%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT246^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n011.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 08:45:17 AM UTC 2026
% Result : Theorem 2.40s 1.46s
% Output : Refutation 2.40s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT246^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.32 % Computer : n011.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Tue Feb 24 06:11:32 EST 2026
% 0.12/0.32 % CPUTime :
% 0.12/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.43 ---- Original DTF file ---
% 0.17/0.44 thf(spec,logic,$$dhol).
% 0.17/0.44 %------------------------------------------------------------------------------
% 0.17/0.44 % File : DAT246^1 : TPTP v9.2.1. Released v7.0.0.
% 0.17/0.44 % Domain : Data Structures
% 0.17/0.44 % Problem : Infinite streams (sequences/lists) 156
% 0.17/0.44 % Version : [Bla16] axioms : Especial.
% 0.17/0.44 % English :
% 0.17/0.44
% 0.17/0.44 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.17/0.44 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.17/0.44 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.17/0.44 % Source : [Bla16]
% 0.17/0.44 % Names : stream__156.p [Bla16]
% 0.17/0.44
% 0.17/0.44 % Status : Theorem
% 0.17/0.44 % Rating : 0.33 v9.1.0, 0.00 v7.5.0, 0.67 v7.2.0, 1.00 v7.1.0
% 0.17/0.44 % Syntax : Number of formulae : 319 ( 68 unt; 44 typ; 0 def)
% 0.17/0.44 % Number of atoms : 875 ( 188 equ; 0 cnn)
% 0.17/0.44 % Maximal formula atoms : 13 ( 3 avg)
% 0.17/0.44 % Number of connectives : 3635 ( 86 ~; 21 |; 39 &;2963 @)
% 0.17/0.44 % ( 0 <=>; 526 =>; 0 <=; 0 <~>)
% 0.17/0.44 % Maximal formula depth : 21 ( 9 avg)
% 0.17/0.44 % Number of types : 3 ( 2 usr)
% 0.17/0.44 % Number of type conns : 163 ( 163 >; 0 *; 0 +; 0 <<)
% 0.17/0.44 % Number of symbols : 43 ( 42 usr; 3 con; 0-4 aty)
% 0.17/0.44 % Number of variables : 991 ( 41 ^; 873 !; 41 ?; 991 :)
% 0.17/0.44 % ( 36 !>; 0 ?*; 0 @-; 0 @+)
% 0.17/0.44 % SPC : TH1_THM_EQU_NAR
% 0.17/0.44
% 0.17/0.44 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.17/0.44 % 2016-07-13 14:41:00.835
% 0.17/0.44 %------------------------------------------------------------------------------
% 0.17/0.44 %----Could-be-implicit typings (5)
% 0.17/0.44 thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream,type,
% 0.17/0.44 stream170649215stream: $tType > $tType ).
% 0.17/0.44
% 0.17/0.44 thf(ty_t_List_Olist,type,
% 0.17/0.44 list: $tType > $tType ).
% 0.17/0.44
% 0.17/0.44 thf(ty_t_Nat_Onat,type,
% 0.17/0.44 nat: $tType ).
% 0.17/0.44
% 0.17/0.44 thf(ty_t_itself,type,
% 0.17/0.44 itself: $tType > $tType ).
% 0.17/0.44
% 0.17/0.44 thf(ty_tf_a,type,
% 0.17/0.44 a: $tType ).
% 0.17/0.44
% 0.17/0.44 %----Explicit typings (39)
% 0.17/0.44 thf(sy_cl_HOL_Otype,type,
% 0.17/0.44 type:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Groups_Ozero,type,
% 0.17/0.44 zero:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Groups_Ominus,type,
% 0.17/0.44 minus:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Oord,type,
% 0.17/0.44 ord:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Oorder,type,
% 0.17/0.44 order:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Groups_Ogroup__add,type,
% 0.17/0.44 group_add:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Ono__bot,type,
% 0.17/0.44 no_bot:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Ono__top,type,
% 0.17/0.44 no_top:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Olinorder,type,
% 0.17/0.44 linorder:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Opreorder,type,
% 0.17/0.44 preorder:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Owellorder,type,
% 0.17/0.44 wellorder:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Odense__order,type,
% 0.17/0.44 dense_order:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.17/0.44 dense_linorder:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.17/0.44 ordered_ab_group_add:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.17/0.44 cancel146912293up_add:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.17/0.44 canoni770627133id_add:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.17/0.44 condit1656338222tinuum:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.17/0.44 condit1037483654norder:
% 0.17/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.17/0.44 minus_minus:
% 0.17/0.44 !>[A: $tType] : ( A > A > A ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.17/0.44 zero_zero:
% 0.17/0.44 !>[A: $tType] : A ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 0.17/0.44 hilbert_GreatestM:
% 0.17/0.44 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_List_Ocount__list,type,
% 0.17/0.44 count_list:
% 0.17/0.44 !>[A: $tType] : ( ( list @ A ) > A > nat ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_List_Olinorder__class_Osorted,type,
% 0.17/0.44 linorder_sorted:
% 0.17/0.44 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_List_Olist_Otl,type,
% 0.17/0.44 tl:
% 0.17/0.44 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_List_Olist__ex,type,
% 0.17/0.44 list_ex:
% 0.17/0.44 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_List_Onth,type,
% 0.17/0.44 nth:
% 0.17/0.44 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_List_Oremdups__adj,type,
% 0.17/0.44 remdups_adj:
% 0.17/0.44 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_List_Orev,type,
% 0.17/0.44 rev:
% 0.17/0.44 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_List_OtakeWhile,type,
% 0.17/0.44 takeWhile:
% 0.17/0.44 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Nat_OSuc,type,
% 0.17/0.44 suc: nat > nat ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.17/0.44 size_size:
% 0.17/0.44 !>[A: $tType] : ( A > nat ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.17/0.44 ord_less:
% 0.17/0.44 !>[A: $tType] : ( A > A > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.17/0.44 ord_less_eq:
% 0.17/0.44 !>[A: $tType] : ( A > A > $o ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Pure_Otype,type,
% 0.17/0.44 type2:
% 0.17/0.44 !>[A: $tType] : ( itself @ A ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Oshift,type,
% 0.17/0.44 stream1035003186_shift:
% 0.17/0.44 !>[A: $tType] : ( ( list @ A ) > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osnth,type,
% 0.17/0.44 stream370371455e_snth:
% 0.17/0.44 !>[A: $tType] : ( ( stream170649215stream @ A ) > nat > A ) ).
% 0.17/0.44
% 0.17/0.44 thf(sy_v_n,type,
% 0.17/0.44 n: nat ).
% 0.17/0.44
% 0.17/0.44 thf(sy_v_s,type,
% 0.17/0.44 s: stream170649215stream @ a ).
% 0.17/0.44
% 0.17/0.44 thf(sy_v_xs,type,
% 0.17/0.44 xs: list @ a ).
