 2.42/1.49 thf(fact_52_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (Y2 != X2)))))))).
% 2.42/1.49 thf(fact_53_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y2:A,X2:A]: (((~ ((((ord_less @ A) @ Y2) @ X2))) => ((~ ((((ord_less @ A) @ X2) @ Y2))) = (X2 = Y2))))))))).
% 2.42/1.49 thf(fact_54_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X4) => (P @ Y5)))) => (P @ X4)))) => (P @ A2)))))))).
% 2.42/1.49 thf(fact_55_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X2)))))))))).
% 2.42/1.49 thf(fact_56_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (X2 != Y2)))))))).
% 2.42/1.49 thf(fact_57_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (~ ((((ord_less @ A) @ A2) @ B)))))))))).
% 2.42/1.49 thf(fact_58_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((B = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.42/1.49 thf(fact_59_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.42/1.49 thf(fact_60_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((~ ((((ord_less @ A) @ X2) @ X2))))))))).
% 2.42/1.49 thf(fact_61_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) | ((X2 = Y2) | (((ord_less @ A) @ Y2) @ X2))))))))).
% 2.42/1.49 thf(fact_62_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X2) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X2) @ Z2))))))))).
% 2.42/1.49 thf(fact_63_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 2.42/1.49 thf(fact_64_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X2)))))))))).
% 2.42/1.49 thf(fact_65_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (X2 != Y2)))))))).
% 2.42/1.49 thf(fact_66_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (? [Z3:A]: (((((ord_less @ A) @ X2) @ Z3) & (((ord_less @ A) @ Z3) @ Y2))))))))))).
% 2.42/1.49 thf(fact_67_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 2.42/1.49 thf(fact_68_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((X2 != Y2) = ((((ord_less @ A) @ X2) @ Y2) | (((ord_less @ A) @ Y2) @ X2))))))))).
% 2.42/1.49 thf(fact_69_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((X2 != Y2) => ((~ ((((ord_less @ A) @ X2) @ Y2))) => (((ord_less @ A) @ Y2) @ X2))))))))).
% 2.42/1.49 thf(fact_70_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X2:A]: ((? [X_1:A]: ((((ord_less @ A) @ X2) @ X_1))))))))).
% 2.42/1.49 thf(fact_71_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X2:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X2))))))))).
% 2.42/1.49 thf(fact_72_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ C2) @ (F @ B)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C2) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.42/1.49 thf(fact_73_order__less__subst1, axiom, (! [A:$tType,B4:$tType]: ((((order @ B4) & (order @ A)) => (! [A2:A,F:(B4 > A),B:B4,C:B4]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less @ B4) @ B) @ C) => ((! [X4:B4,Y4:B4]: (((((ord_less @ B4) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.42/1.49 thf(fact_74_ord__less__eq__subst, axiom, (! [A:$tType,B4:$tType]: ((((ord @ B4) & (ord @ A)) => (! [A2:A,B:A,F:(A > B4),C:B4]: (((((ord_less @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ B4) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ B4) @ (F @ A2)) @ C)))))))))).
% 2.42/1.49 thf(fact_75_ord__eq__less__subst, axiom, (! [A:$tType,B4:$tType]: ((((ord @ B4) & (ord @ A)) => (! [A2:A,F:(B4 > A),B:B4,C:B4]: (((A2 = (F @ B)) => ((((ord_less @ B4) @ B) @ C) => ((! [X4:B4,Y4:B4]: (((((ord_less @ B4) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.42/1.49 thf(fact_76_measure__induct__rule, axiom, (! [B4:$tType,A:$tType]: (((wellorder @ B4) => (! [F:(A > B4),P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y5:A]: (((((ord_less @ B4) @ (F @ Y5)) @ (F @ X4)) => (P @ Y5)))) => (P @ X4)))) => (P @ A2)))))))).
% 2.42/1.49 thf(fact_77_measure__induct, axiom, (! [B4:$tType,A:$tType]: (((wellorder @ B4) => (! [F:(A > B4),P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y5:A]: (((((ord_less @ B4) @ (F @ Y5)) @ (F @ X4)) => (P @ Y5)))) => (P @ X4)))) => (P @ A2)))))))).
% 2.42/1.49 thf(fact_78_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X2:A]: (((! [X4:A]: (((~ ((P @ X4))) => (? [Y5:A]: (((((ord_less @ nat) @ (V @ Y5)) @ (V @ X4)) & (~ ((P @ Y5))))))))) => (P @ X2))))).
% 2.42/1.49 thf(fact_79_linorder__neqE__nat, axiom, (! [X2:nat,Y2:nat]: (((X2 != Y2) => ((~ ((((ord_less @ nat) @ X2) @ Y2))) => (((ord_less @ nat) @ Y2) @ X2)))))).
% 2.42/1.49 thf(fact_80_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P @ M3))))))))) => (P @ N))))).
% 2.42/1.49 thf(fact_81_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P @ M3)))) => (P @ N3)))) => (P @ N))))).
% 2.42/1.49 thf(fact_82_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.42/1.49 thf(fact_83_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.42/1.49 thf(fact_84_less__not__refl2, axiom, (! [N:nat,M2:nat]: (((((ord_less @ nat) @ N) @ M2) => (M2 != N))))).
% 2.42/1.49 thf(fact_85_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.42/1.49 thf(fact_86_nat__neq__iff, axiom, (! [M2:nat,N:nat]: (((M2 != N) = ((((ord_less @ nat) @ M2) @ N) | (((ord_less @ nat) @ N) @ M2)))))).
% 2.42/1.49 thf(fact_87_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B:nat]: (((P @ K) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B)))) => (? [X4:nat]: (((P @ X4) & (! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ X4)))))))))))).
% 2.42/1.49 thf(fact_88_nat__le__linear, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) | (((ord_less_eq @ nat) @ N) @ M2))))).
% 2.42/1.49 thf(fact_89_le__antisym, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less_eq @ nat) @ N) @ M2) => (M2 = N)))))).
% 2.42/1.49 thf(fact_90_eq__imp__le, axiom, (! [M2:nat,N:nat]: (((M2 = N) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_91_le__trans, axiom, (! [I3:nat,J2:nat,K:nat]: (((((ord_less_eq @ nat) @ I3) @ J2) => ((((ord_less_eq @ nat) @ J2) @ K) => (((ord_less_eq @ nat) @ I3) @ K)))))).
% 2.42/1.49 thf(fact_92_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.42/1.49 thf(fact_93_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X2:A,Y2:A]: (((((size_size @ A) @ X2) != ((size_size @ A) @ Y2)) => (X2 != Y2)))))))).
% 2.42/1.49 thf(fact_94_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys)) => (Xs != Ys))))).
% 2.42/1.49 thf(fact_95_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs3:(list @ A)]: ((((size_size @ (list @ A)) @ Xs3) = N)))))).
% 2.42/1.49 thf(fact_96_list__desc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_desc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ J)) @ (((nth @ A) @ Xs2) @ I2))))))))))))))).
% 2.42/1.49 thf(fact_97_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((A2 != B) => ((((ord_less_eq @ A) @ A2) @ B) => (((ord_less @ A) @ A2) @ B))))))))).
% 2.42/1.49 thf(fact_98_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (((ord_less_eq @ A) @ B) @ A2)))))))).
% 2.42/1.49 thf(fact_99_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) & (A3 != B2))))))))).
% 2.42/1.49 thf(fact_100_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) | (A3 = B2))))))))).
% 2.42/1.49 thf(fact_101_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.42/1.49 thf(fact_102_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X2) @ Y2) => ((! [W:A]: (((((ord_less @ A) @ X2) @ W) => ((((ord_less @ A) @ W) @ Y2) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.42/1.49 thf(fact_103_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X2:A,Y2:A]: (((((ord_less @ A) @ Z2) @ X2) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X2) => (((ord_less_eq @ A) @ Y2) @ W))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.42/1.49 thf(fact_104_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.42/1.49 thf(fact_105_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.42/1.49 thf(fact_106_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) & (A3 != B2))))))))).
% 2.42/1.49 thf(fact_107_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) | (A3 = B2))))))))).
% 2.42/1.49 thf(fact_108_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.42/1.49 thf(fact_109_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.42/1.49 thf(fact_110_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y2:A,X2:A]: (((~ ((((ord_less_eq @ A) @ Y2) @ X2))) => (((ord_less @ A) @ X2) @ Y2)))))))).
% 2.42/1.49 thf(fact_111_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X3))))))))))).
% 2.42/1.49 thf(fact_112_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less @ A) @ X2) @ Y2) | (X2 = Y2))))))))).
% 2.42/1.49 thf(fact_113_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) | (((ord_less @ A) @ Y2) @ X2)))))))).
% 2.42/1.49 thf(fact_114_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y2:A,Z2:A]: (((! [X4:A]: (((((ord_less @ A) @ X4) @ Y2) => (((ord_less_eq @ A) @ X4) @ Z2)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.42/1.49 thf(fact_115_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,Y2:A]: (((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((ord_less_eq @ A) @ Y2) @ X4)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.42/1.49 thf(fact_116_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X2) @ Z2))))))))).
% 2.42/1.49 thf(fact_117_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X2) @ Z2))))))))).
% 2.42/1.49 thf(fact_118_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less_eq @ A) @ X2) @ Y2)))))))).
% 2.42/1.49 thf(fact_119_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((~ ((((ord_less @ A) @ X2) @ Y2))) = (X2 = Y2))))))))).
% 2.42/1.49 thf(fact_120_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) => ((((ord_less_eq @ A) @ X2) @ Y2) = (X2 = Y2))))))))).
% 2.42/1.49 thf(fact_121_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((A2 != B) => (((ord_less @ A) @ A2) @ B))))))))).
% 2.42/1.49 thf(fact_122_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) = (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.42/1.49 thf(fact_123_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y2))) = (((ord_less @ A) @ Y2) @ X2)))))))).
% 2.42/1.49 thf(fact_124_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C2) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.42/1.49 thf(fact_125_order__less__le__subst1, axiom, (! [A:$tType,B4:$tType]: ((((order @ B4) & (order @ A)) => (! [A2:A,F:(B4 > A),B:B4,C:B4]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B4) @ B) @ C) => ((! [X4:B4,Y4:B4]: (((((ord_less_eq @ B4) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.42/1.49 thf(fact_126_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ C2) @ (F @ B)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C2) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.42/1.49 thf(fact_127_order__le__less__subst1, axiom, (! [A:$tType,B4:$tType]: ((((order @ B4) & (order @ A)) => (! [A2:A,F:(B4 > A),B:B4,C:B4]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less @ B4) @ B) @ C) => ((! [X4:B4,Y4:B4]: (((((ord_less @ B4) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.42/1.49 thf(fact_128_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (X3 != Y3))))))))).
% 2.42/1.49 thf(fact_129_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) | (X3 = Y3))))))))).
% 2.42/1.49 thf(fact_130_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.42/1.49 thf(fact_131_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X2:A]: (((((ord_less_eq @ A) @ Y2) @ X2) => (~ ((((ord_less @ A) @ X2) @ Y2)))))))))).
% 2.42/1.49 thf(fact_132_list__strict__asc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_asc @ A) @ Xs) => (! [J4:nat]: (((((ord_less @ nat) @ J4) @ ((size_size @ (list @ A)) @ Xs)) => (! [I4:nat]: (((((ord_less_eq @ nat) @ I4) @ J4) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I4)) @ (((nth @ A) @ Xs) @ J4))))))))))))))).
% 2.42/1.49 thf(fact_133_list__asc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => ((list_asc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J))))))))))))))).
% 2.42/1.49 thf(fact_134_list__strict__desc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_strict_desc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ A) @ (((nth @ A) @ Xs2) @ J)) @ (((nth @ A) @ Xs2) @ I2))))))))))))))).
% 2.42/1.49 thf(fact_135_list__strict__asc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_strict_asc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J))))))))))))))).
% 2.42/1.49 thf(fact_136_list__asc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_asc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J))))))))))))))).
% 2.42/1.49 thf(fact_137_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),B:nat]: (((P @ K) => ((! [Y4:A]: (((P @ Y4) => (((ord_less @ nat) @ (F @ Y4)) @ B)))) => (? [X4:A]: (((P @ X4) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (F @ Y5)) @ (F @ X4))))))))))))).
% 2.42/1.49 thf(fact_138_nat__descend__induct, axiom, (! [N:nat,P:(nat > $o),M2:nat]: (((! [K2:nat]: (((((ord_less @ nat) @ N) @ K2) => (P @ K2)))) => ((! [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) => ((! [I4:nat]: (((((ord_less @ nat) @ K2) @ I4) => (P @ I4)))) => (P @ K2))))) => (P @ M2)))))).
% 2.42/1.49 thf(fact_139_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A2:A,B:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B) => ((P @ A2) => ((~ ((P @ B))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B) & ((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less @ A) @ X5) @ C3)) => (P @ X5)))) & (! [D:A]: (((! [X4:A]: ((((((ord_less_eq @ A) @ A2) @ X4) & (((ord_less @ A) @ X4) @ D)) => (P @ X4)))) => (((ord_less_eq @ A) @ D) @ C3)))))))))))))))))).
% 2.42/1.49 thf(fact_140_le__greater__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [N2:A,A3:A]: ((! [X3:A]: (((((ord_less @ A) @ A3) @ X3) => (N2 != X3))))))))))).
% 2.42/1.49 thf(fact_141_greater__le__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [A3:A,N2:A]: ((! [X3:A]: (((((ord_less_eq @ A) @ X3) @ A3) => (N2 != X3))))))))))).
% 2.42/1.49 thf(fact_142_less__ge__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [N2:A,A3:A]: ((! [X3:A]: (((((ord_less_eq @ A) @ A3) @ X3) => (N2 != X3))))))))))).
% 2.42/1.49 thf(fact_143_less__fun__def, axiom, (! [B4:$tType,A:$tType]: (((ord @ B4) => ((ord_less @ (A > B4)) = (^ [F2:(A > B4),G2:(A > B4)]: (((((ord_less_eq @ (A > B4)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B4)) @ G2) @ F2))))))))))).
% 2.42/1.49 thf(fact_144_list__strict__asc__imp__list__asc, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_asc @ A) @ Xs) => ((list_asc @ A) @ Xs)))))))).
% 2.42/1.49 thf(fact_145_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A2:A]: ((? [B3:A]: (((((ord_less @ A) @ A2) @ B3) | (((ord_less @ A) @ B3) @ A2)))))))))).
% 2.42/1.49 thf(fact_146_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => (? [X4:A]: (((P @ X4) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ X4)) @ (M2 @ Y5)))))))))))).
% 2.42/1.49 thf(fact_147_list__strict__desc__imp__list__desc, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_desc @ A) @ Xs) => ((list_desc @ A) @ Xs)))))))).
% 2.42/1.49 thf(fact_148_ge__less__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A3:A,N2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ A3) => (N2 != X3))))))))))).
% 2.42/1.49 thf(fact_149_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X5)))))))))))))).
% 2.42/1.49 thf(fact_150_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less_eq @ A) @ X5) @ T)))))))))))).
% 2.42/1.49 thf(fact_151_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less_eq @ A) @ T) @ X5)))))))))))).
% 2.42/1.49 thf(fact_152_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((P @ X4) = (P4 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P @ X5) & (Q @ X5)) = ((P4 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.42/1.49 thf(fact_153_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((P @ X4) = (P4 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P @ X5) | (Q @ X5)) = ((P4 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.42/1.49 thf(fact_154_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.42/1.49 thf(fact_155_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.42/1.49 thf(fact_156_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less @ A) @ X5) @ T)))))))))))))).
% 2.42/1.49 thf(fact_157_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less @ A) @ T) @ X5)))))))))))).
% 2.42/1.49 thf(fact_158_pinf_I11_J, axiom, (! [C2:$tType,D2:$tType]: (((ord @ C2) => (! [F3:D2]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ Z3) @ X5) => (F3 = F3)))))))))))).
% 2.42/1.49 thf(fact_159_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((P @ X4) = (P4 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P @ X5) & (Q @ X5)) = ((P4 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.42/1.49 thf(fact_160_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((P @ X4) = (P4 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P @ X5) | (Q @ X5)) = ((P4 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.42/1.49 thf(fact_161_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.42/1.49 thf(fact_162_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.42/1.49 thf(fact_163_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less @ A) @ X5) @ T)))))))))))).
% 2.42/1.49 thf(fact_164_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less @ A) @ T) @ X5)))))))))))))).
% 2.42/1.49 thf(fact_165_minf_I11_J, axiom, (! [C2:$tType,D2:$tType]: (((ord @ C2) => (! [F3:D2]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ X5) @ Z3) => (F3 = F3)))))))))))).
% 2.42/1.49 thf(fact_166_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ T)))))))))))))).
% 2.42/1.49 thf(fact_167_verit__comp__simplify1_I3_J, axiom, (! [B4:$tType]: (((linorder @ B4) => (! [B5:B4,A5:B4]: (((~ ((((ord_less_eq @ B4) @ B5) @ A5))) = (((ord_less @ B4) @ A5) @ B5)))))))).
% 2.42/1.49 thf(fact_168_sorted__iff__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => ((linorder_sorted @ A) = (^ [Xs2:(list @ A)]: ((! [I2:nat,J:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J))))))))))))).
% 2.42/1.49 thf(fact_169_sorted__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),I3:nat,J2:nat]: ((((linorder_sorted @ A) @ Xs) => ((((ord_less_eq @ nat) @ I3) @ J2) => ((((ord_less @ nat) @ J2) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I3)) @ (((nth @ A) @ Xs) @ J2))))))))))).
% 2.42/1.49 thf(fact_170_list__ord__le__sorted__eq, axiom, (! [A:$tType]: (((linorder @ A) => ((list_asc @ A) = (linorder_sorted @ A)))))).
% 2.42/1.49 thf(fact_171_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((A2 = B) | ((~ ((((ord_less_eq @ A) @ A2) @ B))) | (~ ((((ord_less_eq @ A) @ B) @ A2))))))))))).
% 2.42/1.49 thf(fact_172_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.42/1.49 thf(fact_173_sorted__iff__nth__mono__less, axiom, (! [A:$tType]: (((linorder @ A) => ((linorder_sorted @ A) = (^ [Xs2:(list @ A)]: ((! [I2:nat,J:nat]: (((((ord_less @ nat) @ I2) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J))))))))))))).
% 2.42/1.49 thf(fact_174_sorted__rev__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),I3:nat,J2:nat]: ((((linorder_sorted @ A) @ ((rev @ A) @ Xs)) => ((((ord_less_eq @ nat) @ I3) @ J2) => ((((ord_less @ nat) @ J2) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ J2)) @ (((nth @ A) @ Xs) @ I3))))))))))).