% 0.17/0.44
% 0.17/0.44 %----Relevant facts (254)
% 0.17/0.44 thf(fact_0_shift__left__inj,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A,S1: stream170649215stream @ A,S2: stream170649215stream @ A] :
% 0.17/0.44 ( ( ( stream1035003186_shift @ A @ Xs @ S1 )
% 0.17/0.44 = ( stream1035003186_shift @ A @ Xs @ S2 ) )
% 0.17/0.44 = ( S1 = S2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % shift_left_inj
% 0.17/0.44 thf(fact_1_shift__snth__less,axiom,
% 0.17/0.44 ! [A: $tType,P: nat,Xs: list @ A,S: stream170649215stream @ A] :
% 0.17/0.44 ( ( ord_less @ nat @ P @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.17/0.44 => ( ( stream370371455e_snth @ A @ ( stream1035003186_shift @ A @ Xs @ S ) @ P )
% 0.17/0.44 = ( nth @ A @ Xs @ P ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % shift_snth_less
% 0.17/0.44 thf(fact_2_nth__equalityI,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.17/0.44 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.17/0.44 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.17/0.44 => ( ! [I: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.17/0.44 => ( ( nth @ A @ Xs @ I )
% 0.17/0.44 = ( nth @ A @ Ys @ I ) ) )
% 0.17/0.44 => ( Xs = Ys ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % nth_equalityI
% 0.17/0.44 thf(fact_3_Skolem__list__nth,axiom,
% 0.17/0.44 ! [A: $tType,K: nat,P2: nat > A > $o] :
% 0.17/0.44 ( ( ! [I2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I2 @ K )
% 0.17/0.44 => ( ^ [P3: A > $o] :
% 0.17/0.44 ? [X: A] : ( P3 @ X )
% 0.17/0.44 @ ( P2 @ I2 ) ) ) )
% 0.17/0.44 = ( ? [Xs2: list @ A] :
% 0.17/0.44 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.17/0.44 = K )
% 0.17/0.44 & ! [I2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I2 @ K )
% 0.17/0.44 => ( P2 @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Skolem_list_nth
% 0.17/0.44 thf(fact_4_list__eq__iff__nth__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ^ [Y: list @ A,Z: list @ A] : ( Y = Z ) )
% 0.17/0.44 = ( ^ [Xs2: list @ A,Ys2: list @ A] :
% 0.17/0.44 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.17/0.44 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.17/0.44 & ! [I2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.17/0.44 => ( ( nth @ A @ Xs2 @ I2 )
% 0.17/0.44 = ( nth @ A @ Ys2 @ I2 ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % list_eq_iff_nth_eq
% 0.17/0.44 thf(fact_5_minus__apply,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( minus @ B @ ( type2 @ B ) )
% 0.17/0.44 => ( ( minus_minus @ ( A > B ) )
% 0.17/0.44 = ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minus_apply
% 0.17/0.44 thf(fact_6_diff__less__mono2,axiom,
% 0.17/0.44 ! [M: nat,N: nat,L: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 => ( ( ord_less @ nat @ M @ L )
% 0.17/0.44 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_less_mono2
% 0.17/0.44 thf(fact_7_less__imp__diff__less,axiom,
% 0.17/0.44 ! [J: nat,K: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ J @ K )
% 0.17/0.44 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_imp_diff_less
% 0.17/0.44 thf(fact_8_length__induct,axiom,
% 0.17/0.44 ! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
% 0.17/0.44 ( ! [Xs3: list @ A] :
% 0.17/0.44 ( ! [Ys3: list @ A] :
% 0.17/0.44 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.17/0.44 => ( P2 @ Ys3 ) )
% 0.17/0.44 => ( P2 @ Xs3 ) )
% 0.17/0.44 => ( P2 @ Xs ) ) ).
% 0.17/0.44
% 0.17/0.44 % length_induct
% 0.17/0.44 thf(fact_9_diff__strict__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,D: A,C: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less @ A @ D @ C )
% 0.17/0.44 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_strict_mono
% 0.17/0.44 thf(fact_10_diff__eq__diff__less,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A,D: A] :
% 0.17/0.44 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.17/0.44 = ( minus_minus @ A @ C @ D ) )
% 0.17/0.44 => ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_eq_diff_less
% 0.17/0.44 thf(fact_11_diff__strict__left__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A,C: A] :
% 0.17/0.44 ( ( ord_less @ A @ B3 @ A3 )
% 0.17/0.44 => ( ord_less @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_strict_left_mono
% 0.17/0.44 thf(fact_12_diff__strict__right__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_strict_right_mono
% 0.17/0.44 thf(fact_13_shift__snth__ge,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A,P: nat,S: stream170649215stream @ A] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ P )
% 0.17/0.44 => ( ( stream370371455e_snth @ A @ ( stream1035003186_shift @ A @ Xs @ S ) @ P )
% 0.17/0.44 = ( stream370371455e_snth @ A @ S @ ( minus_minus @ nat @ P @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % shift_snth_ge
% 0.17/0.44 thf(fact_14_diff__diff__cancel,axiom,
% 0.17/0.44 ! [I3: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ I3 @ N )
% 0.17/0.44 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I3 ) )
% 0.17/0.44 = I3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_diff_cancel
% 0.17/0.44 thf(fact_15_le__refl,axiom,
% 0.17/0.44 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.17/0.44
% 0.17/0.44 % le_refl
% 0.17/0.44 thf(fact_16_le__trans,axiom,
% 0.17/0.44 ! [I3: nat,J: nat,K: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ I3 @ J )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ J @ K )
% 0.17/0.44 => ( ord_less_eq @ nat @ I3 @ K ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_trans
% 0.17/0.44 thf(fact_17_eq__imp__le,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( M = N )
% 0.17/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % eq_imp_le
% 0.17/0.44 thf(fact_18_le__antisym,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ N @ M )
% 0.17/0.44 => ( M = N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_antisym
% 0.17/0.44 thf(fact_19_nat__le__linear,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat_le_linear
% 0.17/0.44 thf(fact_20_diff__eq__diff__less__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A,D: A] :
% 0.17/0.44 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.17/0.44 = ( minus_minus @ A @ C @ D ) )
% 0.17/0.44 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_eq_diff_less_eq
% 0.17/0.44 thf(fact_21_diff__right__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_right_mono
% 0.17/0.44 thf(fact_22_diff__left__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A,C: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.17/0.44 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_left_mono
% 0.17/0.44 thf(fact_23_diff__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,D: A,C: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ D @ C )
% 0.17/0.44 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_mono
% 0.17/0.44 thf(fact_24_less__mono__imp__le__mono,axiom,
% 0.17/0.44 ! [F: nat > nat,I3: nat,J: nat] :
% 0.17/0.44 ( ! [I: nat,J2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I @ J2 )
% 0.17/0.44 => ( ord_less @ nat @ ( F @ I ) @ ( F @ J2 ) ) )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ I3 @ J )
% 0.17/0.44 => ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_mono_imp_le_mono
% 0.17/0.44 thf(fact_25_le__neq__implies__less,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 => ( ( M != N )
% 0.17/0.44 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_neq_implies_less
% 0.17/0.44 thf(fact_26_less__or__eq__imp__le,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 | ( M = N ) )
% 0.17/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_or_eq_imp_le
% 0.17/0.44 thf(fact_27_le__eq__less__or__eq,axiom,
% 0.17/0.44 ( ( ord_less_eq @ nat )
% 0.17/0.44 = ( ^ [M2: nat,N2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M2 @ N2 )
% 0.17/0.44 | ( M2 = N2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_eq_less_or_eq
% 0.17/0.44 thf(fact_28_less__imp__le__nat,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_imp_le_nat
% 0.17/0.44 thf(fact_29_nat__less__le,axiom,
% 0.17/0.44 ( ( ord_less @ nat )
% 0.17/0.44 = ( ^ [M2: nat,N2: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.17/0.44 & ( M2 != N2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat_less_le
% 0.17/0.44 thf(fact_30_diff__le__mono2,axiom,
% 0.17/0.44 ! [M: nat,N: nat,L: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_le_mono2
% 0.17/0.44 thf(fact_31_le__diff__iff_H,axiom,
% 0.17/0.44 ! [A3: nat,C: nat,B3: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ A3 @ C )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ B3 @ C )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A3 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 0.17/0.44 = ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_diff_iff'
% 0.17/0.44 thf(fact_32_diff__le__self,axiom,
% 0.17/0.44 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.17/0.44
% 0.17/0.44 % diff_le_self
% 0.17/0.44 thf(fact_33_diff__le__mono,axiom,
% 0.17/0.44 ! [M: nat,N: nat,L: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_le_mono
% 0.17/0.44 thf(fact_34_Nat_Odiff__diff__eq,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ K @ M )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ K @ N )
% 0.17/0.44 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.17/0.44 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Nat.diff_diff_eq
% 0.17/0.44 thf(fact_35_le__diff__iff,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ K @ M )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ K @ N )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.17/0.44 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_diff_iff
% 0.17/0.44 thf(fact_36_eq__diff__iff,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ K @ M )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ K @ N )
% 0.17/0.44 => ( ( ( minus_minus @ nat @ M @ K )
% 0.17/0.44 = ( minus_minus @ nat @ N @ K ) )
% 0.17/0.44 = ( M = N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % eq_diff_iff
% 0.17/0.44 thf(fact_37_diff__less__mono,axiom,
% 0.17/0.44 ! [A3: nat,B3: nat,C: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ C @ A3 )
% 0.17/0.44 => ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_less_mono
% 0.17/0.44 thf(fact_38_less__diff__iff,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ K @ M )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ K @ N )
% 0.17/0.44 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.17/0.44 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_diff_iff
% 0.17/0.44 thf(fact_39_diff__right__commute,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,C: A,B3: A] :
% 0.17/0.44 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B3 )
% 0.17/0.44 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_right_commute
% 0.17/0.44 thf(fact_40_diff__eq__diff__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( group_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A,D: A] :
% 0.17/0.44 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.17/0.44 = ( minus_minus @ A @ C @ D ) )
% 0.17/0.44 => ( ( A3 = B3 )
% 0.17/0.44 = ( C = D ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_eq_diff_eq
% 0.17/0.44 thf(fact_41_fun__diff__def,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( minus @ B @ ( type2 @ B ) )
% 0.17/0.44 => ( ( minus_minus @ ( A > B ) )
% 0.17/0.44 = ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % fun_diff_def
% 0.17/0.44 thf(fact_42_infinite__descent__measure,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,V: A > nat,X3: A] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ~ ( P2 @ X4 )
% 0.17/0.44 => ? [Y2: A] :
% 0.17/0.44 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X4 ) )
% 0.17/0.44 & ~ ( P2 @ Y2 ) ) )
% 0.17/0.44 => ( P2 @ X3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % infinite_descent_measure
% 0.17/0.44 thf(fact_43_measure__induct__rule,axiom,
% 0.17/0.44 ! [A: $tType,F: A > nat,P2: A > $o,A3: A] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ! [Y2: A] :
% 0.17/0.44 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X4 ) )
% 0.17/0.44 => ( P2 @ Y2 ) )
% 0.17/0.44 => ( P2 @ X4 ) )
% 0.17/0.44 => ( P2 @ A3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % measure_induct_rule
% 0.17/0.44 thf(fact_44_linorder__neqE__nat,axiom,
% 0.17/0.44 ! [X3: nat,Y3: nat] :
% 0.17/0.44 ( ( X3 != Y3 )
% 0.17/0.44 => ( ~ ( ord_less @ nat @ X3 @ Y3 )
% 0.17/0.44 => ( ord_less @ nat @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linorder_neqE_nat
% 0.17/0.44 thf(fact_45_ext,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( F @ X4 )
% 0.17/0.44 = ( G @ X4 ) )
% 0.17/0.44 => ( F = G ) ) ).