% 2.42/1.49 thf(fact_175_sorted__rev__iff__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: ((((linorder_sorted @ A) @ ((rev @ A) @ Xs)) = (! [I2:nat,J:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ J)) @ (((nth @ A) @ Xs) @ I2))))))))))))).
% 2.42/1.49 thf(fact_176_sorted__iff__nth__Suc, axiom, (! [A:$tType]: (((linorder @ A) => ((linorder_sorted @ A) = (^ [Xs2:(list @ A)]: ((! [I2:nat]: (((((ord_less @ nat) @ (suc @ I2)) @ ((size_size @ (list @ A)) @ Xs2)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ (suc @ I2))))))))))))).
% 2.42/1.49 thf(fact_177_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.42/1.49 thf(fact_178_nat_Oinject, axiom, (! [X22:nat,Y22:nat]: ((((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))).
% 2.42/1.49 thf(fact_179_rev__is__rev__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((rev @ A) @ Xs) = ((rev @ A) @ Ys)) = (Xs = Ys))))).
% 2.42/1.49 thf(fact_180_rev__rev__ident, axiom, (! [A:$tType,Xs:(list @ A)]: ((((rev @ A) @ ((rev @ A) @ Xs)) = Xs)))).
% 2.42/1.49 thf(fact_181_Suc__less__eq, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M2)) @ (suc @ N)) = (((ord_less @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_182_Suc__mono, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (((ord_less @ nat) @ (suc @ M2)) @ (suc @ N)))))).
% 2.42/1.49 thf(fact_183_lessI, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (suc @ N))))).
% 2.42/1.49 thf(fact_184_Suc__le__mono, axiom, (! [N:nat,M2:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M2)) = (((ord_less_eq @ nat) @ N) @ M2))))).
% 2.42/1.49 thf(fact_185_length__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((rev @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))).
% 2.42/1.49 thf(fact_186_Suc__leD, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M2)) @ N) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_187_le__SucE, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M2) @ N))) => (M2 = (suc @ N))))))).
% 2.42/1.49 thf(fact_188_le__SucI, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less_eq @ nat) @ M2) @ (suc @ N)))))).
% 2.42/1.49 thf(fact_189_Suc__le__D, axiom, (! [N:nat,M4:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M4) => (? [M5:nat]: ((M4 = (suc @ M5)))))))).
% 2.42/1.49 thf(fact_190_le__Suc__eq, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M2) @ N) | (M2 = (suc @ N))))))).
% 2.42/1.49 thf(fact_191_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 2.42/1.49 thf(fact_192_not__less__eq__eq, axiom, (! [M2:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M2) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M2))))).
% 2.42/1.49 thf(fact_193_full__nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N3) => (P @ M3)))) => (P @ N3)))) => (P @ N))))).
% 2.42/1.49 thf(fact_194_nat__induct__at__least, axiom, (! [M2:nat,N:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ M2) @ N) => ((P @ M2) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ M2) @ N3) => ((P @ N3) => (P @ (suc @ N3)))))) => (P @ N))))))).
% 2.42/1.49 thf(fact_195_transitive__stepwise__le, axiom, (! [M2:nat,N:nat,R:(nat > (nat > $o))]: (((((ord_less_eq @ nat) @ M2) @ N) => ((! [X4:nat]: (((R @ X4) @ X4))) => ((! [X4:nat,Y4:nat,Z3:nat]: ((((R @ X4) @ Y4) => (((R @ Y4) @ Z3) => ((R @ X4) @ Z3))))) => ((! [N3:nat]: (((R @ N3) @ (suc @ N3)))) => ((R @ M2) @ N)))))))).
% 2.42/1.49 thf(fact_196_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.42/1.49 thf(fact_197_Suc__inject, axiom, (! [X2:nat,Y2:nat]: ((((suc @ X2) = (suc @ Y2)) => (X2 = Y2))))).
% 2.42/1.49 thf(fact_198_rev__swap, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((rev @ A) @ Xs) = Ys) = (Xs = ((rev @ A) @ Ys)))))).
% 2.42/1.49 thf(fact_199_not__less__less__Suc__eq, axiom, (! [N:nat,M2:nat]: (((~ ((((ord_less @ nat) @ N) @ M2))) => ((((ord_less @ nat) @ N) @ (suc @ M2)) = (N = M2)))))).
% 2.42/1.49 thf(fact_200_strict__inc__induct, axiom, (! [I3:nat,J2:nat,P:(nat > $o)]: (((((ord_less @ nat) @ I3) @ J2) => ((! [I:nat]: (((J2 = (suc @ I)) => (P @ I)))) => ((! [I:nat]: (((((ord_less @ nat) @ I) @ J2) => ((P @ (suc @ I)) => (P @ I))))) => (P @ I3))))))).
% 2.42/1.49 thf(fact_201_less__Suc__induct, axiom, (! [I3:nat,J2:nat,P:(nat > (nat > $o))]: (((((ord_less @ nat) @ I3) @ J2) => ((! [I:nat]: (((P @ I) @ (suc @ I)))) => ((! [I:nat,J3:nat,K2:nat]: (((((ord_less @ nat) @ I) @ J3) => ((((ord_less @ nat) @ J3) @ K2) => (((P @ I) @ J3) => (((P @ J3) @ K2) => ((P @ I) @ K2))))))) => ((P @ I3) @ J2))))))).
% 2.42/1.49 thf(fact_202_less__trans__Suc, axiom, (! [I3:nat,J2:nat,K:nat]: (((((ord_less @ nat) @ I3) @ J2) => ((((ord_less @ nat) @ J2) @ K) => (((ord_less @ nat) @ (suc @ I3)) @ K)))))).
% 2.42/1.49 thf(fact_203_Suc__less__SucD, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M2)) @ (suc @ N)) => (((ord_less @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_204_less__antisym, axiom, (! [N:nat,M2:nat]: (((~ ((((ord_less @ nat) @ N) @ M2))) => ((((ord_less @ nat) @ N) @ (suc @ M2)) => (M2 = N)))))).
% 2.42/1.49 thf(fact_205_Suc__less__eq2, axiom, (! [N:nat,M2:nat]: (((((ord_less @ nat) @ (suc @ N)) @ M2) = (? [M6:nat]: (((M2 = (suc @ M6)) & (((ord_less @ nat) @ N) @ M6)))))))).
% 2.42/1.49 thf(fact_206_All__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ (suc @ N)) => (P @ I2)))) = ((P @ N) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ N) => (P @ I2))))))))).
% 2.42/1.49 thf(fact_207_not__less__eq, axiom, (! [M2:nat,N:nat]: (((~ ((((ord_less @ nat) @ M2) @ N))) = (((ord_less @ nat) @ N) @ (suc @ M2)))))).
% 2.42/1.49 thf(fact_208_less__Suc__eq, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ (suc @ N)) = ((((ord_less @ nat) @ M2) @ N) | (M2 = N)))))).
% 2.42/1.49 thf(fact_209_Ex__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((? [I2:nat]: (((((ord_less @ nat) @ I2) @ (suc @ N)) & (P @ I2)))) = ((P @ N) | (? [I2:nat]: (((((ord_less @ nat) @ I2) @ N) & (P @ I2))))))))).
% 2.42/1.49 thf(fact_210_less__SucI, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (((ord_less @ nat) @ M2) @ (suc @ N)))))).
% 2.42/1.49 thf(fact_211_less__SucE, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ (suc @ N)) => ((~ ((((ord_less @ nat) @ M2) @ N))) => (M2 = N)))))).
% 2.42/1.49 thf(fact_212_Suc__lessI, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (((suc @ M2) != N) => (((ord_less @ nat) @ (suc @ M2)) @ N)))))).
% 2.42/1.49 thf(fact_213_Suc__lessE, axiom, (! [I3:nat,K:nat]: (((((ord_less @ nat) @ (suc @ I3)) @ K) => (~ ((! [J3:nat]: (((((ord_less @ nat) @ I3) @ J3) => (K != (suc @ J3))))))))))).
% 2.42/1.49 thf(fact_214_Suc__lessD, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M2)) @ N) => (((ord_less @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_215_Nat_OlessE, axiom, (! [I3:nat,K:nat]: (((((ord_less @ nat) @ I3) @ K) => ((K != (suc @ I3)) => (~ ((! [J3:nat]: (((((ord_less @ nat) @ I3) @ J3) => (K != (suc @ J3)))))))))))).
% 2.42/1.49 thf(fact_216_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,M2:nat]: (((! [N3:nat]: ((((ord_less @ A) @ (F @ N3)) @ (F @ (suc @ N3))))) => ((((ord_less @ A) @ (F @ N)) @ (F @ M2)) = (((ord_less @ nat) @ N) @ M2))))))))).
% 2.42/1.49 thf(fact_217_lift__Suc__mono__less, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N4:nat]: (((! [N3:nat]: ((((ord_less @ A) @ (F @ N3)) @ (F @ (suc @ N3))))) => ((((ord_less @ nat) @ N) @ N4) => (((ord_less @ A) @ (F @ N)) @ (F @ N4)))))))))).
% 2.42/1.49 thf(fact_218_lift__Suc__antimono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N4:nat]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F @ (suc @ N3))) @ (F @ N3)))) => ((((ord_less_eq @ nat) @ N) @ N4) => (((ord_less_eq @ A) @ (F @ N4)) @ (F @ N)))))))))).
% 2.42/1.49 thf(fact_219_lift__Suc__mono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N4:nat]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F @ N3)) @ (F @ (suc @ N3))))) => ((((ord_less_eq @ nat) @ N) @ N4) => (((ord_less_eq @ A) @ (F @ N)) @ (F @ N4)))))))))).
% 2.42/1.49 thf(fact_220_le__imp__less__Suc, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less @ nat) @ M2) @ (suc @ N)))))).
% 2.42/1.49 thf(fact_221_less__eq__Suc__le, axiom, ((ord_less @ nat) = (^ [N2:nat]: (((ord_less_eq @ nat) @ (suc @ N2)))))).
% 2.42/1.49 thf(fact_222_less__Suc__eq__le, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ (suc @ N)) = (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_223_le__less__Suc__eq, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less @ nat) @ N) @ (suc @ M2)) = (N = M2)))))).
% 2.42/1.49 thf(fact_224_Suc__le__lessD, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M2)) @ N) => (((ord_less @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_225_inc__induct, axiom, (! [I3:nat,J2:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I3) @ J2) => ((P @ J2) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ I3) @ N3) => ((((ord_less @ nat) @ N3) @ J2) => ((P @ (suc @ N3)) => (P @ N3)))))) => (P @ I3))))))).
% 2.42/1.49 thf(fact_226_dec__induct, axiom, (! [I3:nat,J2:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I3) @ J2) => ((P @ I3) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ I3) @ N3) => ((((ord_less @ nat) @ N3) @ J2) => ((P @ N3) => (P @ (suc @ N3))))))) => (P @ J2))))))).
% 2.42/1.49 thf(fact_227_Suc__le__eq, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M2)) @ N) = (((ord_less @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_228_Suc__leI, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (((ord_less_eq @ nat) @ (suc @ M2)) @ N))))).
% 2.42/1.49 thf(fact_229_sorted__rev__iff__nth__Suc, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: ((((linorder_sorted @ A) @ ((rev @ A) @ Xs)) = (! [I2:nat]: (((((ord_less @ nat) @ (suc @ I2)) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ (suc @ I2))) @ (((nth @ A) @ Xs) @ I2)))))))))))).
% 2.42/1.49 thf(fact_230_nat__induct_H, axiom, (! [P:(nat > $o),N0:nat,N:nat]: (((P @ N0) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ N0) @ N3) => ((P @ N3) => (P @ (suc @ N3)))))) => ((((ord_less_eq @ nat) @ N0) @ N) => (P @ N))))))).
% 2.42/1.49 thf(fact_231_rev__nth, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((rev @ A) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (suc @ N)))))))).
% 2.42/1.49 thf(fact_232_list__ex__length, axiom, (! [A:$tType]: (((list_ex @ A) = (^ [P3:(A > $o),Xs2:(list @ A)]: ((? [N2:nat]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs2)) & (P3 @ (((nth @ A) @ Xs2) @ N2))))))))))).
% 2.42/1.49 thf(fact_233_Suc__diff__diff, axiom, (! [M2:nat,N:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ (suc @ M2)) @ N)) @ (suc @ K)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M2) @ N)) @ K))))).
% 2.42/1.49 thf(fact_234_diff__Suc__Suc, axiom, (! [M2:nat,N:nat]: (((((minus_minus @ nat) @ (suc @ M2)) @ (suc @ N)) = (((minus_minus @ nat) @ M2) @ N))))).
% 2.42/1.49 thf(fact_235_diff__diff__cancel, axiom, (! [I3:nat,N:nat]: (((((ord_less_eq @ nat) @ I3) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I3)) = I3))))).
% 2.42/1.49 thf(fact_236_list__ex__rev, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((list_ex @ A) @ P) @ ((rev @ A) @ Xs)) = (((list_ex @ A) @ P) @ Xs))))).
% 2.42/1.49 thf(fact_237_i__append__nth2, axiom, (! [A:$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)))))))).
% 2.42/1.49 thf(fact_238_zero__induct__lemma, axiom, (! [P:(nat > $o),K:nat,I3:nat]: (((P @ K) => ((! [N3:nat]: (((P @ (suc @ N3)) => (P @ N3)))) => (P @ (((minus_minus @ nat) @ K) @ I3))))))).
% 2.42/1.49 thf(fact_239_less__diff__imp__less, axiom, (! [I3:nat,J2:nat,M2:nat]: (((((ord_less @ nat) @ I3) @ (((minus_minus @ nat) @ J2) @ M2)) => (((ord_less @ nat) @ I3) @ J2))))).
% 2.42/1.49 thf(fact_240_nat__diff__left__cancel__eq1, axiom, (! [K:nat,M2:nat,N:nat]: ((((((minus_minus @ nat) @ K) @ M2) = (((minus_minus @ nat) @ K) @ N)) => ((((ord_less @ nat) @ M2) @ K) => (M2 = N)))))).
% 2.42/1.49 thf(fact_241_nat__diff__left__cancel__eq2, axiom, (! [K:nat,M2:nat,N:nat]: ((((((minus_minus @ nat) @ K) @ M2) = (((minus_minus @ nat) @ K) @ N)) => ((((ord_less @ nat) @ N) @ K) => (M2 = N)))))).
% 2.42/1.49 thf(fact_242_nat__diff__left__cancel__less, axiom, (! [K:nat,M2:nat,N:nat]: (((((ord_less @ nat) @ (((minus_minus @ nat) @ K) @ M2)) @ (((minus_minus @ nat) @ K) @ N)) => (((ord_less @ nat) @ N) @ M2))))).
% 2.42/1.49 thf(fact_243_nat__diff__right__cancel__eq1, axiom, (! [M2:nat,K:nat,N:nat]: ((((((minus_minus @ nat) @ M2) @ K) = (((minus_minus @ nat) @ N) @ K)) => ((((ord_less @ nat) @ K) @ M2) => (M2 = N)))))).
% 2.42/1.49 thf(fact_244_nat__diff__right__cancel__eq2, axiom, (! [M2:nat,K:nat,N:nat]: ((((((minus_minus @ nat) @ M2) @ K) = (((minus_minus @ nat) @ N) @ K)) => ((((ord_less @ nat) @ K) @ N) => (M2 = N)))))).
% 2.42/1.49 thf(fact_245_nat__diff__right__cancel__less, axiom, (! [N:nat,K:nat,M2:nat]: (((((ord_less @ nat) @ (((minus_minus @ nat) @ N) @ K)) @ (((minus_minus @ nat) @ M2) @ K)) => (((ord_less @ nat) @ N) @ M2))))).
% 2.42/1.49 thf(fact_246_diff__less__mono2, axiom, (! [M2:nat,N:nat,L:nat]: (((((ord_less @ nat) @ M2) @ N) => ((((ord_less @ nat) @ M2) @ L) => (((ord_less @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M2))))))).
% 2.42/1.49 thf(fact_247_less__imp__diff__less, axiom, (! [J2:nat,K:nat,N:nat]: (((((ord_less @ nat) @ J2) @ K) => (((ord_less @ nat) @ (((minus_minus @ nat) @ J2) @ N)) @ K))))).
% 2.42/1.49 thf(fact_248_diff__commute, axiom, (! [I3:nat,J2:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I3) @ J2)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I3) @ K)) @ J2))))).
% 2.42/1.49 thf(fact_249_diff__le__mono2, axiom, (! [M2:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M2)))))).
% 2.42/1.49 thf(fact_250_le__diff__iff_H, axiom, (! [A2:nat,C:nat,B:nat]: (((((ord_less_eq @ nat) @ A2) @ C) => ((((ord_less_eq @ nat) @ B) @ C) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C) @ A2)) @ (((minus_minus @ nat) @ C) @ B)) = (((ord_less_eq @ nat) @ B) @ A2))))))).
% 2.42/1.49 thf(fact_251_diff__le__self, axiom, (! [M2:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M2) @ N)) @ M2)))).
% 2.42/1.49 thf(fact_252_diff__le__mono, axiom, (! [M2:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M2) @ L)) @ (((minus_minus @ nat) @ N) @ L)))))).
% 2.42/1.49 thf(fact_253_Nat_Odiff__diff__eq, axiom, (! [K:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M2) => ((((ord_less_eq @ nat) @ K) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M2) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M2) @ N))))))).
% 2.42/1.49 thf(fact_254_le__diff__iff, axiom, (! [K:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M2) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M2) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less_eq @ nat) @ M2) @ N))))))).
% 2.42/1.49 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: (((preorder @ A7) => (preorder @ (A6 > A7)))))).
% 2.42/1.49 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: (((order @ A7) => (order @ (A6 > A7)))))).
% 2.42/1.49 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: (((ord @ A7) => (ord @ (A6 > A7)))))).
% 2.42/1.49 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ nat)).
% 2.42/1.49 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.42/1.49 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.42/1.49 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.42/1.49 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 2.42/1.49 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.42/1.49 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.42/1.49 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 2.42/1.49 thf(tcon_HOL_Obool___Orderings_Opreorder_4, axiom, (preorder @ $o)).
% 2.42/1.49 thf(tcon_HOL_Obool___Orderings_Olinorder_5, axiom, (linorder @ $o)).
% 2.42/1.49 thf(tcon_HOL_Obool___Orderings_Oorder_6, axiom, (order @ $o)).
% 2.42/1.49 thf(tcon_HOL_Obool___Orderings_Oord_7, axiom, (ord @ $o)).
% 2.42/1.49 thf(tcon_List_Olist___Nat_Osize_8, axiom, (! [A6:$tType]: ((size @ (list @ A6))))).
% 2.42/1.49 thf(conj_0, hypothesis, (! [X5:nat]: ((((((listIn521021761append @ a) @ xs) @ f) @ X5) = ((((listIn521021761append @ a) @ ys) @ g) @ X5))))).
% 2.42/1.49 thf(conj_1, hypothesis, (((ord_less_eq @ nat) @ ((size_size @ (list @ a)) @ xs)) @ ((size_size @ (list @ a)) @ ys))).
% 2.42/1.49 thf(conj_2, conjecture, (! [I:nat]: (((~ ((((ord_less @ nat) @ I) @ ((size_size @ (list @ a)) @ xs)))) | ((((nth @ a) @ xs) @ I) = (((nth @ a) @ ys) @ I)))))).
% 2.42/1.49 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2DTF11294.p
% 2.42/1.49 ------------------------
% 2.69/1.51 ---- Cleaned THF ---
% 2.69/1.51 thf(ty_t_List_Olist,type,
% 2.69/1.51 list: $tType > $tType ).
% 2.69/1.51
% 2.69/1.51 thf(ty_t_Nat_Onat,type,
% 2.69/1.51 nat: $tType ).
% 2.69/1.51
% 2.69/1.51 thf(ty_tf_a,type,
% 2.69/1.51 a: $tType ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_HOL_Otype,type,
% 2.69/1.51 type:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Nat_Osize,type,
% 2.69/1.51 size:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Oord,type,
% 2.69/1.51 ord:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Oorder,type,
% 2.69/1.51 order:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Ono__bot,type,
% 2.69/1.51 no_bot:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Ono__top,type,
% 2.69/1.51 no_top:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Olinorder,type,
% 2.69/1.51 linorder:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Opreorder,type,
% 2.69/1.51 preorder:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Owellorder,type,
% 2.69/1.51 wellorder:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Odense__order,type,
% 2.69/1.51 dense_order:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.69/1.51 dense_linorder:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.69/1.51 condit1656338222tinuum:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.69/1.51 condit1037483654norder:
% 2.69/1.51 !>[A: $tType] : $o ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.69/1.51 minus_minus:
% 2.69/1.51 !>[A: $tType] : ( A > A > A ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_List2_Olist__asc,type,
% 2.69/1.51 list_asc:
% 2.69/1.51 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_List2_Olist__desc,type,
% 2.69/1.51 list_desc:
% 2.69/1.51 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_List2_Olist__strict__asc,type,
% 2.69/1.51 list_strict_asc:
% 2.69/1.51 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_List2_Olist__strict__desc,type,
% 2.69/1.51 list_strict_desc:
% 2.69/1.51 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
% 2.69/1.51 listIn521021761append:
% 2.69/1.51 !>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_List_Olinorder__class_Osorted,type,
% 2.69/1.51 linorder_sorted:
% 2.69/1.51 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_List_Olist__ex,type,
% 2.69/1.51 list_ex:
% 2.69/1.51 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_List_Onth,type,
% 2.69/1.51 nth:
% 2.69/1.51 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_List_Orev,type,
% 2.69/1.51 rev:
% 2.69/1.51 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_Nat_OSuc,type,
% 2.69/1.51 suc: nat > nat ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.69/1.51 size_size:
% 2.69/1.51 !>[A: $tType] : ( A > nat ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.69/1.51 ord_less:
% 2.69/1.51 !>[A: $tType] : ( A > A > $o ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.69/1.51 ord_less_eq:
% 2.69/1.51 !>[A: $tType] : ( A > A > $o ) ).
% 2.69/1.51
% 2.69/1.51 thf(sy_v_f,type,
% 2.69/1.51 f: nat > a ).
% 2.69/1.51
% 2.69/1.51 thf(sy_v_g,type,
% 2.69/1.51 g: nat > a ).
% 2.69/1.51
% 2.69/1.51 thf(sy_v_xs,type,
% 2.69/1.51 xs: list @ a ).
% 2.69/1.51
% 2.69/1.51 thf(sy_v_ys,type,
% 2.69/1.51 ys: list @ a ).
% 2.69/1.51
% 2.69/1.51 thf(fact_0_i__append__eq__i__append__conv,axiom,
% 2.69/1.51 ! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
% 2.69/1.51 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.69/1.51 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.69/1.51 => ( ( ( listIn521021761append @ A @ Xs @ F )
% 2.69/1.51 = ( listIn521021761append @ A @ Ys @ G ) )
% 2.69/1.51 = ( ( Xs = Ys )
% 2.69/1.51 & ( F = G ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_1_i__append__nth1,axiom,
% 2.69/1.51 ! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
% 2.69/1.51 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.69/1.51 => ( ( listIn521021761append @ A @ Xs @ F @ N )
% 2.69/1.51 = ( nth @ A @ Xs @ N ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_2_nth__equalityI,axiom,
% 2.69/1.51 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.69/1.51 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.69/1.51 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.69/1.51 => ( ! [I: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.69/1.51 => ( ( nth @ A @ Xs @ I )
% 2.69/1.51 = ( nth @ A @ Ys @ I ) ) )
% 2.69/1.51 => ( Xs = Ys ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_3_Skolem__list__nth,axiom,
% 2.69/1.51 ! [A: $tType,K: nat,P: nat > A > $o] :
% 2.69/1.51 ( ( ! [I2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I2 @ K )
% 2.69/1.51 => ? [X: A] : ( P @ I2 @ X ) ) )
% 2.69/1.51 = ( ? [Xs2: list @ A] :
% 2.69/1.51 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.69/1.51 = K )
% 2.69/1.51 & ! [I2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I2 @ K )
% 2.69/1.51 => ( P @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_4_list__eq__iff__nth__eq,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( ^ [Y: list @ A,Z: list @ A] : ( Y = Z ) )
% 2.69/1.51 = ( ^ [Xs2: list @ A,Ys2: list @ A] :
% 2.69/1.51 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.69/1.51 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 2.69/1.51 & ! [I2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.51 => ( ( nth @ A @ Xs2 @ I2 )
% 2.69/1.51 = ( nth @ A @ Ys2 @ I2 ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_5_order__refl,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_6_list__desc__trans__le,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ( ( list_desc @ A )
% 2.69/1.51 = ( ^ [Xs2: list @ A] :
% 2.69/1.51 ! [J: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.51 => ! [I2: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.69/1.51 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_7_length__induct,axiom,
% 2.69/1.51 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.69/1.51 ( ! [Xs3: list @ A] :
% 2.69/1.51 ( ! [Ys3: list @ A] :
% 2.69/1.51 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.69/1.51 => ( P @ Ys3 ) )
% 2.69/1.51 => ( P @ Xs3 ) )
% 2.69/1.51 => ( P @ Xs ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_8_nat__less__le,axiom,
% 2.69/1.51 ( ( ord_less @ nat )
% 2.69/1.51 = ( ^ [M: nat,N2: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ M @ N2 )
% 2.69/1.51 & ( M != N2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_9_less__imp__le__nat,axiom,
% 2.69/1.51 ! [M2: nat,N: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.51 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_10_le__eq__less__or__eq,axiom,
% 2.69/1.51 ( ( ord_less_eq @ nat )
% 2.69/1.51 = ( ^ [M: nat,N2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ M @ N2 )
% 2.69/1.51 | ( M = N2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_11_less__or__eq__imp__le,axiom,
% 2.69/1.51 ! [M2: nat,N: nat] :
% 2.69/1.51 ( ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.51 | ( M2 = N ) )
% 2.69/1.51 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_12_le__neq__implies__less,axiom,
% 2.69/1.51 ! [M2: nat,N: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.51 => ( ( M2 != N )
% 2.69/1.51 => ( ord_less @ nat @ M2 @ N ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_13_less__mono__imp__le__mono,axiom,
% 2.69/1.51 ! [F: nat > nat,I3: nat,J2: nat] :
% 2.69/1.51 ( ! [I: nat,J3: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I @ J3 )
% 2.69/1.51 => ( ord_less @ nat @ ( F @ I ) @ ( F @ J3 ) ) )
% 2.69/1.51 => ( ( ord_less_eq @ nat @ I3 @ J2 )
% 2.69/1.51 => ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_14_dual__order_Oantisym,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [B: A,A2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ B @ A2 )
% 2.69/1.51 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( A2 = B ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_15_dual__order_Oeq__iff,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.69/1.51 = ( ^ [A3: A,B2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ B2 @ A3 )
% 2.69/1.51 & ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_16_dual__order_Otrans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [B: A,A2: A,C: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ B @ A2 )
% 2.69/1.51 => ( ( ord_less_eq @ A @ C @ B )
% 2.69/1.51 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_17_linorder__wlog,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [P: A > A > $o,A2: A,B: A] :
% 2.69/1.51 ( ! [A4: A,B3: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.69/1.51 => ( P @ A4 @ B3 ) )
% 2.69/1.51 => ( ! [A4: A,B3: A] :
% 2.69/1.51 ( ( P @ B3 @ A4 )
% 2.69/1.51 => ( P @ A4 @ B3 ) )
% 2.69/1.51 => ( P @ A2 @ B ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_18_dual__order_Orefl,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_19_order__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A,Z2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.69/1.51 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_20_order__class_Oorder_Oantisym,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.69/1.51 => ( A2 = B ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_21_ord__le__eq__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( ord @ A )
% 2.69/1.51 => ! [A2: A,B: A,C: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ( B = C )
% 2.69/1.51 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_22_ord__eq__le__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( ord @ A )
% 2.69/1.51 => ! [A2: A,B: A,C: A] :
% 2.69/1.51 ( ( A2 = B )
% 2.69/1.51 => ( ( ord_less_eq @ A @ B @ C )
% 2.69/1.51 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_23_order__class_Oorder_Oeq__iff,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.69/1.51 = ( ^ [A3: A,B2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.69/1.51 & ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_24_antisym__conv,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [Y2: A,X2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.69/1.51 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 = ( X2 = Y2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_25_le__cases3,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A,Z2: A] :
% 2.69/1.51 ( ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.69/1.51 => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.69/1.51 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.69/1.51 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.69/1.51 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.69/1.51 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.69/1.51 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
% 2.69/1.51 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.69/1.51 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 2.69/1.51 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 2.69/1.51 => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_26_order_Otrans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A,C: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less_eq @ A @ B @ C )
% 2.69/1.51 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_27_le__cases,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_28_eq__refl,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( X2 = Y2 )
% 2.69/1.51 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_29_linear,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_30_antisym,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.69/1.51 => ( X2 = Y2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_31_eq__iff,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.69/1.51 = ( ^ [X3: A,Y3: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.51 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_32_ord__le__eq__subst,axiom,
% 2.69/1.51 ! [A: $tType,B4: $tType] :
% 2.69/1.51 ( ( ( ord @ B4 )
% 2.69/1.51 & ( ord @ A ) )
% 2.69/1.51 => ! [A2: A,B: A,F: A > B4,C: B4] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ( ( F @ B )
% 2.69/1.51 = C )
% 2.69/1.51 => ( ! [X4: A,Y4: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less_eq @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less_eq @ B4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_33_ord__eq__le__subst,axiom,
% 2.69/1.51 ! [A: $tType,B4: $tType] :
% 2.69/1.51 ( ( ( ord @ B4 )
% 2.69/1.51 & ( ord @ A ) )
% 2.69/1.51 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 2.69/1.51 ( ( A2
% 2.69/1.51 = ( F @ B ) )
% 2.69/1.51 => ( ( ord_less_eq @ B4 @ B @ C )
% 2.69/1.51 => ( ! [X4: B4,Y4: B4] :
% 2.69/1.51 ( ( ord_less_eq @ B4 @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_34_order__subst2,axiom,
% 2.69/1.51 ! [A: $tType,C2: $tType] :
% 2.69/1.51 ( ( ( order @ C2 )
% 2.69/1.51 & ( order @ A ) )
% 2.69/1.51 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 2.69/1.51 => ( ! [X4: A,Y4: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_35_order__subst1,axiom,
% 2.69/1.51 ! [A: $tType,B4: $tType] :
% 2.69/1.51 ( ( ( order @ B4 )
% 2.69/1.51 & ( order @ A ) )
% 2.69/1.51 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 2.69/1.51 => ( ( ord_less_eq @ B4 @ B @ C )
% 2.69/1.51 => ( ! [X4: B4,Y4: B4] :
% 2.69/1.51 ( ( ord_less_eq @ B4 @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_36_le__fun__def,axiom,
% 2.69/1.51 ! [B4: $tType,A: $tType] :
% 2.69/1.51 ( ( ord @ B4 )
% 2.69/1.51 => ( ( ord_less_eq @ ( A > B4 ) )
% 2.69/1.51 = ( ^ [F2: A > B4,G2: A > B4] :
% 2.69/1.51 ! [X3: A] : ( ord_less_eq @ B4 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_37_le__funI,axiom,
% 2.69/1.51 ! [B4: $tType,A: $tType] :
% 2.69/1.51 ( ( ord @ B4 )
% 2.69/1.51 => ! [F: A > B4,G: A > B4] :
% 2.69/1.51 ( ! [X4: A] : ( ord_less_eq @ B4 @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.69/1.51 => ( ord_less_eq @ ( A > B4 ) @ F @ G ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_38_le__funE,axiom,
% 2.69/1.51 ! [B4: $tType,A: $tType] :
% 2.69/1.51 ( ( ord @ B4 )
% 2.69/1.51 => ! [F: A > B4,G: A > B4,X2: A] :
% 2.69/1.51 ( ( ord_less_eq @ ( A > B4 ) @ F @ G )
% 2.69/1.51 => ( ord_less_eq @ B4 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_39_le__funD,axiom,
% 2.69/1.51 ! [B4: $tType,A: $tType] :
% 2.69/1.51 ( ( ord @ B4 )
% 2.69/1.51 => ! [F: A > B4,G: A > B4,X2: A] :
% 2.69/1.51 ( ( ord_less_eq @ ( A > B4 ) @ F @ G )
% 2.69/1.51 => ( ord_less_eq @ B4 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_40_dual__order_Ostrict__implies__not__eq,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [B: A,A2: A] :
% 2.69/1.51 ( ( ord_less @ A @ B @ A2 )
% 2.69/1.51 => ( A2 != B ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_41_order_Ostrict__implies__not__eq,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( A2 != B ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_42_not__less__iff__gr__or__eq,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 2.69/1.51 = ( ( ord_less @ A @ Y2 @ X2 )
% 2.69/1.51 | ( X2 = Y2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_43_dual__order_Ostrict__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [B: A,A2: A,C: A] :
% 2.69/1.51 ( ( ord_less @ A @ B @ A2 )
% 2.69/1.51 => ( ( ord_less @ A @ C @ B )
% 2.69/1.51 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_44_linorder__less__wlog,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [P: A > A > $o,A2: A,B: A] :
% 2.69/1.51 ( ! [A4: A,B3: A] :
% 2.69/1.51 ( ( ord_less @ A @ A4 @ B3 )
% 2.69/1.51 => ( P @ A4 @ B3 ) )
% 2.69/1.51 => ( ! [A4: A] : ( P @ A4 @ A4 )
% 2.69/1.51 => ( ! [A4: A,B3: A] :
% 2.69/1.51 ( ( P @ B3 @ A4 )
% 2.69/1.51 => ( P @ A4 @ B3 ) )
% 2.69/1.51 => ( P @ A2 @ B ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_45_ext,axiom,
% 2.69/1.51 ! [B4: $tType,A: $tType,F: A > B4,G: A > B4] :
% 2.69/1.51 ( ! [X4: A] :
% 2.69/1.51 ( ( F @ X4 )
% 2.69/1.51 = ( G @ X4 ) )
% 2.69/1.51 => ( F = G ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_46_exists__least__iff,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( wellorder @ A )
% 2.69/1.51 => ( ( ^ [P2: A > $o] :
% 2.69/1.51 ? [X: A] : ( P2 @ X ) )
% 2.69/1.51 = ( ^ [P3: A > $o] :
% 2.69/1.51 ? [N2: A] :
% 2.69/1.51 ( ( P3 @ N2 )
% 2.69/1.51 & ! [M: A] :
% 2.69/1.51 ( ( ord_less @ A @ M @ N2 )
% 2.69/1.51 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_47_less__imp__not__less,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_48_order_Ostrict__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A,C: A] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less @ A @ B @ C )
% 2.69/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_49_dual__order_Oirrefl,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A] :
% 2.69/1.51 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_50_linorder__cases,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( X2 != Y2 )
% 2.69/1.51 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_51_less__imp__triv,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A,P: $o] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ord_less @ A @ Y2 @ X2 )
% 2.69/1.51 => P ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_52_less__imp__not__eq2,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( Y2 != X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_53_antisym__conv3,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [Y2: A,X2: A] :
% 2.69/1.51 ( ~ ( ord_less @ A @ Y2 @ X2 )
% 2.69/1.51 => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 2.69/1.51 = ( X2 = Y2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_54_less__induct,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( wellorder @ A )
% 2.69/1.51 => ! [P: A > $o,A2: A] :
% 2.69/1.51 ( ! [X4: A] :
% 2.69/1.51 ( ! [Y5: A] :
% 2.69/1.51 ( ( ord_less @ A @ Y5 @ X4 )
% 2.69/1.51 => ( P @ Y5 ) )
% 2.69/1.51 => ( P @ X4 ) )
% 2.69/1.51 => ( P @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_55_less__not__sym,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_56_less__imp__not__eq,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( X2 != Y2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_57_dual__order_Oasym,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [B: A,A2: A] :
% 2.69/1.51 ( ( ord_less @ A @ B @ A2 )
% 2.69/1.51 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_58_ord__less__eq__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( ord @ A )
% 2.69/1.51 => ! [A2: A,B: A,C: A] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( ( B = C )
% 2.69/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_59_ord__eq__less__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( ord @ A )
% 2.69/1.51 => ! [A2: A,B: A,C: A] :
% 2.69/1.51 ( ( A2 = B )
% 2.69/1.51 => ( ( ord_less @ A @ B @ C )
% 2.69/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_60_less__irrefl,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A] :
% 2.69/1.51 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_61_less__linear,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 | ( X2 = Y2 )