% 0.17/0.44
% 0.17/0.44 % ext
% 0.17/0.44 thf(fact_46_infinite__descent,axiom,
% 0.17/0.44 ! [P2: nat > $o,N: nat] :
% 0.17/0.44 ( ! [N3: nat] :
% 0.17/0.44 ( ~ ( P2 @ N3 )
% 0.17/0.44 => ? [M3: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M3 @ N3 )
% 0.17/0.44 & ~ ( P2 @ M3 ) ) )
% 0.17/0.44 => ( P2 @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % infinite_descent
% 0.17/0.44 thf(fact_47_nat__less__induct,axiom,
% 0.17/0.44 ! [P2: nat > $o,N: nat] :
% 0.17/0.44 ( ! [N3: nat] :
% 0.17/0.44 ( ! [M3: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M3 @ N3 )
% 0.17/0.44 => ( P2 @ M3 ) )
% 0.17/0.44 => ( P2 @ N3 ) )
% 0.17/0.44 => ( P2 @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat_less_induct
% 0.17/0.44 thf(fact_48_less__irrefl__nat,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ~ ( ord_less @ nat @ N @ N ) ).
% 0.17/0.44
% 0.17/0.44 % less_irrefl_nat
% 0.17/0.44 thf(fact_49_measure__induct,axiom,
% 0.17/0.44 ! [A: $tType,F: A > nat,P2: A > $o,A3: A] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ! [Y2: A] :
% 0.17/0.44 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X4 ) )
% 0.17/0.44 => ( P2 @ Y2 ) )
% 0.17/0.44 => ( P2 @ X4 ) )
% 0.17/0.44 => ( P2 @ A3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % measure_induct
% 0.17/0.44 thf(fact_50_less__not__refl3,axiom,
% 0.17/0.44 ! [S: nat,T: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ S @ T )
% 0.17/0.44 => ( S != T ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_not_refl3
% 0.17/0.44 thf(fact_51_less__not__refl2,axiom,
% 0.17/0.44 ! [N: nat,M: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ N @ M )
% 0.17/0.44 => ( M != N ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_not_refl2
% 0.17/0.44 thf(fact_52_less__not__refl,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ~ ( ord_less @ nat @ N @ N ) ).
% 0.17/0.44
% 0.17/0.44 % less_not_refl
% 0.17/0.44 thf(fact_53_nat__neq__iff,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( M != N )
% 0.17/0.44 = ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat_neq_iff
% 0.17/0.44 thf(fact_54_neq__if__length__neq,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.17/0.44 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.17/0.44 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.17/0.44 => ( Xs != Ys ) ) ).
% 0.17/0.44
% 0.17/0.44 % neq_if_length_neq
% 0.17/0.44 thf(fact_55_Ex__list__of__length,axiom,
% 0.17/0.44 ! [A: $tType,N: nat] :
% 0.17/0.44 ? [Xs3: list @ A] :
% 0.17/0.44 ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.17/0.44 = N ) ).
% 0.17/0.44
% 0.17/0.44 % Ex_list_of_length
% 0.17/0.44 thf(fact_56_diff__commute,axiom,
% 0.17/0.44 ! [I3: nat,J: nat,K: nat] :
% 0.17/0.44 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J ) @ K )
% 0.17/0.44 = ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ K ) @ J ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_commute
% 0.17/0.44 thf(fact_57_order__refl,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_refl
% 0.17/0.44 thf(fact_58_ex__has__greatest__nat,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
% 0.17/0.44 ( ( P2 @ K )
% 0.17/0.44 => ( ! [Y4: A] :
% 0.17/0.44 ( ( P2 @ Y4 )
% 0.17/0.44 => ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
% 0.17/0.44 => ? [X4: A] :
% 0.17/0.44 ( ( P2 @ X4 )
% 0.17/0.44 & ! [Y2: A] :
% 0.17/0.44 ( ( P2 @ Y2 )
% 0.17/0.44 => ( ord_less_eq @ nat @ ( M @ Y2 ) @ ( M @ X4 ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ex_has_greatest_nat
% 0.17/0.44 thf(fact_59_complete__interval,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,P2: A > $o] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( P2 @ A3 )
% 0.17/0.44 => ( ~ ( P2 @ B3 )
% 0.17/0.44 => ? [C2: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.17/0.44 & ( ord_less_eq @ A @ C2 @ B3 )
% 0.17/0.44 & ! [X5: A] :
% 0.17/0.44 ( ( ( ord_less_eq @ A @ A3 @ X5 )
% 0.17/0.44 & ( ord_less @ A @ X5 @ C2 ) )
% 0.17/0.44 => ( P2 @ X5 ) )
% 0.17/0.44 & ! [D2: A] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( ( ord_less_eq @ A @ A3 @ X4 )
% 0.17/0.44 & ( ord_less @ A @ X4 @ D2 ) )
% 0.17/0.44 => ( P2 @ X4 ) )
% 0.17/0.44 => ( ord_less_eq @ A @ D2 @ C2 ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % complete_interval
% 0.17/0.44 thf(fact_60_order_Onot__eq__order__implies__strict,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A] :
% 0.17/0.44 ( ( A3 != B3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.not_eq_order_implies_strict
% 0.17/0.44 thf(fact_61_dual__order_Ostrict__implies__order,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A] :
% 0.17/0.44 ( ( ord_less @ A @ B3 @ A3 )
% 0.17/0.44 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.strict_implies_order
% 0.17/0.44 thf(fact_62_dual__order_Ostrict__iff__order,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( ord_less @ A )
% 0.17/0.44 = ( ^ [B4: A,A4: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.17/0.44 & ( A4 != B4 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.strict_iff_order
% 0.17/0.44 thf(fact_63_dual__order_Oorder__iff__strict,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( ord_less_eq @ A )
% 0.17/0.44 = ( ^ [B4: A,A4: A] :
% 0.17/0.44 ( ( ord_less @ A @ B4 @ A4 )
% 0.17/0.44 | ( A4 = B4 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.order_iff_strict
% 0.17/0.44 thf(fact_64_order_Ostrict__implies__order,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.strict_implies_order
% 0.17/0.44 thf(fact_65_dense__le__bounded,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A,Z2: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ! [W: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ W )
% 0.17/0.44 => ( ( ord_less @ A @ W @ Y3 )
% 0.17/0.44 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.17/0.44 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dense_le_bounded
% 0.17/0.44 thf(fact_66_dense__ge__bounded,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Z2: A,X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z2 @ X3 )
% 0.17/0.44 => ( ! [W: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z2 @ W )
% 0.17/0.44 => ( ( ord_less @ A @ W @ X3 )
% 0.17/0.44 => ( ord_less_eq @ A @ Y3 @ W ) ) )
% 0.17/0.44 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dense_ge_bounded
% 0.17/0.44 thf(fact_67_dual__order_Ostrict__trans2,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A,C: A] :
% 0.17/0.44 ( ( ord_less @ A @ B3 @ A3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.17/0.44 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.strict_trans2
% 0.17/0.44 thf(fact_68_dual__order_Oantisym,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( A3 = B3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.antisym
% 0.17/0.44 thf(fact_69_dual__order_Otrans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A,C: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.17/0.44 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.trans
% 0.17/0.44 thf(fact_70_linorder__wlog,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [P2: A > A > $o,A3: A,B3: A] :
% 0.17/0.44 ( ! [A5: A,B5: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.17/0.44 => ( P2 @ A5 @ B5 ) )
% 0.17/0.44 => ( ! [A5: A,B5: A] :
% 0.17/0.44 ( ( P2 @ B5 @ A5 )
% 0.17/0.44 => ( P2 @ A5 @ B5 ) )
% 0.17/0.44 => ( P2 @ A3 @ B3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linorder_wlog
% 0.17/0.44 thf(fact_71_dual__order_Orefl,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.refl
% 0.17/0.44 thf(fact_72_order__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A,Z2: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.17/0.44 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_trans
% 0.17/0.44 thf(fact_73_order__class_Oorder_Oantisym,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.17/0.44 => ( A3 = B3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_class.order.antisym
% 0.17/0.44 thf(fact_74_ord__le__eq__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ord @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( B3 = C )
% 0.17/0.44 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ord_le_eq_trans
% 0.17/0.44 thf(fact_75_ord__eq__le__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ord @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( A3 = B3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.17/0.44 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ord_eq_le_trans
% 0.17/0.44 thf(fact_76_antisym__conv,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Y3: A,X3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 = ( X3 = Y3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % antisym_conv
% 0.17/0.44 thf(fact_77_le__cases3,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A,Z2: A] :
% 0.17/0.44 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.17/0.44 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.17/0.44 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.17/0.44 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 0.17/0.44 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.17/0.44 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.17/0.44 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 0.17/0.44 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.17/0.44 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 0.17/0.44 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 0.17/0.44 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_cases3