% 2.69/1.51 | ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_62_less__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A,Z2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.69/1.51 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_63_less__asym_H,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [A2: A,B: A] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_64_less__asym,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_65_less__imp__neq,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( X2 != Y2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_66_dense,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( dense_order @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ? [Z3: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Z3 )
% 2.69/1.51 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_67_order_Oasym,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_68_neq__iff,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( X2 != Y2 )
% 2.69/1.51 = ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 | ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_69_neqE,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( X2 != Y2 )
% 2.69/1.51 => ( ~ ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_70_gt__ex,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( no_top @ A )
% 2.69/1.51 => ! [X2: A] :
% 2.69/1.51 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_71_lt__ex,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( no_bot @ A )
% 2.69/1.51 => ! [X2: A] :
% 2.69/1.51 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_72_order__less__subst2,axiom,
% 2.69/1.51 ! [A: $tType,C2: $tType] :
% 2.69/1.51 ( ( ( order @ C2 )
% 2.69/1.51 & ( order @ A ) )
% 2.69/1.51 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 2.69/1.51 => ( ! [X4: A,Y4: A] :
% 2.69/1.51 ( ( ord_less @ A @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_73_order__less__subst1,axiom,
% 2.69/1.51 ! [A: $tType,B4: $tType] :
% 2.69/1.51 ( ( ( order @ B4 )
% 2.69/1.51 & ( order @ A ) )
% 2.69/1.51 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 2.69/1.51 => ( ( ord_less @ B4 @ B @ C )
% 2.69/1.51 => ( ! [X4: B4,Y4: B4] :
% 2.69/1.51 ( ( ord_less @ B4 @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_74_ord__less__eq__subst,axiom,
% 2.69/1.51 ! [A: $tType,B4: $tType] :
% 2.69/1.51 ( ( ( ord @ B4 )
% 2.69/1.51 & ( ord @ A ) )
% 2.69/1.51 => ! [A2: A,B: A,F: A > B4,C: B4] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( ( ( F @ B )
% 2.69/1.51 = C )
% 2.69/1.51 => ( ! [X4: A,Y4: A] :
% 2.69/1.51 ( ( ord_less @ A @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less @ B4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_75_ord__eq__less__subst,axiom,
% 2.69/1.51 ! [A: $tType,B4: $tType] :
% 2.69/1.51 ( ( ( ord @ B4 )
% 2.69/1.51 & ( ord @ A ) )
% 2.69/1.51 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 2.69/1.51 ( ( A2
% 2.69/1.51 = ( F @ B ) )
% 2.69/1.51 => ( ( ord_less @ B4 @ B @ C )
% 2.69/1.51 => ( ! [X4: B4,Y4: B4] :
% 2.69/1.51 ( ( ord_less @ B4 @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_76_measure__induct__rule,axiom,
% 2.69/1.51 ! [B4: $tType,A: $tType] :
% 2.69/1.51 ( ( wellorder @ B4 )
% 2.69/1.51 => ! [F: A > B4,P: A > $o,A2: A] :
% 2.69/1.51 ( ! [X4: A] :
% 2.69/1.51 ( ! [Y5: A] :
% 2.69/1.51 ( ( ord_less @ B4 @ ( F @ Y5 ) @ ( F @ X4 ) )
% 2.69/1.51 => ( P @ Y5 ) )
% 2.69/1.51 => ( P @ X4 ) )
% 2.69/1.51 => ( P @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_77_measure__induct,axiom,
% 2.69/1.51 ! [B4: $tType,A: $tType] :
% 2.69/1.51 ( ( wellorder @ B4 )
% 2.69/1.51 => ! [F: A > B4,P: A > $o,A2: A] :
% 2.69/1.51 ( ! [X4: A] :
% 2.69/1.51 ( ! [Y5: A] :
% 2.69/1.51 ( ( ord_less @ B4 @ ( F @ Y5 ) @ ( F @ X4 ) )
% 2.69/1.51 => ( P @ Y5 ) )
% 2.69/1.51 => ( P @ X4 ) )
% 2.69/1.51 => ( P @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_78_infinite__descent__measure,axiom,
% 2.69/1.51 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 2.69/1.51 ( ! [X4: A] :
% 2.69/1.51 ( ~ ( P @ X4 )
% 2.69/1.51 => ? [Y5: A] :
% 2.69/1.51 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X4 ) )
% 2.69/1.51 & ~ ( P @ Y5 ) ) )
% 2.69/1.51 => ( P @ X2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_79_linorder__neqE__nat,axiom,
% 2.69/1.51 ! [X2: nat,Y2: nat] :
% 2.69/1.51 ( ( X2 != Y2 )
% 2.69/1.51 => ( ~ ( ord_less @ nat @ X2 @ Y2 )
% 2.69/1.51 => ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_80_infinite__descent,axiom,
% 2.69/1.51 ! [P: nat > $o,N: nat] :
% 2.69/1.51 ( ! [N3: nat] :
% 2.69/1.51 ( ~ ( P @ N3 )
% 2.69/1.51 => ? [M3: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ M3 @ N3 )
% 2.69/1.51 & ~ ( P @ M3 ) ) )
% 2.69/1.51 => ( P @ N ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_81_nat__less__induct,axiom,
% 2.69/1.51 ! [P: nat > $o,N: nat] :
% 2.69/1.51 ( ! [N3: nat] :
% 2.69/1.51 ( ! [M3: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ M3 @ N3 )
% 2.69/1.51 => ( P @ M3 ) )
% 2.69/1.51 => ( P @ N3 ) )
% 2.69/1.51 => ( P @ N ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_82_less__irrefl__nat,axiom,
% 2.69/1.51 ! [N: nat] :
% 2.69/1.51 ~ ( ord_less @ nat @ N @ N ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_83_less__not__refl3,axiom,
% 2.69/1.51 ! [S: nat,T: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ S @ T )
% 2.69/1.51 => ( S != T ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_84_less__not__refl2,axiom,
% 2.69/1.51 ! [N: nat,M2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ N @ M2 )
% 2.69/1.51 => ( M2 != N ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_85_less__not__refl,axiom,
% 2.69/1.51 ! [N: nat] :
% 2.69/1.51 ~ ( ord_less @ nat @ N @ N ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_86_nat__neq__iff,axiom,
% 2.69/1.51 ! [M2: nat,N: nat] :
% 2.69/1.51 ( ( M2 != N )
% 2.69/1.51 = ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.51 | ( ord_less @ nat @ N @ M2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_87_Nat_Oex__has__greatest__nat,axiom,
% 2.69/1.51 ! [P: nat > $o,K: nat,B: nat] :
% 2.69/1.51 ( ( P @ K )
% 2.69/1.51 => ( ! [Y4: nat] :
% 2.69/1.51 ( ( P @ Y4 )
% 2.69/1.51 => ( ord_less_eq @ nat @ Y4 @ B ) )
% 2.69/1.51 => ? [X4: nat] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 & ! [Y5: nat] :
% 2.69/1.51 ( ( P @ Y5 )
% 2.69/1.51 => ( ord_less_eq @ nat @ Y5 @ X4 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_88_nat__le__linear,axiom,
% 2.69/1.51 ! [M2: nat,N: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.51 | ( ord_less_eq @ nat @ N @ M2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_89_le__antisym,axiom,
% 2.69/1.51 ! [M2: nat,N: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.51 => ( ( ord_less_eq @ nat @ N @ M2 )
% 2.69/1.51 => ( M2 = N ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_90_eq__imp__le,axiom,
% 2.69/1.51 ! [M2: nat,N: nat] :
% 2.69/1.51 ( ( M2 = N )
% 2.69/1.51 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_91_le__trans,axiom,
% 2.69/1.51 ! [I3: nat,J2: nat,K: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ I3 @ J2 )
% 2.69/1.51 => ( ( ord_less_eq @ nat @ J2 @ K )
% 2.69/1.51 => ( ord_less_eq @ nat @ I3 @ K ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_92_le__refl,axiom,
% 2.69/1.51 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_93_size__neq__size__imp__neq,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( size @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ( size_size @ A @ X2 )
% 2.69/1.51 != ( size_size @ A @ Y2 ) )
% 2.69/1.51 => ( X2 != Y2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_94_neq__if__length__neq,axiom,
% 2.69/1.51 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.69/1.51 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.69/1.51 != ( size_size @ ( list @ A ) @ Ys ) )
% 2.69/1.51 => ( Xs != Ys ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_95_Ex__list__of__length,axiom,
% 2.69/1.51 ! [A: $tType,N: nat] :
% 2.69/1.51 ? [Xs3: list @ A] :
% 2.69/1.51 ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.69/1.51 = N ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_96_list__desc__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ( ( list_desc @ A )
% 2.69/1.51 = ( ^ [Xs2: list @ A] :
% 2.69/1.51 ! [J: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.51 => ! [I2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I2 @ J )
% 2.69/1.51 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_97_order_Onot__eq__order__implies__strict,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A] :
% 2.69/1.51 ( ( A2 != B )
% 2.69/1.51 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_98_dual__order_Ostrict__implies__order,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [B: A,A2: A] :
% 2.69/1.51 ( ( ord_less @ A @ B @ A2 )
% 2.69/1.51 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_99_dual__order_Ostrict__iff__order,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ord_less @ A )
% 2.69/1.51 = ( ^ [B2: A,A3: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ B2 @ A3 )
% 2.69/1.51 & ( A3 != B2 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_100_dual__order_Oorder__iff__strict,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ord_less_eq @ A )
% 2.69/1.51 = ( ^ [B2: A,A3: A] :
% 2.69/1.51 ( ( ord_less @ A @ B2 @ A3 )
% 2.69/1.51 | ( A3 = B2 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_101_order_Ostrict__implies__order,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_102_dense__le__bounded,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( dense_linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A,Z2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ! [W: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ W )
% 2.69/1.51 => ( ( ord_less @ A @ W @ Y2 )
% 2.69/1.51 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.69/1.51 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_103_dense__ge__bounded,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( dense_linorder @ A )
% 2.69/1.51 => ! [Z2: A,X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z2 @ X2 )
% 2.69/1.51 => ( ! [W: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z2 @ W )
% 2.69/1.51 => ( ( ord_less @ A @ W @ X2 )
% 2.69/1.51 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 2.69/1.51 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_104_dual__order_Ostrict__trans2,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [B: A,A2: A,C: A] :
% 2.69/1.51 ( ( ord_less @ A @ B @ A2 )
% 2.69/1.51 => ( ( ord_less_eq @ A @ C @ B )
% 2.69/1.51 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_105_dual__order_Ostrict__trans1,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [B: A,A2: A,C: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ B @ A2 )
% 2.69/1.51 => ( ( ord_less @ A @ C @ B )
% 2.69/1.51 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_106_order_Ostrict__iff__order,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ord_less @ A )
% 2.69/1.51 = ( ^ [A3: A,B2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.69/1.51 & ( A3 != B2 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_107_order_Oorder__iff__strict,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ord_less_eq @ A )
% 2.69/1.51 = ( ^ [A3: A,B2: A] :
% 2.69/1.51 ( ( ord_less @ A @ A3 @ B2 )
% 2.69/1.51 | ( A3 = B2 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_108_order_Ostrict__trans2,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A,C: A] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less_eq @ A @ B @ C )
% 2.69/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_109_order_Ostrict__trans1,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A,C: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less @ A @ B @ C )
% 2.69/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_110_not__le__imp__less,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [Y2: A,X2: A] :
% 2.69/1.51 ( ~ ( ord_less_eq @ A @ Y2 @ X2 )
% 2.69/1.51 => ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_111_less__le__not__le,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ( ( ord_less @ A )
% 2.69/1.51 = ( ^ [X3: A,Y3: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.51 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_112_le__imp__less__or__eq,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 | ( X2 = Y2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_113_le__less__linear,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 | ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_114_dense__le,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( dense_linorder @ A )
% 2.69/1.51 => ! [Y2: A,Z2: A] :
% 2.69/1.51 ( ! [X4: A] :
% 2.69/1.51 ( ( ord_less @ A @ X4 @ Y2 )
% 2.69/1.51 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.69/1.51 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_115_dense__ge,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( dense_linorder @ A )
% 2.69/1.51 => ! [Z2: A,Y2: A] :
% 2.69/1.51 ( ! [X4: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z2 @ X4 )
% 2.69/1.51 => ( ord_less_eq @ A @ Y2 @ X4 ) )
% 2.69/1.51 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_116_less__le__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A,Z2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.69/1.51 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_117_le__less__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A,Z2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.69/1.51 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_118_less__imp__le,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_119_antisym__conv2,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 2.69/1.51 = ( X2 = Y2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_120_antisym__conv1,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.51 = ( X2 = Y2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_121_le__neq__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [A2: A,B: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ( A2 != B )
% 2.69/1.51 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_122_not__less,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 2.69/1.51 = ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_123_not__le,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ( ~ ( ord_less_eq @ A @ X2 @ Y2 ) )
% 2.69/1.51 = ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_124_order__less__le__subst2,axiom,
% 2.69/1.51 ! [A: $tType,C2: $tType] :
% 2.69/1.51 ( ( ( order @ C2 )
% 2.69/1.51 & ( order @ A ) )
% 2.69/1.51 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 2.69/1.51 => ( ! [X4: A,Y4: A] :
% 2.69/1.51 ( ( ord_less @ A @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_125_order__less__le__subst1,axiom,
% 2.69/1.51 ! [A: $tType,B4: $tType] :
% 2.69/1.51 ( ( ( order @ B4 )
% 2.69/1.51 & ( order @ A ) )
% 2.69/1.51 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 2.69/1.51 => ( ( ord_less_eq @ B4 @ B @ C )
% 2.69/1.51 => ( ! [X4: B4,Y4: B4] :
% 2.69/1.51 ( ( ord_less_eq @ B4 @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_126_order__le__less__subst2,axiom,
% 2.69/1.51 ! [A: $tType,C2: $tType] :
% 2.69/1.51 ( ( ( order @ C2 )
% 2.69/1.51 & ( order @ A ) )
% 2.69/1.51 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.51 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 2.69/1.51 => ( ! [X4: A,Y4: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_127_order__le__less__subst1,axiom,
% 2.69/1.51 ! [A: $tType,B4: $tType] :
% 2.69/1.51 ( ( ( order @ B4 )
% 2.69/1.51 & ( order @ A ) )
% 2.69/1.51 => ! [A2: A,F: B4 > A,B: B4,C: B4] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 2.69/1.51 => ( ( ord_less @ B4 @ B @ C )
% 2.69/1.51 => ( ! [X4: B4,Y4: B4] :
% 2.69/1.51 ( ( ord_less @ B4 @ X4 @ Y4 )
% 2.69/1.51 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.69/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_128_less__le,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ord_less @ A )
% 2.69/1.51 = ( ^ [X3: A,Y3: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.51 & ( X3 != Y3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_129_le__less,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ( ( ord_less_eq @ A )
% 2.69/1.51 = ( ^ [X3: A,Y3: A] :
% 2.69/1.51 ( ( ord_less @ A @ X3 @ Y3 )
% 2.69/1.51 | ( X3 = Y3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_130_leI,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [X2: A,Y2: A] :
% 2.69/1.51 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 2.69/1.51 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_131_leD,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( order @ A )
% 2.69/1.51 => ! [Y2: A,X2: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.69/1.51 => ~ ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_132_list__strict__asc__trans__le,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [Xs: list @ A] :
% 2.69/1.51 ( ( list_strict_asc @ A @ Xs )
% 2.69/1.51 => ! [J4: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.69/1.51 => ! [I4: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ I4 @ J4 )
% 2.69/1.51 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_133_list__asc__trans__le,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ( ( list_asc @ A )
% 2.69/1.51 = ( ^ [Xs2: list @ A] :
% 2.69/1.51 ! [J: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.51 => ! [I2: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.69/1.51 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_134_list__strict__desc__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ( ( list_strict_desc @ A )
% 2.69/1.51 = ( ^ [Xs2: list @ A] :
% 2.69/1.51 ! [J: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.51 => ! [I2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I2 @ J )
% 2.69/1.51 => ( ord_less @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_135_list__strict__asc__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ( ( list_strict_asc @ A )
% 2.69/1.51 = ( ^ [Xs2: list @ A] :
% 2.69/1.51 ! [J: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.51 => ! [I2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I2 @ J )
% 2.69/1.51 => ( ord_less @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_136_list__asc__trans,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ( ( list_asc @ A )
% 2.69/1.51 = ( ^ [Xs2: list @ A] :
% 2.69/1.51 ! [J: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.51 => ! [I2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ I2 @ J )
% 2.69/1.51 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_137_Lattices__Big_Oex__has__greatest__nat,axiom,
% 2.69/1.51 ! [A: $tType,P: A > $o,K: A,F: A > nat,B: nat] :
% 2.69/1.51 ( ( P @ K )
% 2.69/1.51 => ( ! [Y4: A] :
% 2.69/1.51 ( ( P @ Y4 )
% 2.69/1.51 => ( ord_less @ nat @ ( F @ Y4 ) @ B ) )
% 2.69/1.51 => ? [X4: A] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 & ! [Y5: A] :
% 2.69/1.51 ( ( P @ Y5 )
% 2.69/1.51 => ( ord_less_eq @ nat @ ( F @ Y5 ) @ ( F @ X4 ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_138_nat__descend__induct,axiom,
% 2.69/1.51 ! [N: nat,P: nat > $o,M2: nat] :
% 2.69/1.51 ( ! [K2: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ N @ K2 )
% 2.69/1.51 => ( P @ K2 ) )
% 2.69/1.51 => ( ! [K2: nat] :
% 2.69/1.51 ( ( ord_less_eq @ nat @ K2 @ N )
% 2.69/1.51 => ( ! [I4: nat] :
% 2.69/1.51 ( ( ord_less @ nat @ K2 @ I4 )
% 2.69/1.51 => ( P @ I4 ) )
% 2.69/1.51 => ( P @ K2 ) ) )