% 0.17/0.44 thf(fact_78_order_Otrans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.17/0.44 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.trans
% 0.17/0.44 thf(fact_79_le__cases,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_cases
% 0.17/0.44 thf(fact_80_eq__refl,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( X3 = Y3 )
% 0.17/0.44 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % eq_refl
% 0.17/0.44 thf(fact_81_linear,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linear
% 0.17/0.44 thf(fact_82_antisym,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.17/0.44 => ( X3 = Y3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % antisym
% 0.17/0.44 thf(fact_83_eq__iff,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.17/0.44 = ( ^ [X2: A,Y5: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 0.17/0.44 & ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % eq_iff
% 0.17/0.44 thf(fact_84_ord__le__eq__subst,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 & ( ord @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,B3: A,F: A > B,C: B] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ( F @ B3 )
% 0.17/0.44 = C )
% 0.17/0.44 => ( ! [X4: A,Y4: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ord_le_eq_subst
% 0.17/0.44 thf(fact_85_ord__eq__le__subst,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 & ( ord @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.17/0.44 ( ( A3
% 0.17/0.44 = ( F @ B3 ) )
% 0.17/0.44 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.17/0.44 => ( ! [X4: B,Y4: B] :
% 0.17/0.44 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ord_eq_le_subst
% 0.17/0.44 thf(fact_86_order__subst2,axiom,
% 0.17/0.44 ! [A: $tType,C3: $tType] :
% 0.17/0.44 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.17/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
% 0.17/0.44 => ( ! [X4: A,Y4: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less_eq @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_subst2
% 0.17/0.44 thf(fact_87_order__subst1,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( order @ B @ ( type2 @ B ) )
% 0.17/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.17/0.44 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.17/0.44 => ( ! [X4: B,Y4: B] :
% 0.17/0.44 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_subst1
% 0.17/0.44 thf(fact_88_le__fun__def,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 => ( ( ord_less_eq @ ( A > B ) )
% 0.17/0.44 = ( ^ [F2: A > B,G2: A > B] :
% 0.17/0.44 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_fun_def
% 0.17/0.44 thf(fact_89_le__funI,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 => ! [F: A > B,G: A > B] :
% 0.17/0.44 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.17/0.44 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_funI
% 0.17/0.44 thf(fact_90_le__funE,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 => ! [F: A > B,G: A > B,X3: A] :
% 0.17/0.44 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.17/0.44 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_funE
% 0.17/0.44 thf(fact_91_le__funD,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 => ! [F: A > B,G: A > B,X3: A] :
% 0.17/0.44 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.17/0.44 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_funD
% 0.17/0.44 thf(fact_92_ord__eq__less__subst,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 & ( ord @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.17/0.44 ( ( A3
% 0.17/0.44 = ( F @ B3 ) )
% 0.17/0.44 => ( ( ord_less @ B @ B3 @ C )
% 0.17/0.44 => ( ! [X4: B,Y4: B] :
% 0.17/0.44 ( ( ord_less @ B @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ord_eq_less_subst
% 0.17/0.44 thf(fact_93_ord__less__eq__subst,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 & ( ord @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,B3: A,F: A > B,C: B] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ( F @ B3 )
% 0.17/0.44 = C )
% 0.17/0.44 => ( ! [X4: A,Y4: A] :
% 0.17/0.44 ( ( ord_less @ A @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ord_less_eq_subst
% 0.17/0.44 thf(fact_94_order__less__subst1,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( order @ B @ ( type2 @ B ) )
% 0.17/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.17/0.44 => ( ( ord_less @ B @ B3 @ C )
% 0.17/0.44 => ( ! [X4: B,Y4: B] :
% 0.17/0.44 ( ( ord_less @ B @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_less_subst1
% 0.17/0.44 thf(fact_95_order__less__subst2,axiom,
% 0.17/0.44 ! [A: $tType,C3: $tType] :
% 0.17/0.44 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.17/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less @ C3 @ ( F @ B3 ) @ C )
% 0.17/0.44 => ( ! [X4: A,Y4: A] :
% 0.17/0.44 ( ( ord_less @ A @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_less_subst2
% 0.17/0.44 thf(fact_96_lt__ex,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A] :
% 0.17/0.44 ? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % lt_ex
% 0.17/0.44 thf(fact_97_gt__ex,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( no_top @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A] :
% 0.17/0.44 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 0.17/0.44
% 0.17/0.44 % gt_ex
% 0.17/0.44 thf(fact_98_neqE,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( X3 != Y3 )
% 0.17/0.44 => ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % neqE
% 0.17/0.44 thf(fact_99_neq__iff,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( X3 != Y3 )
% 0.17/0.44 = ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 | ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % neq_iff
% 0.17/0.44 thf(fact_100_order_Oasym,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.asym
% 0.17/0.44 thf(fact_101_dense,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ? [Z3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Z3 )
% 0.17/0.44 & ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dense
% 0.17/0.44 thf(fact_102_less__imp__neq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( X3 != Y3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_imp_neq
% 0.17/0.44 thf(fact_103_less__asym,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_asym
% 0.17/0.44 thf(fact_104_less__asym_H,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_asym'
% 0.17/0.44 thf(fact_105_less__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A,Z2: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ord_less @ A @ Y3 @ Z2 )
% 0.17/0.44 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_trans
% 0.17/0.44 thf(fact_106_less__linear,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 | ( X3 = Y3 )
% 0.17/0.44 | ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_linear
% 0.17/0.44 thf(fact_107_less__irrefl,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A] :
% 0.17/0.44 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_irrefl
% 0.17/0.44 thf(fact_108_ord__eq__less__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ord @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( A3 = B3 )
% 0.17/0.44 => ( ( ord_less @ A @ B3 @ C )
% 0.17/0.44 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ord_eq_less_trans
% 0.17/0.44 thf(fact_109_ord__less__eq__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( ord @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( B3 = C )
% 0.17/0.44 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ord_less_eq_trans
% 0.17/0.44 thf(fact_110_dual__order_Oasym,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A] :
% 0.17/0.44 ( ( ord_less @ A @ B3 @ A3 )
% 0.17/0.44 => ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.asym
% 0.17/0.44 thf(fact_111_less__imp__not__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( X3 != Y3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_imp_not_eq
% 0.17/0.44 thf(fact_112_less__not__sym,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_not_sym
% 0.17/0.44 thf(fact_113_less__induct,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [P2: A > $o,A3: A] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ! [Y2: A] :
% 0.17/0.44 ( ( ord_less @ A @ Y2 @ X4 )
% 0.17/0.44 => ( P2 @ Y2 ) )
% 0.17/0.44 => ( P2 @ X4 ) )
% 0.17/0.44 => ( P2 @ A3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_induct
% 0.17/0.44 thf(fact_114_antisym__conv3,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Y3: A,X3: A] :
% 0.17/0.44 ( ~ ( ord_less @ A @ Y3 @ X3 )
% 0.17/0.44 => ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 0.17/0.44 = ( X3 = Y3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % antisym_conv3
% 0.17/0.44 thf(fact_115_less__imp__not__eq2,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( Y3 != X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_imp_not_eq2
% 0.17/0.44 thf(fact_116_less__imp__triv,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A,P2: $o] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ord_less @ A @ Y3 @ X3 )