% 2.69/1.51 => ( P @ M2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_139_complete__interval,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( condit1037483654norder @ A )
% 2.69/1.51 => ! [A2: A,B: A,P: A > $o] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B )
% 2.69/1.51 => ( ( P @ A2 )
% 2.69/1.51 => ( ~ ( P @ B )
% 2.69/1.51 => ? [C3: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.69/1.51 & ( ord_less_eq @ A @ C3 @ B )
% 2.69/1.51 & ! [X5: A] :
% 2.69/1.51 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 2.69/1.51 & ( ord_less @ A @ X5 @ C3 ) )
% 2.69/1.51 => ( P @ X5 ) )
% 2.69/1.51 & ! [D: A] :
% 2.69/1.51 ( ! [X4: A] :
% 2.69/1.51 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 2.69/1.51 & ( ord_less @ A @ X4 @ D ) )
% 2.69/1.51 => ( P @ X4 ) )
% 2.69/1.51 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_140_le__greater__neq__conv,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ( ( ord_less_eq @ A )
% 2.69/1.51 = ( ^ [N2: A,A3: A] :
% 2.69/1.51 ! [X3: A] :
% 2.69/1.51 ( ( ord_less @ A @ A3 @ X3 )
% 2.69/1.51 => ( N2 != X3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_141_greater__le__neq__conv,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ( ( ord_less @ A )
% 2.69/1.51 = ( ^ [A3: A,N2: A] :
% 2.69/1.51 ! [X3: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ X3 @ A3 )
% 2.69/1.51 => ( N2 != X3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_142_less__ge__neq__conv,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ( ( ord_less @ A )
% 2.69/1.51 = ( ^ [N2: A,A3: A] :
% 2.69/1.51 ! [X3: A] :
% 2.69/1.51 ( ( ord_less_eq @ A @ A3 @ X3 )
% 2.69/1.51 => ( N2 != X3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_143_less__fun__def,axiom,
% 2.69/1.51 ! [B4: $tType,A: $tType] :
% 2.69/1.51 ( ( ord @ B4 )
% 2.69/1.51 => ( ( ord_less @ ( A > B4 ) )
% 2.69/1.51 = ( ^ [F2: A > B4,G2: A > B4] :
% 2.69/1.51 ( ( ord_less_eq @ ( A > B4 ) @ F2 @ G2 )
% 2.69/1.51 & ~ ( ord_less_eq @ ( A > B4 ) @ G2 @ F2 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_144_list__strict__asc__imp__list__asc,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [Xs: list @ A] :
% 2.69/1.51 ( ( list_strict_asc @ A @ Xs )
% 2.69/1.51 => ( list_asc @ A @ Xs ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_145_ex__gt__or__lt,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( condit1656338222tinuum @ A )
% 2.69/1.51 => ! [A2: A] :
% 2.69/1.51 ? [B3: A] :
% 2.69/1.51 ( ( ord_less @ A @ A2 @ B3 )
% 2.69/1.51 | ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_146_ex__has__least__nat,axiom,
% 2.69/1.51 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 2.69/1.51 ( ( P @ K )
% 2.69/1.51 => ? [X4: A] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 & ! [Y5: A] :
% 2.69/1.51 ( ( P @ Y5 )
% 2.69/1.51 => ( ord_less_eq @ nat @ ( M2 @ X4 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_147_list__strict__desc__imp__list__desc,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( preorder @ A )
% 2.69/1.51 => ! [Xs: list @ A] :
% 2.69/1.51 ( ( list_strict_desc @ A @ Xs )
% 2.69/1.51 => ( list_desc @ A @ Xs ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_148_ge__less__neq__conv,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ( ( ord_less_eq @ A )
% 2.69/1.51 = ( ^ [A3: A,N2: A] :
% 2.69/1.51 ! [X3: A] :
% 2.69/1.51 ( ( ord_less @ A @ X3 @ A3 )
% 2.69/1.51 => ( N2 != X3 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_149_minf_I8_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [T: A] :
% 2.69/1.51 ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ X5 @ Z3 )
% 2.69/1.51 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_150_minf_I6_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [T: A] :
% 2.69/1.51 ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ X5 @ Z3 )
% 2.69/1.51 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_151_pinf_I8_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [T: A] :
% 2.69/1.51 ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z3 @ X5 )
% 2.69/1.51 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_152_pinf_I1_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.69/1.51 ( ? [Z4: A] :
% 2.69/1.51 ! [X4: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z4 @ X4 )
% 2.69/1.51 => ( ( P @ X4 )
% 2.69/1.51 = ( P4 @ X4 ) ) )
% 2.69/1.51 => ( ? [Z4: A] :
% 2.69/1.51 ! [X4: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z4 @ X4 )
% 2.69/1.51 => ( ( Q @ X4 )
% 2.69/1.51 = ( Q2 @ X4 ) ) )
% 2.69/1.51 => ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z3 @ X5 )
% 2.69/1.51 => ( ( ( P @ X5 )
% 2.69/1.51 & ( Q @ X5 ) )
% 2.69/1.51 = ( ( P4 @ X5 )
% 2.69/1.51 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_153_pinf_I2_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.69/1.51 ( ? [Z4: A] :
% 2.69/1.51 ! [X4: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z4 @ X4 )
% 2.69/1.51 => ( ( P @ X4 )
% 2.69/1.51 = ( P4 @ X4 ) ) )
% 2.69/1.51 => ( ? [Z4: A] :
% 2.69/1.51 ! [X4: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z4 @ X4 )
% 2.69/1.51 => ( ( Q @ X4 )
% 2.69/1.51 = ( Q2 @ X4 ) ) )
% 2.69/1.51 => ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z3 @ X5 )
% 2.69/1.51 => ( ( ( P @ X5 )
% 2.69/1.51 | ( Q @ X5 ) )
% 2.69/1.51 = ( ( P4 @ X5 )
% 2.69/1.51 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_154_pinf_I3_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [T: A] :
% 2.69/1.51 ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z3 @ X5 )
% 2.69/1.51 => ( X5 != T ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_155_pinf_I4_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [T: A] :
% 2.69/1.51 ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z3 @ X5 )
% 2.69/1.51 => ( X5 != T ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_156_pinf_I5_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [T: A] :
% 2.69/1.51 ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z3 @ X5 )
% 2.69/1.51 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_157_pinf_I7_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [T: A] :
% 2.69/1.51 ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ Z3 @ X5 )
% 2.69/1.51 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_158_pinf_I11_J,axiom,
% 2.69/1.51 ! [C2: $tType,D2: $tType] :
% 2.69/1.51 ( ( ord @ C2 )
% 2.69/1.51 => ! [F3: D2] :
% 2.69/1.51 ? [Z3: C2] :
% 2.69/1.51 ! [X5: C2] :
% 2.69/1.51 ( ( ord_less @ C2 @ Z3 @ X5 )
% 2.69/1.51 => ( F3 = F3 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_159_minf_I1_J,axiom,
% 2.69/1.51 ! [A: $tType] :
% 2.69/1.51 ( ( linorder @ A )
% 2.69/1.51 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.69/1.51 ( ? [Z4: A] :
% 2.69/1.51 ! [X4: A] :
% 2.69/1.51 ( ( ord_less @ A @ X4 @ Z4 )
% 2.69/1.51 => ( ( P @ X4 )
% 2.69/1.51 = ( P4 @ X4 ) ) )
% 2.69/1.51 => ( ? [Z4: A] :
% 2.69/1.51 ! [X4: A] :
% 2.69/1.51 ( ( ord_less @ A @ X4 @ Z4 )
% 2.69/1.51 => ( ( Q @ X4 )
% 2.69/1.51 = ( Q2 @ X4 ) ) )
% 2.69/1.51 => ? [Z3: A] :
% 2.69/1.51 ! [X5: A] :
% 2.69/1.51 ( ( ord_less @ A @ X5 @ Z3 )
% 2.69/1.51 => ( ( ( P @ X5 )
% 2.69/1.52 & ( Q @ X5 ) )
% 2.69/1.52 = ( ( P4 @ X5 )
% 2.69/1.52 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_160_minf_I2_J,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.69/1.52 ( ? [Z4: A] :
% 2.69/1.52 ! [X4: A] :
% 2.69/1.52 ( ( ord_less @ A @ X4 @ Z4 )
% 2.69/1.52 => ( ( P @ X4 )
% 2.69/1.52 = ( P4 @ X4 ) ) )
% 2.69/1.52 => ( ? [Z4: A] :
% 2.69/1.52 ! [X4: A] :
% 2.69/1.52 ( ( ord_less @ A @ X4 @ Z4 )
% 2.69/1.52 => ( ( Q @ X4 )
% 2.69/1.52 = ( Q2 @ X4 ) ) )
% 2.69/1.52 => ? [Z3: A] :
% 2.69/1.52 ! [X5: A] :
% 2.69/1.52 ( ( ord_less @ A @ X5 @ Z3 )
% 2.69/1.52 => ( ( ( P @ X5 )
% 2.69/1.52 | ( Q @ X5 ) )
% 2.69/1.52 = ( ( P4 @ X5 )
% 2.69/1.52 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_161_minf_I3_J,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [T: A] :
% 2.69/1.52 ? [Z3: A] :
% 2.69/1.52 ! [X5: A] :
% 2.69/1.52 ( ( ord_less @ A @ X5 @ Z3 )
% 2.69/1.52 => ( X5 != T ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_162_minf_I4_J,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [T: A] :
% 2.69/1.52 ? [Z3: A] :
% 2.69/1.52 ! [X5: A] :
% 2.69/1.52 ( ( ord_less @ A @ X5 @ Z3 )
% 2.69/1.52 => ( X5 != T ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_163_minf_I5_J,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [T: A] :
% 2.69/1.52 ? [Z3: A] :
% 2.69/1.52 ! [X5: A] :
% 2.69/1.52 ( ( ord_less @ A @ X5 @ Z3 )
% 2.69/1.52 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_164_minf_I7_J,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [T: A] :
% 2.69/1.52 ? [Z3: A] :
% 2.69/1.52 ! [X5: A] :
% 2.69/1.52 ( ( ord_less @ A @ X5 @ Z3 )
% 2.69/1.52 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_165_minf_I11_J,axiom,
% 2.69/1.52 ! [C2: $tType,D2: $tType] :
% 2.69/1.52 ( ( ord @ C2 )
% 2.69/1.52 => ! [F3: D2] :
% 2.69/1.52 ? [Z3: C2] :
% 2.69/1.52 ! [X5: C2] :
% 2.69/1.52 ( ( ord_less @ C2 @ X5 @ Z3 )
% 2.69/1.52 => ( F3 = F3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_166_pinf_I6_J,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [T: A] :
% 2.69/1.52 ? [Z3: A] :
% 2.69/1.52 ! [X5: A] :
% 2.69/1.52 ( ( ord_less @ A @ Z3 @ X5 )
% 2.69/1.52 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_167_verit__comp__simplify1_I3_J,axiom,
% 2.69/1.52 ! [B4: $tType] :
% 2.69/1.52 ( ( linorder @ B4 )
% 2.69/1.52 => ! [B5: B4,A5: B4] :
% 2.69/1.52 ( ( ~ ( ord_less_eq @ B4 @ B5 @ A5 ) )
% 2.69/1.52 = ( ord_less @ B4 @ A5 @ B5 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_168_sorted__iff__nth__mono,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ( ( linorder_sorted @ A )
% 2.69/1.52 = ( ^ [Xs2: list @ A] :
% 2.69/1.52 ! [I2: nat,J: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.69/1.52 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.52 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_169_sorted__nth__mono,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [Xs: list @ A,I3: nat,J2: nat] :
% 2.69/1.52 ( ( linorder_sorted @ A @ Xs )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ I3 @ J2 )
% 2.69/1.52 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.69/1.52 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_170_list__ord__le__sorted__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ( ( list_asc @ A )
% 2.69/1.52 = ( linorder_sorted @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_171_verit__la__disequality,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [A2: A,B: A] :
% 2.69/1.52 ( ( A2 = B )
% 2.69/1.52 | ~ ( ord_less_eq @ A @ A2 @ B )
% 2.69/1.52 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_172_verit__comp__simplify1_I1_J,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A )
% 2.69/1.52 => ! [A2: A] :
% 2.69/1.52 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_173_sorted__iff__nth__mono__less,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ( ( linorder_sorted @ A )
% 2.69/1.52 = ( ^ [Xs2: list @ A] :
% 2.69/1.52 ! [I2: nat,J: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I2 @ J )
% 2.69/1.52 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.52 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_174_sorted__rev__nth__mono,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [Xs: list @ A,I3: nat,J2: nat] :
% 2.69/1.52 ( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ I3 @ J2 )
% 2.69/1.52 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.69/1.52 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J2 ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_175_sorted__rev__iff__nth__mono,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [Xs: list @ A] :
% 2.69/1.52 ( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
% 2.69/1.52 = ( ! [I2: nat,J: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.69/1.52 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.69/1.52 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_176_sorted__iff__nth__Suc,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ( ( linorder_sorted @ A )
% 2.69/1.52 = ( ^ [Xs2: list @ A] :
% 2.69/1.52 ! [I2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.52 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_177_old_Onat_Oinject,axiom,
% 2.69/1.52 ! [Nat: nat,Nat2: nat] :
% 2.69/1.52 ( ( ( suc @ Nat )
% 2.69/1.52 = ( suc @ Nat2 ) )
% 2.69/1.52 = ( Nat = Nat2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_178_nat_Oinject,axiom,
% 2.69/1.52 ! [X22: nat,Y22: nat] :
% 2.69/1.52 ( ( ( suc @ X22 )
% 2.69/1.52 = ( suc @ Y22 ) )
% 2.69/1.52 = ( X22 = Y22 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_179_rev__is__rev__conv,axiom,
% 2.69/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.69/1.52 ( ( ( rev @ A @ Xs )
% 2.69/1.52 = ( rev @ A @ Ys ) )
% 2.69/1.52 = ( Xs = Ys ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_180_rev__rev__ident,axiom,
% 2.69/1.52 ! [A: $tType,Xs: list @ A] :
% 2.69/1.52 ( ( rev @ A @ ( rev @ A @ Xs ) )
% 2.69/1.52 = Xs ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_181_Suc__less__eq,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
% 2.69/1.52 = ( ord_less @ nat @ M2 @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_182_Suc__mono,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.52 => ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_183_lessI,axiom,
% 2.69/1.52 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_184_Suc__le__mono,axiom,
% 2.69/1.52 ! [N: nat,M2: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M2 ) )
% 2.69/1.52 = ( ord_less_eq @ nat @ N @ M2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_185_length__rev,axiom,
% 2.69/1.52 ! [A: $tType,Xs: list @ A] :
% 2.69/1.52 ( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
% 2.69/1.52 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_186_Suc__leD,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
% 2.69/1.52 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_187_le__SucE,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 2.69/1.52 => ( ~ ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 => ( M2
% 2.69/1.52 = ( suc @ N ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_188_le__SucI,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 => ( ord_less_eq @ nat @ M2 @ ( suc @ N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_189_Suc__le__D,axiom,
% 2.69/1.52 ! [N: nat,M4: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
% 2.69/1.52 => ? [M5: nat] :
% 2.69/1.52 ( M4
% 2.69/1.52 = ( suc @ M5 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_190_le__Suc__eq,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 2.69/1.52 = ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 | ( M2
% 2.69/1.52 = ( suc @ N ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_191_Suc__n__not__le__n,axiom,
% 2.69/1.52 ! [N: nat] :
% 2.69/1.52 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_192_not__less__eq__eq,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ~ ( ord_less_eq @ nat @ M2 @ N ) )
% 2.69/1.52 = ( ord_less_eq @ nat @ ( suc @ N ) @ M2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_193_full__nat__induct,axiom,
% 2.69/1.52 ! [P: nat > $o,N: nat] :
% 2.69/1.52 ( ! [N3: nat] :
% 2.69/1.52 ( ! [M3: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
% 2.69/1.52 => ( P @ M3 ) )
% 2.69/1.52 => ( P @ N3 ) )
% 2.69/1.52 => ( P @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_194_nat__induct__at__least,axiom,
% 2.69/1.52 ! [M2: nat,N: nat,P: nat > $o] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 => ( ( P @ M2 )
% 2.69/1.52 => ( ! [N3: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ N3 )
% 2.69/1.52 => ( ( P @ N3 )
% 2.69/1.52 => ( P @ ( suc @ N3 ) ) ) )
% 2.69/1.52 => ( P @ N ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_195_transitive__stepwise__le,axiom,
% 2.69/1.52 ! [M2: nat,N: nat,R: nat > nat > $o] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 => ( ! [X4: nat] : ( R @ X4 @ X4 )
% 2.69/1.52 => ( ! [X4: nat,Y4: nat,Z3: nat] :
% 2.69/1.52 ( ( R @ X4 @ Y4 )
% 2.69/1.52 => ( ( R @ Y4 @ Z3 )
% 2.69/1.52 => ( R @ X4 @ Z3 ) ) )
% 2.69/1.52 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 2.69/1.52 => ( R @ M2 @ N ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_196_n__not__Suc__n,axiom,
% 2.69/1.52 ! [N: nat] :
% 2.69/1.52 ( N
% 2.69/1.52 != ( suc @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_197_Suc__inject,axiom,
% 2.69/1.52 ! [X2: nat,Y2: nat] :
% 2.69/1.52 ( ( ( suc @ X2 )
% 2.69/1.52 = ( suc @ Y2 ) )
% 2.69/1.52 => ( X2 = Y2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_198_rev__swap,axiom,
% 2.69/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.69/1.52 ( ( ( rev @ A @ Xs )
% 2.69/1.52 = Ys )
% 2.69/1.52 = ( Xs
% 2.69/1.52 = ( rev @ A @ Ys ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_199_not__less__less__Suc__eq,axiom,
% 2.69/1.52 ! [N: nat,M2: nat] :
% 2.69/1.52 ( ~ ( ord_less @ nat @ N @ M2 )
% 2.69/1.52 => ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
% 2.69/1.52 = ( N = M2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_200_strict__inc__induct,axiom,
% 2.69/1.52 ! [I3: nat,J2: nat,P: nat > $o] :
% 2.69/1.52 ( ( ord_less @ nat @ I3 @ J2 )
% 2.69/1.52 => ( ! [I: nat] :
% 2.69/1.52 ( ( J2
% 2.69/1.52 = ( suc @ I ) )
% 2.69/1.52 => ( P @ I ) )
% 2.69/1.52 => ( ! [I: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I @ J2 )
% 2.69/1.52 => ( ( P @ ( suc @ I ) )
% 2.69/1.52 => ( P @ I ) ) )
% 2.69/1.52 => ( P @ I3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_201_less__Suc__induct,axiom,
% 2.69/1.52 ! [I3: nat,J2: nat,P: nat > nat > $o] :
% 2.69/1.52 ( ( ord_less @ nat @ I3 @ J2 )
% 2.69/1.52 => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
% 2.69/1.52 => ( ! [I: nat,J3: nat,K2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I @ J3 )
% 2.69/1.52 => ( ( ord_less @ nat @ J3 @ K2 )
% 2.69/1.52 => ( ( P @ I @ J3 )
% 2.69/1.52 => ( ( P @ J3 @ K2 )
% 2.69/1.52 => ( P @ I @ K2 ) ) ) ) )
% 2.69/1.52 => ( P @ I3 @ J2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_202_less__trans__Suc,axiom,
% 2.69/1.52 ! [I3: nat,J2: nat,K: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I3 @ J2 )
% 2.69/1.52 => ( ( ord_less @ nat @ J2 @ K )
% 2.69/1.52 => ( ord_less @ nat @ ( suc @ I3 ) @ K ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_203_Suc__less__SucD,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
% 2.69/1.52 => ( ord_less @ nat @ M2 @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_204_less__antisym,axiom,
% 2.69/1.52 ! [N: nat,M2: nat] :
% 2.69/1.52 ( ~ ( ord_less @ nat @ N @ M2 )
% 2.69/1.52 => ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
% 2.69/1.52 => ( M2 = N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_205_Suc__less__eq2,axiom,
% 2.69/1.52 ! [N: nat,M2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 2.69/1.52 = ( ? [M6: nat] :
% 2.69/1.52 ( ( M2
% 2.69/1.52 = ( suc @ M6 ) )
% 2.69/1.52 & ( ord_less @ nat @ N @ M6 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_206_All__less__Suc,axiom,
% 2.69/1.52 ! [N: nat,P: nat > $o] :
% 2.69/1.52 ( ( ! [I2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 2.69/1.52 => ( P @ I2 ) ) )
% 2.69/1.52 = ( ( P @ N )
% 2.69/1.52 & ! [I2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I2 @ N )
% 2.69/1.52 => ( P @ I2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_207_not__less__eq,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ~ ( ord_less @ nat @ M2 @ N ) )
% 2.69/1.52 = ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_208_less__Suc__eq,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ M2 @ ( suc @ N ) )
% 2.69/1.52 = ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.52 | ( M2 = N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_209_Ex__less__Suc,axiom,
% 2.69/1.52 ! [N: nat,P: nat > $o] :
% 2.69/1.52 ( ( ? [I2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 2.69/1.52 & ( P @ I2 ) ) )
% 2.69/1.52 = ( ( P @ N )
% 2.69/1.52 | ? [I2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I2 @ N )
% 2.69/1.52 & ( P @ I2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_210_less__SucI,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.52 => ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_211_less__SucE,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ M2 @ ( suc @ N ) )
% 2.69/1.52 => ( ~ ( ord_less @ nat @ M2 @ N )
% 2.69/1.52 => ( M2 = N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_212_Suc__lessI,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.52 => ( ( ( suc @ M2 )
% 2.69/1.52 != N )
% 2.69/1.52 => ( ord_less @ nat @ ( suc @ M2 ) @ N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_213_Suc__lessE,axiom,
% 2.69/1.52 ! [I3: nat,K: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( suc @ I3 ) @ K )
% 2.69/1.52 => ~ ! [J3: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I3 @ J3 )
% 2.69/1.52 => ( K
% 2.69/1.52 != ( suc @ J3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_214_Suc__lessD,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( suc @ M2 ) @ N )
% 2.69/1.52 => ( ord_less @ nat @ M2 @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_215_Nat_OlessE,axiom,
% 2.69/1.52 ! [I3: nat,K: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I3 @ K )
% 2.69/1.52 => ( ( K
% 2.69/1.52 != ( suc @ I3 ) )
% 2.69/1.52 => ~ ! [J3: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I3 @ J3 )
% 2.69/1.52 => ( K
% 2.69/1.52 != ( suc @ J3 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_216_lift__Suc__mono__less__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A )
% 2.69/1.52 => ! [F: nat > A,N: nat,M2: nat] :
% 2.69/1.52 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 2.69/1.52 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M2 ) )
% 2.69/1.52 = ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_217_lift__Suc__mono__less,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A )
% 2.69/1.52 => ! [F: nat > A,N: nat,N4: nat] :
% 2.69/1.52 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 2.69/1.52 => ( ( ord_less @ nat @ N @ N4 )
% 2.69/1.52 => ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_218_lift__Suc__antimono__le,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A )
% 2.69/1.52 => ! [F: nat > A,N: nat,N4: nat] :
% 2.69/1.52 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ N @ N4 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_219_lift__Suc__mono__le,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A )
% 2.69/1.52 => ! [F: nat > A,N: nat,N4: nat] :
% 2.69/1.52 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ N @ N4 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_220_le__imp__less__Suc,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 => ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_221_less__eq__Suc__le,axiom,
% 2.69/1.52 ( ( ord_less @ nat )
% 2.69/1.52 = ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_222_less__Suc__eq__le,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ M2 @ ( suc @ N ) )
% 2.69/1.52 = ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_223_le__less__Suc__eq,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 => ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
% 2.69/1.52 = ( N = M2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_224_Suc__le__lessD,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
% 2.69/1.52 => ( ord_less @ nat @ M2 @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_225_inc__induct,axiom,
% 2.69/1.52 ! [I3: nat,J2: nat,P: nat > $o] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ I3 @ J2 )
% 2.69/1.52 => ( ( P @ J2 )
% 2.69/1.52 => ( ! [N3: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ I3 @ N3 )
% 2.69/1.52 => ( ( ord_less @ nat @ N3 @ J2 )
% 2.69/1.52 => ( ( P @ ( suc @ N3 ) )
% 2.69/1.52 => ( P @ N3 ) ) ) )
% 2.69/1.52 => ( P @ I3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_226_dec__induct,axiom,
% 2.69/1.52 ! [I3: nat,J2: nat,P: nat > $o] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ I3 @ J2 )
% 2.69/1.52 => ( ( P @ I3 )
% 2.69/1.52 => ( ! [N3: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ I3 @ N3 )
% 2.69/1.52 => ( ( ord_less @ nat @ N3 @ J2 )
% 2.69/1.52 => ( ( P @ N3 )
% 2.69/1.52 => ( P @ ( suc @ N3 ) ) ) ) )
% 2.69/1.52 => ( P @ J2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_227_Suc__le__eq,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
% 2.69/1.52 = ( ord_less @ nat @ M2 @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_228_Suc__leI,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.52 => ( ord_less_eq @ nat @ ( suc @ M2 ) @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_229_sorted__rev__iff__nth__Suc,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A )
% 2.69/1.52 => ! [Xs: list @ A] :
% 2.69/1.52 ( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
% 2.69/1.52 = ( ! [I2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.69/1.52 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I2 ) ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_230_nat__induct_H,axiom,
% 2.69/1.52 ! [P: nat > $o,N0: nat,N: nat] :
% 2.69/1.52 ( ( P @ N0 )
% 2.69/1.52 => ( ! [N3: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ N0 @ N3 )
% 2.69/1.52 => ( ( P @ N3 )
% 2.69/1.52 => ( P @ ( suc @ N3 ) ) ) )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ N0 @ N )
% 2.69/1.52 => ( P @ N ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_231_rev__nth,axiom,
% 2.69/1.52 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.69/1.52 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.69/1.52 => ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
% 2.69/1.52 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_232_list__ex__length,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( list_ex @ A )
% 2.69/1.52 = ( ^ [P3: A > $o,Xs2: list @ A] :
% 2.69/1.52 ? [N2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.69/1.52 & ( P3 @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_233_Suc__diff__diff,axiom,
% 2.69/1.52 ! [M2: nat,N: nat,K: nat] :
% 2.69/1.52 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
% 2.69/1.52 = ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_234_diff__Suc__Suc,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] :
% 2.69/1.52 ( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
% 2.69/1.52 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_235_diff__diff__cancel,axiom,
% 2.69/1.52 ! [I3: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ I3 @ N )
% 2.69/1.52 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I3 ) )
% 2.69/1.52 = I3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_236_list__ex__rev,axiom,
% 2.69/1.52 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 2.69/1.52 ( ( list_ex @ A @ P @ ( rev @ A @ Xs ) )
% 2.69/1.52 = ( list_ex @ A @ P @ Xs ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_237_i__append__nth2,axiom,
% 2.69/1.52 ! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 2.69/1.52 => ( ( listIn521021761append @ A @ Xs @ F @ N )
% 2.69/1.52 = ( F @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_238_zero__induct__lemma,axiom,
% 2.69/1.52 ! [P: nat > $o,K: nat,I3: nat] :
% 2.69/1.52 ( ( P @ K )
% 2.69/1.52 => ( ! [N3: nat] :
% 2.69/1.52 ( ( P @ ( suc @ N3 ) )
% 2.69/1.52 => ( P @ N3 ) )
% 2.69/1.52 => ( P @ ( minus_minus @ nat @ K @ I3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_239_less__diff__imp__less,axiom,
% 2.69/1.52 ! [I3: nat,J2: nat,M2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ I3 @ ( minus_minus @ nat @ J2 @ M2 ) )
% 2.69/1.52 => ( ord_less @ nat @ I3 @ J2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_240_nat__diff__left__cancel__eq1,axiom,
% 2.69/1.52 ! [K: nat,M2: nat,N: nat] :
% 2.69/1.52 ( ( ( minus_minus @ nat @ K @ M2 )
% 2.69/1.52 = ( minus_minus @ nat @ K @ N ) )
% 2.69/1.52 => ( ( ord_less @ nat @ M2 @ K )
% 2.69/1.52 => ( M2 = N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_241_nat__diff__left__cancel__eq2,axiom,
% 2.69/1.52 ! [K: nat,M2: nat,N: nat] :
% 2.69/1.52 ( ( ( minus_minus @ nat @ K @ M2 )
% 2.69/1.52 = ( minus_minus @ nat @ K @ N ) )
% 2.69/1.52 => ( ( ord_less @ nat @ N @ K )
% 2.69/1.52 => ( M2 = N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_242_nat__diff__left__cancel__less,axiom,
% 2.69/1.52 ! [K: nat,M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( minus_minus @ nat @ K @ M2 ) @ ( minus_minus @ nat @ K @ N ) )
% 2.69/1.52 => ( ord_less @ nat @ N @ M2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_243_nat__diff__right__cancel__eq1,axiom,
% 2.69/1.52 ! [M2: nat,K: nat,N: nat] :
% 2.69/1.52 ( ( ( minus_minus @ nat @ M2 @ K )
% 2.69/1.52 = ( minus_minus @ nat @ N @ K ) )
% 2.69/1.52 => ( ( ord_less @ nat @ K @ M2 )
% 2.69/1.52 => ( M2 = N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_244_nat__diff__right__cancel__eq2,axiom,
% 2.69/1.52 ! [M2: nat,K: nat,N: nat] :
% 2.69/1.52 ( ( ( minus_minus @ nat @ M2 @ K )
% 2.69/1.52 = ( minus_minus @ nat @ N @ K ) )
% 2.69/1.52 => ( ( ord_less @ nat @ K @ N )
% 2.69/1.52 => ( M2 = N ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_245_nat__diff__right__cancel__less,axiom,
% 2.69/1.52 ! [N: nat,K: nat,M2: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M2 @ K ) )
% 2.69/1.52 => ( ord_less @ nat @ N @ M2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_246_diff__less__mono2,axiom,
% 2.69/1.52 ! [M2: nat,N: nat,L: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ M2 @ N )
% 2.69/1.52 => ( ( ord_less @ nat @ M2 @ L )
% 2.69/1.52 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_247_less__imp__diff__less,axiom,
% 2.69/1.52 ! [J2: nat,K: nat,N: nat] :
% 2.69/1.52 ( ( ord_less @ nat @ J2 @ K )
% 2.69/1.52 => ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ N ) @ K ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_248_diff__commute,axiom,
% 2.69/1.52 ! [I3: nat,J2: nat,K: nat] :
% 2.69/1.52 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J2 ) @ K )
% 2.69/1.52 = ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ K ) @ J2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_249_diff__le__mono2,axiom,
% 2.69/1.52 ! [M2: nat,N: nat,L: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_250_le__diff__iff_H,axiom,
% 2.69/1.52 ! [A2: nat,C: nat,B: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ A2 @ C )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ B @ C )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
% 2.69/1.52 = ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_251_diff__le__self,axiom,
% 2.69/1.52 ! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_252_diff__le__mono,axiom,
% 2.69/1.52 ! [M2: nat,N: nat,L: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.69/1.52 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_253_Nat_Odiff__diff__eq,axiom,
% 2.69/1.52 ! [K: nat,M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ K @ M2 )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ K @ N )
% 2.69/1.52 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.69/1.52 = ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_254_le__diff__iff,axiom,
% 2.69/1.52 ! [K: nat,M2: nat,N: nat] :
% 2.69/1.52 ( ( ord_less_eq @ nat @ K @ M2 )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ K @ N )
% 2.69/1.52 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.69/1.52 = ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.69/1.52 ! [A6: $tType,A7: $tType] :
% 2.69/1.52 ( ( preorder @ A7 )
% 2.69/1.52 => ( preorder @ ( A6 > A7 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.69/1.52 ! [A6: $tType,A7: $tType] :
% 2.69/1.52 ( ( order @ A7 )
% 2.69/1.52 => ( order @ ( A6 > A7 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Oord,axiom,
% 2.69/1.52 ! [A6: $tType,A7: $tType] :
% 2.69/1.52 ( ( ord @ A7 )
% 2.69/1.52 => ( ord @ ( A6 > A7 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.69/1.52 condit1037483654norder @ nat ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.69/1.52 wellorder @ nat ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.69/1.52 preorder @ nat ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.69/1.52 linorder @ nat ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.69/1.52 no_top @ nat ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.69/1.52 order @ nat ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.69/1.52 ord @ nat ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 2.69/1.52 size @ nat ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 2.69/1.52 preorder @ $o ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
% 2.69/1.52 linorder @ $o ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
% 2.69/1.52 order @ $o ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
% 2.69/1.52 ord @ $o ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_List_Olist___Nat_Osize_8,axiom,
% 2.69/1.52 ! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(conj_0,hypothesis,
% 2.69/1.52 ! [X5: nat] :
% 2.69/1.52 ( ( listIn521021761append @ a @ xs @ f @ X5 )
% 2.69/1.52 = ( listIn521021761append @ a @ ys @ g @ X5 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(conj_1,hypothesis,
% 2.69/1.52 ord_less_eq @ nat @ ( size_size @ ( list @ a ) @ xs ) @ ( size_size @ ( list @ a ) @ ys ) ).
% 2.69/1.52
% 2.69/1.52 thf(conj_2,conjecture,
% 2.69/1.52 ! [I: nat] :
% 2.69/1.52 ( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ a ) @ xs ) )
% 2.69/1.52 | ( ( nth @ a @ xs @ I )
% 2.69/1.52 = ( nth @ a @ ys @ I ) ) ) ).
% 2.69/1.52 ------------------------
% 2.69/1.53 % (11437)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_11294 for (2999ds/4Mi)
% 2.69/1.53 % (11436)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/183Mi)
% 2.69/1.53 % (11442)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/18Mi)
% 2.69/1.53 % (11437)Instruction limit reached!
% 2.69/1.53 % (11437)------------------------------
% 2.69/1.53 % (11437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.53 % (11437)Termination reason: Unknown
% 2.69/1.53 % (11437)Termination phase: shuffling
% 2.69/1.53
% 2.69/1.53 % (11437)Memory used [KB]: 1407
% 2.69/1.53 % (11437)Time elapsed: 0.003 s
% 2.69/1.53 % (11437)Instructions burned: 6 (million)
% 2.69/1.53 % (11437)------------------------------
% 2.69/1.53 % (11437)------------------------------
% 2.69/1.54 % (11440)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_11294 for (2999ds/2Mi)
% 2.69/1.54 % (11438)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_11294 for (2999ds/27Mi)
% 2.69/1.54 % (11441)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_11294 for (2999ds/275Mi)
% 2.69/1.54 % (11440)Instruction limit reached!
% 2.69/1.54 % (11440)------------------------------
% 2.69/1.54 % (11440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.54 % (11440)Termination reason: Unknown
% 2.69/1.54 % (11440)Termination phase: shuffling
% 2.69/1.54
% 2.69/1.54 % (11440)Memory used [KB]: 1407
% 2.69/1.54 % (11440)Time elapsed: 0.003 s
% 2.69/1.54 % (11440)Instructions burned: 4 (million)
% 2.69/1.54 % (11440)------------------------------
% 2.69/1.54 % (11440)------------------------------
% 2.69/1.54 % (11442)Instruction limit reached!
% 2.69/1.54 % (11442)------------------------------
% 2.69/1.54 % (11442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.54 % (11442)Termination reason: Unknown
% 2.69/1.54 % (11442)Termination phase: shuffling
% 2.69/1.54
% 2.69/1.54 % (11442)Memory used [KB]: 1791
% 2.69/1.54 % (11442)Time elapsed: 0.007 s
% 2.69/1.54 % (11442)Instructions burned: 18 (million)
% 2.69/1.54 % (11442)------------------------------
% 2.69/1.54 % (11442)------------------------------
% 2.69/1.54 % (11439)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/2Mi)
% 2.69/1.54 % (11439)Instruction limit reached!
% 2.69/1.54 % (11439)------------------------------
% 2.69/1.54 % (11439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.54 % (11439)Termination reason: Unknown
% 2.69/1.54 % (11439)Termination phase: shuffling
% 2.69/1.54
% 2.69/1.54 % (11439)Memory used [KB]: 1279
% 2.69/1.54 % (11439)Time elapsed: 0.003 s
% 2.69/1.54 % (11439)Instructions burned: 2 (million)
% 2.69/1.54 % (11439)------------------------------
% 2.69/1.54 % (11439)------------------------------
% 2.69/1.54 % (11445)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/3Mi)
% 2.69/1.54 % (11445)Instruction limit reached!
% 2.69/1.54 % (11445)------------------------------
% 2.69/1.54 % (11445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.54 % (11445)Termination reason: Unknown
% 2.69/1.54 % (11445)Termination phase: shuffling
% 2.69/1.54
% 2.69/1.54 % (11445)Memory used [KB]: 1279
% 2.69/1.54 % (11445)Time elapsed: 0.002 s
% 2.69/1.54 % (11445)Instructions burned: 3 (million)
% 2.69/1.54 % (11445)------------------------------
% 2.69/1.54 % (11445)------------------------------
% 2.69/1.55 % (11446)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_11294 for (2999ds/37Mi)
% 2.69/1.55 % (11447)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_11294 for (2999ds/15Mi)
% 2.69/1.55 % (11438)Instruction limit reached!
% 2.69/1.55 % (11438)------------------------------
% 2.69/1.55 % (11438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.55 % (11438)Termination reason: Unknown
% 2.69/1.55 % (11438)Termination phase: Property scanning
% 2.69/1.55
% 2.69/1.55 % (11438)Memory used [KB]: 1791
% 2.69/1.55 % (11438)Time elapsed: 0.019 s
% 2.69/1.55 % (11438)Instructions burned: 27 (million)
% 2.69/1.55 % (11438)------------------------------
% 2.69/1.55 % (11438)------------------------------
% 2.69/1.56 % (11447)Instruction limit reached!
% 2.69/1.56 % (11447)------------------------------
% 2.69/1.56 % (11447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (11447)Termination reason: Unknown
% 2.69/1.56 % (11447)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (11447)Memory used [KB]: 1663
% 2.69/1.56 % (11447)Time elapsed: 0.009 s
% 2.69/1.56 % (11447)Instructions burned: 15 (million)
% 2.69/1.56 % (11447)------------------------------
% 2.69/1.56 % (11447)------------------------------
% 2.69/1.56 % (11456)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_11294 for (2999ds/1041Mi)
% 2.69/1.57 % (11446)Instruction limit reached!
% 2.69/1.57 % (11446)------------------------------
% 2.69/1.57 % (11446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.57 % (11446)Termination reason: Unknown
% 2.69/1.57 % (11446)Termination phase: shuffling
% 2.69/1.57
% 2.69/1.57 % (11446)Memory used [KB]: 2046
% 2.69/1.57 % (11446)Time elapsed: 0.021 s
% 2.69/1.57 % (11446)Instructions burned: 38 (million)
% 2.69/1.57 % (11446)------------------------------
% 2.69/1.57 % (11446)------------------------------
% 2.69/1.57 % (11457)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/7Mi)
% 2.69/1.57 % (11448)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_11294 for (2999ds/3Mi)
% 2.69/1.57 % (11457)Instruction limit reached!