% 0.17/0.44 => P2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_imp_triv
% 0.17/0.44 thf(fact_117_linorder__cases,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( X3 != Y3 )
% 0.17/0.44 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linorder_cases
% 0.17/0.44 thf(fact_118_dual__order_Oirrefl,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A] :
% 0.17/0.44 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.irrefl
% 0.17/0.44 thf(fact_119_order_Ostrict__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less @ A @ B3 @ C )
% 0.17/0.44 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.strict_trans
% 0.17/0.44 thf(fact_120_less__imp__not__less,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_imp_not_less
% 0.17/0.44 thf(fact_121_dual__order_Ostrict__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A,C: A] :
% 0.17/0.44 ( ( ord_less @ A @ B3 @ A3 )
% 0.17/0.44 => ( ( ord_less @ A @ C @ B3 )
% 0.17/0.44 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.strict_trans
% 0.17/0.44 thf(fact_122_not__less__iff__gr__or__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 0.17/0.44 = ( ( ord_less @ A @ Y3 @ X3 )
% 0.17/0.44 | ( X3 = Y3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_less_iff_gr_or_eq
% 0.17/0.44 thf(fact_123_order_Ostrict__implies__not__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( A3 != B3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.strict_implies_not_eq
% 0.17/0.44 thf(fact_124_dual__order_Ostrict__implies__not__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A] :
% 0.17/0.44 ( ( ord_less @ A @ B3 @ A3 )
% 0.17/0.44 => ( A3 != B3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.strict_implies_not_eq
% 0.17/0.44 thf(fact_125_ex__gt__or__lt,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A] :
% 0.17/0.44 ? [B5: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B5 )
% 0.17/0.44 | ( ord_less @ A @ B5 @ A3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ex_gt_or_lt
% 0.17/0.44 thf(fact_126_ex__has__least__nat,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,K: A,M: A > nat] :
% 0.17/0.44 ( ( P2 @ K )
% 0.17/0.44 => ? [X4: A] :
% 0.17/0.44 ( ( P2 @ X4 )
% 0.17/0.44 & ! [Y2: A] :
% 0.17/0.44 ( ( P2 @ Y2 )
% 0.17/0.44 => ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ex_has_least_nat
% 0.17/0.44 thf(fact_127_leD,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Y3: A,X3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.17/0.44 => ~ ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % leD
% 0.17/0.44 thf(fact_128_leI,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % leI
% 0.17/0.44 thf(fact_129_le__less,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( ord_less_eq @ A )
% 0.17/0.44 = ( ^ [X2: A,Y5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X2 @ Y5 )
% 0.17/0.44 | ( X2 = Y5 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_less
% 0.17/0.44 thf(fact_130_less__le,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( ord_less @ A )
% 0.17/0.44 = ( ^ [X2: A,Y5: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 0.17/0.44 & ( X2 != Y5 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_le
% 0.17/0.44 thf(fact_131_order__le__less__subst1,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( order @ B @ ( type2 @ B ) )
% 0.17/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.17/0.44 => ( ( ord_less @ B @ B3 @ C )
% 0.17/0.44 => ( ! [X4: B,Y4: B] :
% 0.17/0.44 ( ( ord_less @ B @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_le_less_subst1
% 0.17/0.44 thf(fact_132_order__le__less__subst2,axiom,
% 0.17/0.44 ! [A: $tType,C3: $tType] :
% 0.17/0.44 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.17/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less @ C3 @ ( F @ B3 ) @ C )
% 0.17/0.44 => ( ! [X4: A,Y4: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_le_less_subst2
% 0.17/0.44 thf(fact_133_order__less__le__subst1,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( order @ B @ ( type2 @ B ) )
% 0.17/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.17/0.44 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.17/0.44 => ( ! [X4: B,Y4: B] :
% 0.17/0.44 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_less_le_subst1
% 0.17/0.44 thf(fact_134_order__less__le__subst2,axiom,
% 0.17/0.44 ! [A: $tType,C3: $tType] :
% 0.17/0.44 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.17/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.17/0.44 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
% 0.17/0.44 => ( ! [X4: A,Y4: A] :
% 0.17/0.44 ( ( ord_less @ A @ X4 @ Y4 )
% 0.17/0.44 => ( ord_less @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.17/0.44 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order_less_le_subst2
% 0.17/0.44 thf(fact_135_not__le,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ~ ( ord_less_eq @ A @ X3 @ Y3 ) )
% 0.17/0.44 = ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_le
% 0.17/0.44 thf(fact_136_not__less,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 0.17/0.44 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_less
% 0.17/0.44 thf(fact_137_le__neq__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( A3 != B3 )
% 0.17/0.44 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_neq_trans
% 0.17/0.44 thf(fact_138_less__imp__le,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_imp_le
% 0.17/0.44 thf(fact_139_antisym__conv1,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 = ( X3 = Y3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % antisym_conv1
% 0.17/0.44 thf(fact_140_antisym__conv2,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 0.17/0.44 = ( X3 = Y3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % antisym_conv2
% 0.17/0.44 thf(fact_141_le__less__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A,Z2: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ord_less @ A @ Y3 @ Z2 )
% 0.17/0.44 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_less_trans
% 0.17/0.44 thf(fact_142_less__le__trans,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A,Z2: A] :
% 0.17/0.44 ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.17/0.44 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_le_trans
% 0.17/0.44 thf(fact_143_dense__ge,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Z2: A,Y3: A] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z2 @ X4 )
% 0.17/0.44 => ( ord_less_eq @ A @ Y3 @ X4 ) )
% 0.17/0.44 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dense_ge
% 0.17/0.44 thf(fact_144_dense__le,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Y3: A,Z2: A] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ X4 @ Y3 )
% 0.17/0.44 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.17/0.44 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dense_le
% 0.17/0.44 thf(fact_145_le__less__linear,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 | ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_less_linear
% 0.17/0.44 thf(fact_146_le__imp__less__or__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [X3: A,Y3: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.17/0.44 => ( ( ord_less @ A @ X3 @ Y3 )
% 0.17/0.44 | ( X3 = Y3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_imp_less_or_eq
% 0.17/0.44 thf(fact_147_less__le__not__le,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( ord_less @ A )
% 0.17/0.44 = ( ^ [X2: A,Y5: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 0.17/0.44 & ~ ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_le_not_le
% 0.17/0.44 thf(fact_148_not__le__imp__less,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Y3: A,X3: A] :
% 0.17/0.44 ( ~ ( ord_less_eq @ A @ Y3 @ X3 )
% 0.17/0.44 => ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_le_imp_less
% 0.17/0.44 thf(fact_149_order_Ostrict__trans1,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less @ A @ B3 @ C )
% 0.17/0.44 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.strict_trans1
% 0.17/0.44 thf(fact_150_order_Ostrict__trans2,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [A3: A,B3: A,C: A] :
% 0.17/0.44 ( ( ord_less @ A @ A3 @ B3 )
% 0.17/0.44 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.17/0.44 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.strict_trans2
% 0.17/0.44 thf(fact_151_order_Oorder__iff__strict,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( ord_less_eq @ A )
% 0.17/0.44 = ( ^ [A4: A,B4: A] :
% 0.17/0.44 ( ( ord_less @ A @ A4 @ B4 )
% 0.17/0.44 | ( A4 = B4 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.order_iff_strict
% 0.17/0.44 thf(fact_152_order_Ostrict__iff__order,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( ord_less @ A )
% 0.17/0.44 = ( ^ [A4: A,B4: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.17/0.44 & ( A4 != B4 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % order.strict_iff_order
% 0.17/0.44 thf(fact_153_dual__order_Ostrict__trans1,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [B3: A,A3: A,C: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.17/0.44 => ( ( ord_less @ A @ C @ B3 )