% 2.69/1.57 % (11457)------------------------------
% 2.69/1.57 % (11457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.57 % (11457)Termination reason: Unknown
% 2.69/1.57 % (11457)Termination phase: shuffling
% 2.69/1.57
% 2.69/1.57 % (11457)Memory used [KB]: 1535
% 2.69/1.57 % (11457)Time elapsed: 0.005 s
% 2.69/1.57 % (11457)Instructions burned: 8 (million)
% 2.69/1.57 % (11457)------------------------------
% 2.69/1.57 % (11457)------------------------------
% 2.69/1.57 % (11458)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/16Mi)
% 2.69/1.57 % (11448)Instruction limit reached!
% 2.69/1.57 % (11448)------------------------------
% 2.69/1.57 % (11448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.57 % (11448)Termination reason: Unknown
% 2.69/1.57 % (11448)Termination phase: shuffling
% 2.69/1.57
% 2.69/1.57 % (11448)Memory used [KB]: 1407
% 2.69/1.57 % (11448)Time elapsed: 0.005 s
% 2.69/1.57 % (11448)Instructions burned: 3 (million)
% 2.69/1.57 % (11448)------------------------------
% 2.69/1.57 % (11448)------------------------------
% 2.69/1.58 % (11458)Instruction limit reached!
% 2.69/1.58 % (11458)------------------------------
% 2.69/1.58 % (11458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.58 % (11458)Termination reason: Unknown
% 2.69/1.58 % (11458)Termination phase: shuffling
% 2.69/1.58
% 2.69/1.58 % (11458)Memory used [KB]: 1663
% 2.69/1.58 % (11458)Time elapsed: 0.009 s
% 2.69/1.58 % (11458)Instructions burned: 16 (million)
% 2.69/1.58 % (11458)------------------------------
% 2.69/1.58 % (11458)------------------------------
% 2.69/1.59 % (11462)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_11294 for (2999ds/3Mi)
% 2.69/1.59 % (11462)Instruction limit reached!
% 2.69/1.59 % (11462)------------------------------
% 2.69/1.59 % (11462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.59 % (11462)Termination reason: Unknown
% 2.69/1.59 % (11462)Termination phase: shuffling
% 2.69/1.59
% 2.69/1.59 % (11462)Memory used [KB]: 1279
% 2.69/1.59 % (11462)Time elapsed: 0.003 s
% 2.69/1.59 % (11462)Instructions burned: 3 (million)
% 2.69/1.59 % (11462)------------------------------
% 2.69/1.59 % (11462)------------------------------
% 2.69/1.59 % (11465)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_11294 for (2999ds/3Mi)
% 2.69/1.59 % (11465)Instruction limit reached!
% 2.69/1.59 % (11465)------------------------------
% 2.69/1.59 % (11465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.59 % (11465)Termination reason: Unknown
% 2.69/1.59 % (11465)Termination phase: shuffling
% 2.69/1.59
% 2.69/1.59 % (11465)Memory used [KB]: 1279
% 2.69/1.59 % (11465)Time elapsed: 0.002 s
% 2.69/1.59 % (11465)Instructions burned: 3 (million)
% 2.69/1.59 % (11465)------------------------------
% 2.69/1.59 % (11465)------------------------------
% 2.71/1.60 % (11468)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/3Mi)
% 2.71/1.60 % (11471)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/4Mi)
% 2.71/1.60 % (11468)Instruction limit reached!
% 2.71/1.60 % (11468)------------------------------
% 2.71/1.60 % (11468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (11468)Termination reason: Unknown
% 2.71/1.60 % (11468)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (11468)Memory used [KB]: 1407
% 2.71/1.60 % (11468)Time elapsed: 0.004 s
% 2.71/1.60 % (11468)Instructions burned: 4 (million)
% 2.71/1.60 % (11468)------------------------------
% 2.71/1.60 % (11468)------------------------------
% 2.71/1.60 % (11471)Instruction limit reached!
% 2.71/1.60 % (11471)------------------------------
% 2.71/1.60 % (11471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (11471)Termination reason: Unknown
% 2.71/1.60 % (11471)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (11471)Memory used [KB]: 1407
% 2.71/1.60 % (11471)Time elapsed: 0.003 s
% 2.71/1.60 % (11471)Instructions burned: 4 (million)
% 2.71/1.60 % (11471)------------------------------
% 2.71/1.60 % (11471)------------------------------
% 2.71/1.60 % (11466)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/7Mi)
% 2.71/1.60 % (11436)Instruction limit reached!
% 2.71/1.60 % (11436)------------------------------
% 2.71/1.60 % (11436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (11436)Termination reason: Unknown
% 2.71/1.60 % (11436)Termination phase: Saturation
% 2.71/1.60
% 2.71/1.60 % (11436)Memory used [KB]: 7036
% 2.71/1.60 % (11436)Time elapsed: 0.074 s
% 2.71/1.60 % (11436)Instructions burned: 183 (million)
% 2.71/1.60 % (11436)------------------------------
% 2.71/1.60 % (11436)------------------------------
% 2.71/1.61 % (11466)Instruction limit reached!
% 2.71/1.61 % (11466)------------------------------
% 2.71/1.61 % (11466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (11466)Termination reason: Unknown
% 2.71/1.61 % (11466)Termination phase: shuffling
% 2.71/1.61
% 2.71/1.61 % (11466)Memory used [KB]: 1535
% 2.71/1.61 % (11466)Time elapsed: 0.005 s
% 2.71/1.61 % (11466)Instructions burned: 7 (million)
% 2.71/1.61 % (11466)------------------------------
% 2.71/1.61 % (11466)------------------------------
% 2.71/1.61 % (11473)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/18Mi)
% 2.71/1.61 % (11441)First to succeed.
% 2.71/1.61 % (11476)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11294 for (2999ds/6Mi)
% 2.71/1.61 % (11473)Instruction limit reached!
% 2.71/1.61 % (11473)------------------------------
% 2.71/1.61 % (11473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (11473)Termination reason: Unknown
% 2.71/1.61 % (11473)Termination phase: shuffling
% 2.71/1.61
% 2.71/1.61 % (11473)Memory used [KB]: 1791
% 2.71/1.61 % (11473)Time elapsed: 0.009 s
% 2.71/1.61 % (11473)Instructions burned: 18 (million)
% 2.71/1.61 % (11473)------------------------------
% 2.71/1.61 % (11473)------------------------------
% 2.71/1.62 % (11441)Refutation found. Thanks to Tanya!
% 2.71/1.62 % SZS status Theorem for DTF2THF_11294
% 2.71/1.62 % SZS output start Proof for DTF2THF_11294
% 2.71/1.62 thf(type_def_6, type, list: ($tType) > $tType).
% 2.71/1.62 thf(type_def_7, type, nat: $tType).
% 2.71/1.62 thf(type_def_8, type, a: $tType).
% 2.71/1.62 thf(func_def_0, type, list: ($tType) > $tType).
% 2.71/1.62 thf(func_def_1, type, nat: $tType).
% 2.71/1.62 thf(func_def_2, type, a: $tType).
% 2.71/1.62 thf(func_def_3, type, type: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_4, type, size: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_5, type, ord: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_6, type, order: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_7, type, no_bot: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_8, type, no_top: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_9, type, linorder: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_10, type, preorder: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_11, type, wellorder: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_12, type, dense_order: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_13, type, dense_linorder: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_14, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_15, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.71/1.62 thf(func_def_16, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.71/1.62 thf(func_def_17, type, list_asc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.71/1.62 thf(func_def_18, type, list_desc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.71/1.62 thf(func_def_19, type, list_strict_asc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.71/1.62 thf(func_def_20, type, list_strict_desc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.71/1.62 thf(func_def_21, type, listIn521021761append: !>[X0: $tType]:(list @ X0 > (nat > X0) > nat > X0)).
% 2.71/1.62 thf(func_def_22, type, linorder_sorted: !>[X0: $tType]:(list @ X0 > $o)).
% 2.71/1.62 thf(func_def_23, type, list_ex: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 2.71/1.62 thf(func_def_24, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 2.71/1.62 thf(func_def_25, type, rev: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.71/1.62 thf(func_def_26, type, suc: nat > nat).
% 2.71/1.62 thf(func_def_27, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.71/1.62 thf(func_def_28, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.71/1.62 thf(func_def_29, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.71/1.62 thf(func_def_30, type, f: nat > a).
% 2.71/1.62 thf(func_def_31, type, g: nat > a).
% 2.71/1.62 thf(func_def_32, type, xs: list @ a).
% 2.71/1.62 thf(func_def_33, type, ys: list @ a).
% 2.71/1.62 thf(func_def_48, type, sK0: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.71/1.62 thf(func_def_49, type, sK1: !>[X0: $tType]:(nat > list @ X0)).
% 2.71/1.62 thf(func_def_50, type, sK2: (nat > $o) > nat).
% 2.71/1.62 thf(func_def_51, type, sK3: (nat > $o) > nat).
% 2.71/1.62 thf(func_def_52, type, sK4: nat > (nat > $o) > nat).
% 2.71/1.62 thf(func_def_53, type, sK5: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.71/1.62 thf(func_def_54, type, sK6: (nat > $o) > nat).
% 2.71/1.62 thf(func_def_55, type, sK7: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.71/1.62 thf(func_def_56, type, sK8: !>[X0: $tType]:(nat > (X0 > nat) > (X0 > $o) > X0)).
% 2.71/1.62 thf(func_def_57, type, sK9: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.71/1.62 thf(func_def_58, type, sK10: (nat > nat) > nat).
% 2.71/1.62 thf(func_def_59, type, sK11: (nat > nat) > nat).
% 2.71/1.62 thf(func_def_60, type, sK12: !>[X0: $tType]:(nat > (nat > X0 > $o) > X0)).
% 2.71/1.62 thf(func_def_61, type, sK13: !>[X0: $tType]:(list @ X0 > (nat > X0 > $o) > nat > nat)).
% 2.71/1.62 thf(func_def_62, type, sK14: !>[X0: $tType]:((nat > X0 > $o) > nat > list @ X0)).
% 2.71/1.62 thf(func_def_63, type, sK15: !>[X0: $tType]:((nat > X0 > $o) > nat > nat)).
% 2.71/1.62 thf(func_def_64, type, sK16: !>[X0: $tType]:(list @ X0 > list @ X0 > nat)).
% 2.71/1.62 thf(func_def_65, type, sK17: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.71/1.62 thf(func_def_66, type, sK18: (nat > $o) > nat > nat).
% 2.71/1.62 thf(func_def_67, type, sK19: (nat > $o) > nat > nat).
% 2.71/1.62 thf(func_def_68, type, sK20: nat).
% 2.71/1.62 thf(func_def_69, type, sK21: (nat > $o) > nat).
% 2.71/1.62 thf(func_def_71, type, ph23: !>[X0: $tType]:(X0)).
% 2.71/1.62 thf(f1364,plain,(
% 2.71/1.62 $false),
% 2.71/1.62 inference(avatar_sat_refutation,[status(thm)],[f1012,f1252,f1278,f1341,f1363])).
% 2.71/1.62 thf(f1363,plain,(
% 2.71/1.62 ~spl22_2),
% 2.71/1.62 inference(avatar_contradiction_clause,[status(thm)],[f1362])).
% 2.71/1.62 thf(f1362,plain,(
% 2.71/1.62 $false | ~spl22_2),
% 2.71/1.62 inference(subsumption_resolution,[status(thm)],[f1349,f985])).
% 2.71/1.62 thf(f985,plain,(
% 2.71/1.62 ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) = $true)),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f904])).
% 2.71/1.62 thf(f904,plain,(
% 2.71/1.62 ((nth @ a @ ys @ sK20) != (nth @ a @ xs @ sK20)) & ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) = $true)),
% 2.71/1.62 inference(skolemisation,[status(esa),new_symbols(skolem,[sK20])],[f837,f903])).
% 2.71/1.62 thf(f903,plain,(
% 2.71/1.62 ? [X0 : nat] : (((nth @ a @ xs @ X0) != (nth @ a @ ys @ X0)) & ((ord_less @ nat @ X0 @ (size_size @ list @ a @ xs)) = $true)) => (((nth @ a @ ys @ sK20) != (nth @ a @ xs @ sK20)) & ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) = $true))),
% 2.71/1.62 introduced(definition,[],[choice_axiom])).
% 2.71/1.62 thf(f837,plain,(
% 2.71/1.62 ? [X0 : nat] : (((nth @ a @ xs @ X0) != (nth @ a @ ys @ X0)) & ((ord_less @ nat @ X0 @ (size_size @ list @ a @ xs)) = $true))),
% 2.71/1.62 inference(ennf_transformation,[status(thm)],[f791])).
% 2.71/1.62 thf(f791,plain,(
% 2.71/1.62 ~! [X0 : nat] : (((nth @ a @ xs @ X0) = (nth @ a @ ys @ X0)) | ((ord_less @ nat @ X0 @ (size_size @ list @ a @ xs)) != $true))),
% 2.71/1.62 inference(flattening,[status(thm)],[f763])).
% 2.71/1.62 thf(f763,plain,(
% 2.71/1.62 ~! [X0 : nat] : (~((ord_less @ nat @ X0 @ (size_size @ list @ a @ xs)) = $true) | ((nth @ a @ xs @ X0) = (nth @ a @ ys @ X0)))),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f762])).
% 2.71/1.62 thf(f762,plain,(
% 2.71/1.62 ~! [X0 : nat] : (~(ord_less @ nat @ X0 @ (size_size @ list @ a @ xs)) | ((nth @ a @ xs @ X0) = (nth @ a @ ys @ X0)))),
% 2.71/1.62 inference(rectify,[status(thm)],[f275])).
% 2.71/1.62 thf(f275,negated_conjecture,(
% 2.71/1.62 ~! [X6 : nat] : (~(ord_less @ nat @ X6 @ (size_size @ list @ a @ xs)) | ((nth @ a @ xs @ X6) = (nth @ a @ ys @ X6)))),
% 2.71/1.62 inference(negated_conjecture,[status(cth)],[f274])).
% 2.71/1.62 thf(f274,conjecture,(
% 2.71/1.62 ! [X6 : nat] : (~(ord_less @ nat @ X6 @ (size_size @ list @ a @ xs)) | ((nth @ a @ xs @ X6) = (nth @ a @ ys @ X6)))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',conj_2)).
% 2.71/1.62 thf(f1349,plain,(
% 2.71/1.62 ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) != $true) | ~spl22_2),
% 2.71/1.62 inference(superposition,[status(thm)],[f1237,f1011])).
% 2.71/1.62 thf(f1011,plain,(
% 2.71/1.62 ((size_size @ list @ a @ xs) = (size_size @ list @ a @ ys)) | ~spl22_2),
% 2.71/1.62 inference(avatar_component_clause,[status(thm)],[f1009])).
% 2.71/1.62 thf(f1009,definition,(
% 2.71/1.62 spl22_2 <=> ((size_size @ list @ a @ xs) = (size_size @ list @ a @ ys))),
% 2.71/1.62 introduced(definition,[new_symbols(naming,[spl22_2])],[avatar_definition])).
% 2.71/1.62 thf(f1237,plain,(
% 2.71/1.62 ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ ys)) != $true)),
% 2.71/1.62 inference(subsumption_resolution,[status(thm)],[f1234,f985])).
% 2.71/1.62 thf(f1234,plain,(
% 2.71/1.62 ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ ys)) != $true) | ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) != $true)),
% 2.71/1.62 inference(trivial_inequality_removal,[status(thm)],[f1230])).
% 2.71/1.62 thf(f1230,plain,(
% 2.71/1.62 ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) != $true) | ((nth @ a @ xs @ sK20) != (nth @ a @ xs @ sK20)) | ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ ys)) != $true)),
% 2.71/1.62 inference(superposition,[status(thm)],[f986,f1028])).
% 2.71/1.62 thf(f1028,plain,(
% 2.71/1.62 ( ! [X0 : nat] : (((nth @ a @ xs @ X0) = (nth @ a @ ys @ X0)) | ($true != (ord_less @ nat @ X0 @ (size_size @ list @ a @ ys))) | ((ord_less @ nat @ X0 @ (size_size @ list @ a @ xs)) != $true)) )),
% 2.71/1.62 inference(superposition,[status(thm)],[f936,f1016])).
% 2.71/1.62 thf(f1016,plain,(
% 2.71/1.62 ( ! [X0 : nat] : (((nth @ a @ ys @ X0) = (listIn521021761append @ a @ xs @ f @ X0)) | ($true != (ord_less @ nat @ X0 @ (size_size @ list @ a @ ys)))) )),
% 2.71/1.62 inference(superposition,[status(thm)],[f936,f928])).
% 2.71/1.62 thf(f928,plain,(
% 2.71/1.62 ( ! [X0 : nat] : (((listIn521021761append @ a @ ys @ g @ X0) = (listIn521021761append @ a @ xs @ f @ X0))) )),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f790])).
% 2.71/1.62 thf(f790,plain,(
% 2.71/1.62 ! [X0 : nat] : ((listIn521021761append @ a @ ys @ g @ X0) = (listIn521021761append @ a @ xs @ f @ X0))),
% 2.71/1.62 inference(rectify,[status(thm)],[f272])).
% 2.71/1.62 thf(f272,axiom,(
% 2.71/1.62 ! [X57 : nat] : ((listIn521021761append @ a @ xs @ f @ X57) = (listIn521021761append @ a @ ys @ g @ X57))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',conj_0)).
% 2.71/1.62 thf(f936,plain,(
% 2.71/1.62 ( ! [X0 : $tType,X2 : list @ X0,X3 : nat > X0,X1 : nat] : (((listIn521021761append @ X0 @ X2 @ X3 @ X1) = (nth @ X0 @ X2 @ X1)) | ($true != (ord_less @ nat @ X1 @ (size_size @ list @ X0 @ X2)))) )),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f828])).
% 2.71/1.62 thf(f828,plain,(
% 2.71/1.62 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat > X0] : (($true != (ord_less @ nat @ X1 @ (size_size @ list @ X0 @ X2))) | ((listIn521021761append @ X0 @ X2 @ X3 @ X1) = (nth @ X0 @ X2 @ X1)))),
% 2.71/1.62 inference(ennf_transformation,[status(thm)],[f371])).
% 2.71/1.62 thf(f371,plain,(
% 2.71/1.62 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat > X0] : (($true = (ord_less @ nat @ X1 @ (size_size @ list @ X0 @ X2))) => ((listIn521021761append @ X0 @ X2 @ X3 @ X1) = (nth @ X0 @ X2 @ X1)))),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f370])).
% 2.71/1.62 thf(f370,plain,(
% 2.71/1.62 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat > X0] : ((ord_less @ nat @ X1 @ (size_size @ list @ X0 @ X2)) => ((listIn521021761append @ X0 @ X2 @ X3 @ X1) = (nth @ X0 @ X2 @ X1)))),
% 2.71/1.62 inference(rectify,[status(thm)],[f2])).
% 2.71/1.62 thf(f2,axiom,(
% 2.71/1.62 ! [X0 : $tType,X5 : nat,X1 : list @ X0,X3 : nat > X0] : ((ord_less @ nat @ X5 @ (size_size @ list @ X0 @ X1)) => ((listIn521021761append @ X0 @ X1 @ X3 @ X5) = (nth @ X0 @ X1 @ X5)))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',fact_1_i__append__nth1)).
% 2.71/1.62 thf(f986,plain,(
% 2.71/1.62 ((nth @ a @ ys @ sK20) != (nth @ a @ xs @ sK20))),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f904])).
% 2.71/1.62 thf(f1341,plain,(
% 2.71/1.62 spl22_1 | spl22_14),
% 2.71/1.62 inference(avatar_contradiction_clause,[status(thm)],[f1340])).
% 2.71/1.62 thf(f1340,plain,(
% 2.71/1.62 $false | (spl22_1 | spl22_14)),
% 2.71/1.62 inference(subsumption_resolution,[status(thm)],[f1335,f985])).
% 2.71/1.62 thf(f1335,plain,(
% 2.71/1.62 ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) != $true) | (spl22_1 | spl22_14)),
% 2.71/1.62 inference(trivial_inequality_removal,[status(thm)],[f1333])).
% 2.71/1.62 thf(f1333,plain,(
% 2.71/1.62 ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) != $true) | ($true != $true) | (spl22_1 | spl22_14)),
% 2.71/1.62 inference(superposition,[status(thm)],[f1316,f938])).
% 2.71/1.62 thf(f938,plain,(
% 2.71/1.62 ( ! [X0 : nat,X1 : nat] : (((ord_less_eq @ nat @ X0 @ X1) = $true) | ((ord_less @ nat @ X0 @ X1) != $true)) )),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f807])).
% 2.71/1.62 thf(f807,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : ((((ord_less @ nat @ X0 @ X1) != $true) & (X0 != X1)) | ((ord_less_eq @ nat @ X0 @ X1) = $true))),
% 2.71/1.62 inference(ennf_transformation,[status(thm)],[f465])).
% 2.71/1.62 thf(f465,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (((X0 = X1) | ((ord_less @ nat @ X0 @ X1) = $true)) => ((ord_less_eq @ nat @ X0 @ X1) = $true))),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f464])).
% 2.71/1.62 thf(f464,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) | (X0 = X1)) => (ord_less_eq @ nat @ X0 @ X1))),
% 2.71/1.62 inference(rectify,[status(thm)],[f12])).
% 2.71/1.62 thf(f12,axiom,(
% 2.71/1.62 ! [X21 : nat,X5 : nat] : (((ord_less @ nat @ X21 @ X5) | (X5 = X21)) => (ord_less_eq @ nat @ X21 @ X5))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',fact_11_less__or__eq__imp__le)).
% 2.71/1.62 thf(f1316,plain,(
% 2.71/1.62 ($true != (ord_less_eq @ nat @ sK20 @ (size_size @ list @ a @ xs))) | (spl22_1 | spl22_14)),
% 2.71/1.62 inference(trivial_inequality_removal,[status(thm)],[f1309])).
% 2.71/1.62 thf(f1309,plain,(
% 2.71/1.62 ($true != (ord_less_eq @ nat @ sK20 @ (size_size @ list @ a @ xs))) | ($true != $true) | (spl22_1 | spl22_14)),
% 2.71/1.62 inference(superposition,[status(thm)],[f1251,f1207])).
% 2.71/1.62 thf(f1207,plain,(
% 2.71/1.62 ( ! [X0 : nat] : (($true = (ord_less_eq @ nat @ X0 @ (size_size @ list @ a @ ys))) | ((ord_less_eq @ nat @ X0 @ (size_size @ list @ a @ xs)) != $true)) ) | spl22_1),
% 2.71/1.62 inference(trivial_inequality_removal,[status(thm)],[f1180])).
% 2.71/1.62 thf(f1180,plain,(
% 2.71/1.62 ( ! [X0 : nat] : (($true != $true) | ((ord_less_eq @ nat @ X0 @ (size_size @ list @ a @ xs)) != $true) | ($true = (ord_less_eq @ nat @ X0 @ (size_size @ list @ a @ ys)))) ) | spl22_1),
% 2.71/1.62 inference(superposition,[status(thm)],[f1167,f977])).
% 2.71/1.62 thf(f977,plain,(
% 2.71/1.62 ( ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X1 @ X0)) | ((ord_less_eq @ nat @ X0 @ X1) = $true)) )),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f540])).
% 2.71/1.62 thf(f540,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X1 @ X0)) | ((ord_less_eq @ nat @ X0 @ X1) = $true))),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f539])).
% 2.71/1.62 thf(f539,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) | (ord_less_eq @ nat @ X1 @ X0))),
% 2.71/1.62 inference(rectify,[status(thm)],[f89])).
% 2.71/1.62 thf(f89,axiom,(
% 2.71/1.62 ! [X21 : nat,X5 : nat] : ((ord_less_eq @ nat @ X21 @ X5) | (ord_less_eq @ nat @ X5 @ X21))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',fact_88_nat__le__linear)).
% 2.71/1.62 thf(f1167,plain,(
% 2.71/1.62 ( ! [X0 : nat] : (((ord_less_eq @ nat @ (size_size @ list @ a @ ys) @ X0) != $true) | ((ord_less_eq @ nat @ X0 @ (size_size @ list @ a @ xs)) != $true)) ) | spl22_1),
% 2.71/1.62 inference(trivial_inequality_removal,[status(thm)],[f1162])).
% 2.71/1.62 thf(f1162,plain,(
% 2.71/1.62 ( ! [X0 : nat] : (($true != $true) | ((ord_less_eq @ nat @ (size_size @ list @ a @ ys) @ X0) != $true) | ((ord_less_eq @ nat @ X0 @ (size_size @ list @ a @ xs)) != $true)) ) | spl22_1),
% 2.71/1.62 inference(superposition,[status(thm)],[f1007,f935])).
% 2.71/1.62 thf(f935,plain,(
% 2.71/1.62 ( ! [X2 : nat,X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X2)) | ((ord_less_eq @ nat @ X1 @ X2) != $true) | ((ord_less_eq @ nat @ X0 @ X1) != $true)) )),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f843])).
% 2.71/1.62 thf(f843,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat,X2 : nat] : (((ord_less_eq @ nat @ X0 @ X1) != $true) | ($true = (ord_less_eq @ nat @ X0 @ X2)) | ((ord_less_eq @ nat @ X1 @ X2) != $true))),
% 2.71/1.62 inference(flattening,[status(thm)],[f842])).
% 2.71/1.62 thf(f842,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat,X2 : nat] : ((($true = (ord_less_eq @ nat @ X0 @ X2)) | ((ord_less_eq @ nat @ X1 @ X2) != $true)) | ((ord_less_eq @ nat @ X0 @ X1) != $true))),
% 2.71/1.62 inference(ennf_transformation,[status(thm)],[f379])).
% 2.71/1.62 thf(f379,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat,X2 : nat] : (((ord_less_eq @ nat @ X0 @ X1) = $true) => (((ord_less_eq @ nat @ X1 @ X2) = $true) => ($true = (ord_less_eq @ nat @ X0 @ X2))))),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f378])).
% 2.71/1.62 thf(f378,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat,X2 : nat] : ((ord_less_eq @ nat @ X0 @ X1) => ((ord_less_eq @ nat @ X1 @ X2) => (ord_less_eq @ nat @ X0 @ X2)))),
% 2.71/1.62 inference(rectify,[status(thm)],[f92])).
% 2.71/1.62 thf(f92,axiom,(
% 2.71/1.62 ! [X22 : nat,X23 : nat,X7 : nat] : ((ord_less_eq @ nat @ X22 @ X23) => ((ord_less_eq @ nat @ X23 @ X7) => (ord_less_eq @ nat @ X22 @ X7)))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',fact_91_le__trans)).
% 2.71/1.62 thf(f1007,plain,(
% 2.71/1.62 ((ord_less_eq @ nat @ (size_size @ list @ a @ ys) @ (size_size @ list @ a @ xs)) != $true) | spl22_1),
% 2.71/1.62 inference(avatar_component_clause,[status(thm)],[f1005])).
% 2.71/1.62 thf(f1005,definition,(
% 2.71/1.62 spl22_1 <=> ((ord_less_eq @ nat @ (size_size @ list @ a @ ys) @ (size_size @ list @ a @ xs)) = $true)),
% 2.71/1.62 introduced(definition,[new_symbols(naming,[spl22_1])],[avatar_definition])).
% 2.71/1.62 thf(f1251,plain,(
% 2.71/1.62 ((ord_less_eq @ nat @ sK20 @ (size_size @ list @ a @ ys)) != $true) | spl22_14),
% 2.71/1.62 inference(avatar_component_clause,[status(thm)],[f1249])).
% 2.71/1.62 thf(f1249,definition,(
% 2.71/1.62 spl22_14 <=> ((ord_less_eq @ nat @ sK20 @ (size_size @ list @ a @ ys)) = $true)),
% 2.71/1.62 introduced(definition,[new_symbols(naming,[spl22_14])],[avatar_definition])).
% 2.71/1.62 thf(f1278,plain,(
% 2.71/1.62 spl22_1 | ~spl22_13),
% 2.71/1.62 inference(avatar_contradiction_clause,[status(thm)],[f1277])).
% 2.71/1.62 thf(f1277,plain,(
% 2.71/1.62 $false | (spl22_1 | ~spl22_13)),
% 2.71/1.62 inference(subsumption_resolution,[status(thm)],[f1268,f985])).
% 2.71/1.62 thf(f1268,plain,(
% 2.71/1.62 ((ord_less @ nat @ sK20 @ (size_size @ list @ a @ xs)) != $true) | (spl22_1 | ~spl22_13)),
% 2.71/1.62 inference(superposition,[status(thm)],[f1169,f1247])).
% 2.71/1.62 thf(f1247,plain,(
% 2.71/1.62 ((size_size @ list @ a @ ys) = sK20) | ~spl22_13),
% 2.71/1.62 inference(avatar_component_clause,[status(thm)],[f1245])).
% 2.71/1.62 thf(f1245,definition,(
% 2.71/1.62 spl22_13 <=> ((size_size @ list @ a @ ys) = sK20)),
% 2.71/1.62 introduced(definition,[new_symbols(naming,[spl22_13])],[avatar_definition])).
% 2.71/1.62 thf(f1169,plain,(
% 2.71/1.62 ((ord_less @ nat @ (size_size @ list @ a @ ys) @ (size_size @ list @ a @ xs)) != $true) | spl22_1),
% 2.71/1.62 inference(trivial_inequality_removal,[status(thm)],[f1164])).
% 2.71/1.62 thf(f1164,plain,(
% 2.71/1.62 ($true != $true) | ((ord_less @ nat @ (size_size @ list @ a @ ys) @ (size_size @ list @ a @ xs)) != $true) | spl22_1),
% 2.71/1.62 inference(superposition,[status(thm)],[f1007,f919])).
% 2.71/1.62 thf(f919,plain,(
% 2.71/1.62 ( ! [X0 : nat,X1 : nat] : (((ord_less_eq @ nat @ X0 @ X1) = $true) | ((ord_less @ nat @ X0 @ X1) != $true)) )),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f820])).
% 2.71/1.62 thf(f820,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) != $true) | ((ord_less_eq @ nat @ X0 @ X1) = $true))),
% 2.71/1.62 inference(ennf_transformation,[status(thm)],[f534])).
% 2.71/1.62 thf(f534,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) = $true) => ((ord_less_eq @ nat @ X0 @ X1) = $true))),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f533])).
% 2.71/1.62 thf(f533,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : ((ord_less @ nat @ X0 @ X1) => (ord_less_eq @ nat @ X0 @ X1))),
% 2.71/1.62 inference(rectify,[status(thm)],[f10])).
% 2.71/1.62 thf(f10,axiom,(
% 2.71/1.62 ! [X21 : nat,X5 : nat] : ((ord_less @ nat @ X21 @ X5) => (ord_less_eq @ nat @ X21 @ X5))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',fact_9_less__imp__le__nat)).
% 2.71/1.62 thf(f1252,plain,(
% 2.71/1.62 spl22_13 | ~spl22_14),
% 2.71/1.62 inference(avatar_split_clause,[status(thm)],[f1243,f1249,f1245])).
% 2.71/1.62 thf(f1243,plain,(
% 2.71/1.62 ((ord_less_eq @ nat @ sK20 @ (size_size @ list @ a @ ys)) != $true) | ((size_size @ list @ a @ ys) = sK20)),
% 2.71/1.62 inference(trivial_inequality_removal,[status(thm)],[f1238])).
% 2.71/1.62 thf(f1238,plain,(
% 2.71/1.62 ((size_size @ list @ a @ ys) = sK20) | ((ord_less_eq @ nat @ sK20 @ (size_size @ list @ a @ ys)) != $true) | ($true != $true)),
% 2.71/1.62 inference(superposition,[status(thm)],[f1237,f916])).
% 2.71/1.62 thf(f916,plain,(
% 2.71/1.62 ( ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1) | ((ord_less_eq @ nat @ X0 @ X1) != $true)) )),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f822])).
% 2.71/1.62 thf(f822,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) = $true) | ((ord_less_eq @ nat @ X0 @ X1) != $true) | (X0 = X1))),
% 2.71/1.62 inference(flattening,[status(thm)],[f821])).
% 2.71/1.62 thf(f821,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : ((((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1)) | ((ord_less_eq @ nat @ X0 @ X1) != $true))),
% 2.71/1.62 inference(ennf_transformation,[status(thm)],[f624])).
% 2.71/1.62 thf(f624,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (((ord_less_eq @ nat @ X0 @ X1) = $true) => ((X0 != X1) => ((ord_less @ nat @ X0 @ X1) = $true)))),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f623])).
% 2.71/1.62 thf(f623,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) => ((X0 != X1) => (ord_less @ nat @ X0 @ X1)))),
% 2.71/1.62 inference(rectify,[status(thm)],[f13])).
% 2.71/1.62 thf(f13,axiom,(
% 2.71/1.62 ! [X21 : nat,X5 : nat] : ((ord_less_eq @ nat @ X21 @ X5) => ((X5 != X21) => (ord_less @ nat @ X21 @ X5)))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',fact_12_le__neq__implies__less)).
% 2.71/1.62 thf(f1012,plain,(
% 2.71/1.62 ~spl22_1 | spl22_2),
% 2.71/1.62 inference(avatar_split_clause,[status(thm)],[f1003,f1009,f1005])).
% 2.71/1.62 thf(f1003,plain,(
% 2.71/1.62 ((size_size @ list @ a @ xs) = (size_size @ list @ a @ ys)) | ((ord_less_eq @ nat @ (size_size @ list @ a @ ys) @ (size_size @ list @ a @ xs)) != $true)),
% 2.71/1.62 inference(trivial_inequality_removal,[status(thm)],[f1002])).
% 2.71/1.62 thf(f1002,plain,(
% 2.71/1.62 ((size_size @ list @ a @ xs) = (size_size @ list @ a @ ys)) | ((ord_less_eq @ nat @ (size_size @ list @ a @ ys) @ (size_size @ list @ a @ xs)) != $true) | ($true != $true)),
% 2.71/1.62 inference(superposition,[status(thm)],[f910,f920])).
% 2.71/1.62 thf(f920,plain,(
% 2.71/1.62 ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ (size_size @ list @ a @ ys)) = $true)),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f500])).
% 2.71/1.62 thf(f500,plain,(
% 2.71/1.62 ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ (size_size @ list @ a @ ys)) = $true)),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f499])).
% 2.71/1.62 thf(f499,plain,(
% 2.71/1.62 (ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ (size_size @ list @ a @ ys))),
% 2.71/1.62 inference(rectify,[status(thm)],[f273])).
% 2.71/1.62 thf(f273,axiom,(
% 2.71/1.62 (ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ (size_size @ list @ a @ ys))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',conj_1)).
% 2.71/1.62 thf(f910,plain,(
% 2.71/1.62 ( ! [X0 : nat,X1 : nat] : (($true != (ord_less_eq @ nat @ X1 @ X0)) | ((ord_less_eq @ nat @ X0 @ X1) != $true) | (X0 = X1)) )),
% 2.71/1.62 inference(cnf_transformation,[status(thm)],[f850])).
% 2.71/1.62 thf(f850,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : ((X0 = X1) | ($true != (ord_less_eq @ nat @ X1 @ X0)) | ((ord_less_eq @ nat @ X0 @ X1) != $true))),
% 2.71/1.62 inference(flattening,[status(thm)],[f849])).
% 2.71/1.62 thf(f849,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (((X0 = X1) | ($true != (ord_less_eq @ nat @ X1 @ X0))) | ((ord_less_eq @ nat @ X0 @ X1) != $true))),
% 2.71/1.62 inference(ennf_transformation,[status(thm)],[f532])).
% 2.71/1.62 thf(f532,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : (((ord_less_eq @ nat @ X0 @ X1) = $true) => (($true = (ord_less_eq @ nat @ X1 @ X0)) => (X0 = X1)))),
% 2.71/1.62 inference(fool_elimination,[status(thm)],[f531])).
% 2.71/1.62 thf(f531,plain,(
% 2.71/1.62 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) => ((ord_less_eq @ nat @ X1 @ X0) => (X0 = X1)))),
% 2.71/1.62 inference(rectify,[status(thm)],[f90])).
% 2.71/1.62 thf(f90,axiom,(
% 2.71/1.62 ! [X21 : nat,X5 : nat] : ((ord_less_eq @ nat @ X21 @ X5) => ((ord_less_eq @ nat @ X5 @ X21) => (X5 = X21)))),
% 2.71/1.62 file('/export/starexec/sandbox2/tmp/tmp.P5Sig2B7UK/DTF2THF_11294.p',fact_89_le__antisym)).
% 2.71/1.62 % SZS output end Proof for DTF2THF_11294
% 2.71/1.62 % (11441)------------------------------
% 2.71/1.62 % (11441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.62 % (11441)Termination reason: Refutation
% 2.71/1.62
% 2.71/1.62 % (11441)Memory used [KB]: 6780
% 2.71/1.62 % (11441)Time elapsed: 0.078 s
% 2.71/1.62 % (11441)Instructions burned: 159 (million)
% 2.71/1.62 % (11441)------------------------------
% 2.71/1.62 % (11441)------------------------------
% 2.71/1.62 % (11435)Success in time 0.089 s
%------------------------------------------------------------------------------