% 0.17/0.44 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dual_order.strict_trans1
% 0.17/0.44 thf(fact_154_minf_I8_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X5 @ Z3 )
% 0.17/0.44 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(8)
% 0.17/0.44 thf(fact_155_minf_I6_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X5 @ Z3 )
% 0.17/0.44 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(6)
% 0.17/0.44 thf(fact_156_pinf_I8_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z3 @ X5 )
% 0.17/0.44 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(8)
% 0.17/0.44 thf(fact_157_pinf_I6_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z3 @ X5 )
% 0.17/0.44 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(6)
% 0.17/0.44 thf(fact_158_sorted__nth__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Xs: list @ A,I3: nat,J: nat] :
% 0.17/0.44 ( ( linorder_sorted @ A @ Xs )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ I3 @ J )
% 0.17/0.44 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.17/0.44 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % sorted_nth_mono
% 0.17/0.44 thf(fact_159_sorted__nth__monoI,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Xs: list @ A] :
% 0.17/0.44 ( ! [I: nat,J2: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ I @ J2 )
% 0.17/0.44 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.17/0.44 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J2 ) ) ) )
% 0.17/0.44 => ( linorder_sorted @ A @ Xs ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % sorted_nth_monoI
% 0.17/0.44 thf(fact_160_sorted__equals__nth__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ( ( linorder_sorted @ A )
% 0.17/0.44 = ( ^ [Xs2: list @ A] :
% 0.17/0.44 ! [J3: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.17/0.44 => ! [I2: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.17/0.44 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % sorted_equals_nth_mono
% 0.17/0.44 thf(fact_161_less__fun__def,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 => ( ( ord_less @ ( A > B ) )
% 0.17/0.44 = ( ^ [F2: A > B,G2: A > B] :
% 0.17/0.44 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.17/0.44 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_fun_def
% 0.17/0.44 thf(fact_162_pinf_I1_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.17/0.44 ( ? [Z4: A] :
% 0.17/0.44 ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z4 @ X4 )
% 0.17/0.44 => ( ( P2 @ X4 )
% 0.17/0.44 = ( P4 @ X4 ) ) )
% 0.17/0.44 => ( ? [Z4: A] :
% 0.17/0.44 ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z4 @ X4 )
% 0.17/0.44 => ( ( Q @ X4 )
% 0.17/0.44 = ( Q2 @ X4 ) ) )
% 0.17/0.44 => ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z3 @ X5 )
% 0.17/0.44 => ( ( ( P2 @ X5 )
% 0.17/0.44 & ( Q @ X5 ) )
% 0.17/0.44 = ( ( P4 @ X5 )
% 0.17/0.44 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(1)
% 0.17/0.44 thf(fact_163_pinf_I2_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.17/0.44 ( ? [Z4: A] :
% 0.17/0.44 ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z4 @ X4 )
% 0.17/0.44 => ( ( P2 @ X4 )
% 0.17/0.44 = ( P4 @ X4 ) ) )
% 0.17/0.44 => ( ? [Z4: A] :
% 0.17/0.44 ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z4 @ X4 )
% 0.17/0.44 => ( ( Q @ X4 )
% 0.17/0.44 = ( Q2 @ X4 ) ) )
% 0.17/0.44 => ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z3 @ X5 )
% 0.17/0.44 => ( ( ( P2 @ X5 )
% 0.17/0.44 | ( Q @ X5 ) )
% 0.17/0.44 = ( ( P4 @ X5 )
% 0.17/0.44 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(2)
% 0.17/0.44 thf(fact_164_pinf_I3_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z3 @ X5 )
% 0.17/0.44 => ( X5 != T ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(3)
% 0.17/0.44 thf(fact_165_pinf_I4_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z3 @ X5 )
% 0.17/0.44 => ( X5 != T ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(4)
% 0.17/0.44 thf(fact_166_pinf_I5_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z3 @ X5 )
% 0.17/0.44 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(5)
% 0.17/0.44 thf(fact_167_pinf_I7_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ Z3 @ X5 )
% 0.17/0.44 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(7)
% 0.17/0.44 thf(fact_168_pinf_I11_J,axiom,
% 0.17/0.44 ! [C3: $tType,D3: $tType] :
% 0.17/0.44 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 0.17/0.44 => ! [F3: D3] :
% 0.17/0.44 ? [Z3: C3] :
% 0.17/0.44 ! [X5: C3] :
% 0.17/0.44 ( ( ord_less @ C3 @ Z3 @ X5 )
% 0.17/0.44 => ( F3 = F3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pinf(11)
% 0.17/0.44 thf(fact_169_minf_I1_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.17/0.44 ( ? [Z4: A] :
% 0.17/0.44 ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ X4 @ Z4 )
% 0.17/0.44 => ( ( P2 @ X4 )
% 0.17/0.44 = ( P4 @ X4 ) ) )
% 0.17/0.44 => ( ? [Z4: A] :
% 0.17/0.44 ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ X4 @ Z4 )
% 0.17/0.44 => ( ( Q @ X4 )
% 0.17/0.44 = ( Q2 @ X4 ) ) )
% 0.17/0.44 => ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X5 @ Z3 )
% 0.17/0.44 => ( ( ( P2 @ X5 )
% 0.17/0.44 & ( Q @ X5 ) )
% 0.17/0.44 = ( ( P4 @ X5 )
% 0.17/0.44 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(1)
% 0.17/0.44 thf(fact_170_minf_I2_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.17/0.44 ( ? [Z4: A] :
% 0.17/0.44 ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ X4 @ Z4 )
% 0.17/0.44 => ( ( P2 @ X4 )
% 0.17/0.44 = ( P4 @ X4 ) ) )
% 0.17/0.44 => ( ? [Z4: A] :
% 0.17/0.44 ! [X4: A] :
% 0.17/0.44 ( ( ord_less @ A @ X4 @ Z4 )
% 0.17/0.44 => ( ( Q @ X4 )
% 0.17/0.44 = ( Q2 @ X4 ) ) )
% 0.17/0.44 => ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X5 @ Z3 )
% 0.17/0.44 => ( ( ( P2 @ X5 )
% 0.17/0.44 | ( Q @ X5 ) )
% 0.17/0.44 = ( ( P4 @ X5 )
% 0.17/0.44 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(2)
% 0.17/0.44 thf(fact_171_minf_I3_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X5 @ Z3 )
% 0.17/0.44 => ( X5 != T ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(3)
% 0.17/0.44 thf(fact_172_minf_I4_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X5 @ Z3 )
% 0.17/0.44 => ( X5 != T ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(4)
% 0.17/0.44 thf(fact_173_minf_I5_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X5 @ Z3 )
% 0.17/0.44 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(5)
% 0.17/0.44 thf(fact_174_minf_I7_J,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [T: A] :
% 0.17/0.44 ? [Z3: A] :
% 0.17/0.44 ! [X5: A] :
% 0.17/0.44 ( ( ord_less @ A @ X5 @ Z3 )
% 0.17/0.44 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(7)
% 0.17/0.44 thf(fact_175_minf_I11_J,axiom,
% 0.17/0.44 ! [C3: $tType,D3: $tType] :
% 0.17/0.44 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 0.17/0.44 => ! [F3: D3] :
% 0.17/0.44 ? [Z3: C3] :
% 0.17/0.44 ! [X5: C3] :
% 0.17/0.44 ( ( ord_less @ C3 @ X5 @ Z3 )
% 0.17/0.44 => ( F3 = F3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % minf(11)
% 0.17/0.44 thf(fact_176_sorted__rev__nth__mono,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Xs: list @ A,I3: nat,J: nat] :
% 0.17/0.44 ( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ I3 @ J )
% 0.17/0.44 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.17/0.44 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % sorted_rev_nth_mono
% 0.17/0.44 thf(fact_177_list__ex__length,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( list_ex @ A )
% 0.17/0.44 = ( ^ [P5: A > $o,Xs2: list @ A] :
% 0.17/0.44 ? [N2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.17/0.44 & ( P5 @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % list_ex_length
% 0.17/0.44 thf(fact_178_length__takeWhile__less__P__nth,axiom,
% 0.17/0.44 ! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
% 0.17/0.44 ( ! [I: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I @ J )
% 0.17/0.44 => ( P2 @ ( nth @ A @ Xs @ I ) ) )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.17/0.44 => ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % length_takeWhile_less_P_nth
% 0.17/0.44 thf(fact_179_rev__rev__ident,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A] :
% 0.17/0.44 ( ( rev @ A @ ( rev @ A @ Xs ) )
% 0.17/0.44 = Xs ) ).
% 0.17/0.44
% 0.17/0.44 % rev_rev_ident
% 0.17/0.44 thf(fact_180_rev__is__rev__conv,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.17/0.44 ( ( ( rev @ A @ Xs )
% 0.17/0.44 = ( rev @ A @ Ys ) )
% 0.17/0.44 = ( Xs = Ys ) ) ).
% 0.17/0.44
% 0.17/0.44 % rev_is_rev_conv
% 0.17/0.44 thf(fact_181_takeWhile__idem,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,Xs: list @ A] :
% 0.17/0.44 ( ( takeWhile @ A @ P2 @ ( takeWhile @ A @ P2 @ Xs ) )
% 0.17/0.44 = ( takeWhile @ A @ P2 @ Xs ) ) ).
% 0.17/0.44
% 0.17/0.44 % takeWhile_idem
% 0.17/0.44 thf(fact_182_length__rev,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A] :
% 0.17/0.44 ( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
% 0.17/0.44 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.17/0.44
% 0.17/0.44 % length_rev
% 0.17/0.44 thf(fact_183_list__ex__rev,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,Xs: list @ A] :
% 0.17/0.44 ( ( list_ex @ A @ P2 @ ( rev @ A @ Xs ) )
% 0.17/0.44 = ( list_ex @ A @ P2 @ Xs ) ) ).
% 0.17/0.44
% 0.17/0.44 % list_ex_rev
% 0.17/0.44 thf(fact_184_sorted__takeWhile,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Xs: list @ A,P2: A > $o] :
% 0.17/0.44 ( ( linorder_sorted @ A @ Xs )
% 0.17/0.44 => ( linorder_sorted @ A @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % sorted_takeWhile
% 0.17/0.44 thf(fact_185_rev__swap,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.17/0.44 ( ( ( rev @ A @ Xs )
% 0.17/0.44 = Ys )
% 0.17/0.44 = ( Xs
% 0.17/0.44 = ( rev @ A @ Ys ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % rev_swap
% 0.17/0.44 thf(fact_186_length__takeWhile__le,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.17/0.44
% 0.17/0.44 % length_takeWhile_le
% 0.17/0.44 thf(fact_187_takeWhile__nth,axiom,
% 0.17/0.44 ! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
% 0.17/0.44 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) )
% 0.17/0.44 => ( ( nth @ A @ ( takeWhile @ A @ P2 @ Xs ) @ J )
% 0.17/0.44 = ( nth @ A @ Xs @ J ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % takeWhile_nth
% 0.17/0.44 thf(fact_188_nth__length__takeWhile,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,Xs: list @ A] :
% 0.17/0.44 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.17/0.44 => ~ ( P2 @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % nth_length_takeWhile
% 0.17/0.44 thf(fact_189_rev__nth,axiom,
% 0.17/0.44 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.17/0.44 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.17/0.44 => ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
% 0.17/0.44 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % rev_nth
% 0.17/0.44 thf(fact_190_count__le__length,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.17/0.44
% 0.17/0.44 % count_le_length
% 0.17/0.44 thf(fact_191_GreatestM__nat__lemma,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
% 0.17/0.44 ( ( P2 @ K )
% 0.17/0.44 => ( ! [Y4: A] :
% 0.17/0.44 ( ( P2 @ Y4 )
% 0.17/0.44 => ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
% 0.17/0.44 => ( ( P2 @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) )
% 0.17/0.44 & ! [Y2: A] :
% 0.17/0.44 ( ( P2 @ Y2 )
% 0.17/0.44 => ( ord_less_eq @ nat @ ( M @ Y2 ) @ ( M @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % GreatestM_nat_lemma
% 0.17/0.44 thf(fact_192_old_Onat_Oinject,axiom,
% 0.17/0.44 ! [Nat: nat,Nat2: nat] :
% 0.17/0.44 ( ( ( suc @ Nat )
% 0.17/0.44 = ( suc @ Nat2 ) )
% 0.17/0.44 = ( Nat = Nat2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % old.nat.inject
% 0.17/0.44 thf(fact_193_nat_Oinject,axiom,
% 0.17/0.44 ! [X22: nat,Y22: nat] :
% 0.17/0.44 ( ( ( suc @ X22 )
% 0.17/0.44 = ( suc @ Y22 ) )
% 0.17/0.44 = ( X22 = Y22 ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat.inject
% 0.17/0.44 thf(fact_194_Suc__less__eq,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.17/0.44 = ( ord_less @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_less_eq
% 0.17/0.44 thf(fact_195_Suc__mono,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_mono
% 0.17/0.44 thf(fact_196_lessI,axiom,
% 0.17/0.44 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % lessI
% 0.17/0.44 thf(fact_197_Suc__le__mono,axiom,
% 0.17/0.44 ! [N: nat,M: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.17/0.44 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_le_mono
% 0.17/0.44 thf(fact_198_diff__Suc__Suc,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.17/0.44 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_Suc_Suc
% 0.17/0.44 thf(fact_199_Suc__diff__diff,axiom,
% 0.17/0.44 ! [M: nat,N: nat,K: nat] :
% 0.17/0.44 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.17/0.44 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_diff_diff
% 0.17/0.44 thf(fact_200_not__less__less__Suc__eq,axiom,
% 0.17/0.44 ! [N: nat,M: nat] :
% 0.17/0.44 ( ~ ( ord_less @ nat @ N @ M )
% 0.17/0.44 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.17/0.44 = ( N = M ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_less_less_Suc_eq
% 0.17/0.44 thf(fact_201_strict__inc__induct,axiom,
% 0.17/0.44 ! [I3: nat,J: nat,P2: nat > $o] :
% 0.17/0.44 ( ( ord_less @ nat @ I3 @ J )
% 0.17/0.44 => ( ! [I: nat] :
% 0.17/0.44 ( ( J
% 0.17/0.44 = ( suc @ I ) )
% 0.17/0.44 => ( P2 @ I ) )
% 0.17/0.44 => ( ! [I: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I @ J )
% 0.17/0.44 => ( ( P2 @ ( suc @ I ) )
% 0.17/0.44 => ( P2 @ I ) ) )
% 0.17/0.44 => ( P2 @ I3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % strict_inc_induct
% 0.17/0.44 thf(fact_202_less__Suc__induct,axiom,
% 0.17/0.44 ! [I3: nat,J: nat,P2: nat > nat > $o] :
% 0.17/0.44 ( ( ord_less @ nat @ I3 @ J )
% 0.17/0.44 => ( ! [I: nat] : ( P2 @ I @ ( suc @ I ) )
% 0.17/0.44 => ( ! [I: nat,J2: nat,K2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I @ J2 )
% 0.17/0.44 => ( ( ord_less @ nat @ J2 @ K2 )
% 0.17/0.44 => ( ( P2 @ I @ J2 )
% 0.17/0.44 => ( ( P2 @ J2 @ K2 )
% 0.17/0.44 => ( P2 @ I @ K2 ) ) ) ) )
% 0.17/0.44 => ( P2 @ I3 @ J ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_Suc_induct
% 0.17/0.44 thf(fact_203_less__trans__Suc,axiom,
% 0.17/0.44 ! [I3: nat,J: nat,K: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I3 @ J )
% 0.17/0.44 => ( ( ord_less @ nat @ J @ K )
% 0.17/0.44 => ( ord_less @ nat @ ( suc @ I3 ) @ K ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_trans_Suc
% 0.17/0.44 thf(fact_204_Suc__less__SucD,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.17/0.44 => ( ord_less @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_less_SucD
% 0.17/0.44 thf(fact_205_less__antisym,axiom,
% 0.17/0.44 ! [N: nat,M: nat] :
% 0.17/0.44 ( ~ ( ord_less @ nat @ N @ M )
% 0.17/0.44 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.17/0.44 => ( M = N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_antisym
% 0.17/0.44 thf(fact_206_Suc__less__eq2,axiom,
% 0.17/0.44 ! [N: nat,M: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.17/0.44 = ( ? [M4: nat] :
% 0.17/0.44 ( ( M
% 0.17/0.44 = ( suc @ M4 ) )
% 0.17/0.44 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_less_eq2
% 0.17/0.44 thf(fact_207_not__less__eq,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 0.17/0.44 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_less_eq
% 0.17/0.44 thf(fact_208_less__Suc__eq,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.17/0.44 = ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 | ( M = N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_Suc_eq
% 0.17/0.44 thf(fact_209_less__SucI,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_SucI
% 0.17/0.44 thf(fact_210_less__SucE,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.17/0.44 => ( ~ ( ord_less @ nat @ M @ N )
% 0.17/0.44 => ( M = N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_SucE
% 0.17/0.44 thf(fact_211_Suc__lessI,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 => ( ( ( suc @ M )
% 0.17/0.44 != N )
% 0.17/0.44 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_lessI
% 0.17/0.44 thf(fact_212_Suc__lessE,axiom,
% 0.17/0.44 ! [I3: nat,K: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ ( suc @ I3 ) @ K )
% 0.17/0.44 => ~ ! [J2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I3 @ J2 )
% 0.17/0.44 => ( K
% 0.17/0.44 != ( suc @ J2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_lessE
% 0.17/0.44 thf(fact_213_Suc__lessD,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 0.17/0.44 => ( ord_less @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_lessD
% 0.17/0.44 thf(fact_214_lessE,axiom,
% 0.17/0.44 ! [I3: nat,K: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I3 @ K )
% 0.17/0.44 => ( ( K
% 0.17/0.44 != ( suc @ I3 ) )
% 0.17/0.44 => ~ ! [J2: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ I3 @ J2 )
% 0.17/0.44 => ( K
% 0.17/0.44 != ( suc @ J2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % lessE
% 0.17/0.44 thf(fact_215_lift__Suc__mono__less,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [F: nat > A,N: nat,N4: nat] :
% 0.17/0.44 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.17/0.44 => ( ( ord_less @ nat @ N @ N4 )
% 0.17/0.44 => ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % lift_Suc_mono_less
% 0.17/0.44 thf(fact_216_lift__Suc__mono__less__iff,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [F: nat > A,N: nat,M: nat] :
% 0.17/0.44 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.17/0.44 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 0.17/0.44 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % lift_Suc_mono_less_iff
% 0.17/0.44 thf(fact_217_lift__Suc__antimono__le,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [F: nat > A,N: nat,N4: nat] :
% 0.17/0.44 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ N @ N4 )
% 0.17/0.44 => ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % lift_Suc_antimono_le
% 0.17/0.44 thf(fact_218_lift__Suc__mono__le,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [F: nat > A,N: nat,N4: nat] :
% 0.17/0.44 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.17/0.44 => ( ( ord_less_eq @ nat @ N @ N4 )
% 0.17/0.44 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % lift_Suc_mono_le
% 0.17/0.44 thf(fact_219_le__imp__less__Suc,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_imp_less_Suc
% 0.17/0.44 thf(fact_220_less__eq__Suc__le,axiom,
% 0.17/0.44 ( ( ord_less @ nat )
% 0.17/0.44 = ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_eq_Suc_le
% 0.17/0.44 thf(fact_221_less__Suc__eq__le,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.17/0.44 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_Suc_eq_le
% 0.17/0.44 thf(fact_222_le__less__Suc__eq,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.17/0.44 = ( N = M ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_less_Suc_eq
% 0.17/0.44 thf(fact_223_Suc__le__lessD,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.17/0.44 => ( ord_less @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_le_lessD
% 0.17/0.44 thf(fact_224_inc__induct,axiom,
% 0.17/0.44 ! [I3: nat,J: nat,P2: nat > $o] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ I3 @ J )
% 0.17/0.44 => ( ( P2 @ J )
% 0.17/0.44 => ( ! [N3: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ I3 @ N3 )
% 0.17/0.44 => ( ( ord_less @ nat @ N3 @ J )
% 0.17/0.44 => ( ( P2 @ ( suc @ N3 ) )
% 0.17/0.44 => ( P2 @ N3 ) ) ) )
% 0.17/0.44 => ( P2 @ I3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inc_induct
% 0.17/0.44 thf(fact_225_dec__induct,axiom,
% 0.17/0.44 ! [I3: nat,J: nat,P2: nat > $o] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ I3 @ J )
% 0.17/0.44 => ( ( P2 @ I3 )
% 0.17/0.44 => ( ! [N3: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ I3 @ N3 )
% 0.17/0.44 => ( ( ord_less @ nat @ N3 @ J )
% 0.17/0.44 => ( ( P2 @ N3 )
% 0.17/0.44 => ( P2 @ ( suc @ N3 ) ) ) ) )
% 0.17/0.44 => ( P2 @ J ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dec_induct
% 0.17/0.44 thf(fact_226_Suc__le__eq,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.17/0.44 = ( ord_less @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_le_eq
% 0.17/0.44 thf(fact_227_Suc__leI,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ M @ N )
% 0.17/0.44 => ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_leI
% 0.17/0.44 thf(fact_228_n__not__Suc__n,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( N
% 0.17/0.44 != ( suc @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % n_not_Suc_n
% 0.17/0.44 thf(fact_229_Suc__inject,axiom,
% 0.17/0.44 ! [X3: nat,Y3: nat] :
% 0.17/0.44 ( ( ( suc @ X3 )
% 0.17/0.44 = ( suc @ Y3 ) )
% 0.17/0.44 => ( X3 = Y3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_inject
% 0.17/0.44 thf(fact_230_zero__induct__lemma,axiom,
% 0.17/0.44 ! [P2: nat > $o,K: nat,I3: nat] :
% 0.17/0.44 ( ( P2 @ K )
% 0.17/0.44 => ( ! [N3: nat] :
% 0.17/0.44 ( ( P2 @ ( suc @ N3 ) )
% 0.17/0.44 => ( P2 @ N3 ) )
% 0.17/0.44 => ( P2 @ ( minus_minus @ nat @ K @ I3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_induct_lemma
% 0.17/0.44 thf(fact_231_full__nat__induct,axiom,
% 0.17/0.44 ! [P2: nat > $o,N: nat] :
% 0.17/0.44 ( ! [N3: nat] :
% 0.17/0.44 ( ! [M3: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
% 0.17/0.44 => ( P2 @ M3 ) )
% 0.17/0.44 => ( P2 @ N3 ) )
% 0.17/0.44 => ( P2 @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % full_nat_induct
% 0.17/0.44 thf(fact_232_not__less__eq__eq,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 0.17/0.44 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_less_eq_eq
% 0.17/0.44 thf(fact_233_Suc__n__not__le__n,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_n_not_le_n
% 0.17/0.44 thf(fact_234_le__Suc__eq,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.17/0.44 = ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 | ( M
% 0.17/0.44 = ( suc @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_Suc_eq
% 0.17/0.44 thf(fact_235_Suc__le__D,axiom,
% 0.17/0.44 ! [N: nat,M5: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
% 0.17/0.44 => ? [M6: nat] :
% 0.17/0.44 ( M5
% 0.17/0.44 = ( suc @ M6 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_le_D
% 0.17/0.44 thf(fact_236_le__SucI,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_SucI
% 0.17/0.44 thf(fact_237_le__SucE,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.17/0.44 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.17/0.44 => ( M
% 0.17/0.44 = ( suc @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_SucE
% 0.17/0.44 thf(fact_238_Suc__leD,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.17/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_leD
% 0.17/0.44 thf(fact_239_diff__less__Suc,axiom,
% 0.17/0.44 ! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_less_Suc
% 0.17/0.44 thf(fact_240_Suc__diff__Suc,axiom,
% 0.17/0.44 ! [N: nat,M: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ N @ M )
% 0.17/0.44 => ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
% 0.17/0.44 = ( minus_minus @ nat @ M @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_diff_Suc
% 0.17/0.44 thf(fact_241_Suc__diff__le,axiom,
% 0.17/0.44 ! [N: nat,M: nat] :
% 0.17/0.44 ( ( ord_less_eq @ nat @ N @ M )
% 0.17/0.44 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 0.17/0.44 = ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_diff_le
% 0.17/0.44 thf(fact_242_GreatestM__natI,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
% 0.17/0.44 ( ( P2 @ K )
% 0.17/0.44 => ( ! [Y4: A] :
% 0.17/0.44 ( ( P2 @ Y4 )
% 0.17/0.44 => ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
% 0.17/0.44 => ( P2 @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % GreatestM_natI
% 0.17/0.44 thf(fact_243_GreatestMI2,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.44 => ! [P2: A > $o,X3: A,M: A > B,Q: A > $o] :
% 0.17/0.44 ( ( P2 @ X3 )
% 0.17/0.44 => ( ! [Y4: A] :
% 0.17/0.44 ( ( P2 @ Y4 )
% 0.17/0.44 => ( ord_less_eq @ B @ ( M @ Y4 ) @ ( M @ X3 ) ) )
% 0.17/0.44 => ( ! [X4: A] :
% 0.17/0.44 ( ( P2 @ X4 )
% 0.17/0.44 => ( ! [Y2: A] :
% 0.17/0.44 ( ( P2 @ Y2 )
% 0.17/0.44 => ( ord_less_eq @ B @ ( M @ Y2 ) @ ( M @ X4 ) ) )
% 0.17/0.44 => ( Q @ X4 ) ) )
% 0.17/0.44 => ( Q @ ( hilbert_GreatestM @ A @ B @ M @ P2 ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % GreatestMI2
% 0.17/0.44 thf(fact_244_GreatestM__nat__le,axiom,
% 0.17/0.44 ! [A: $tType,P2: A > $o,X3: A,M: A > nat,B3: nat] :
% 0.17/0.44 ( ( P2 @ X3 )
% 0.17/0.44 => ( ! [Y4: A] :
% 0.17/0.44 ( ( P2 @ Y4 )
% 0.17/0.44 => ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
% 0.17/0.44 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % GreatestM_nat_le
% 0.17/0.44 thf(fact_245_prod__decode__aux_Oinduct,axiom,
% 0.17/0.44 ! [P2: nat > nat > $o,A0: nat,A1: nat] :
% 0.17/0.44 ( ! [K2: nat,M6: nat] :
% 0.17/0.44 ( ( ~ ( ord_less_eq @ nat @ M6 @ K2 )
% 0.17/0.44 => ( P2 @ ( suc @ K2 ) @ ( minus_minus @ nat @ M6 @ ( suc @ K2 ) ) ) )
% 0.17/0.44 => ( P2 @ K2 @ M6 ) )
% 0.17/0.44 => ( P2 @ A0 @ A1 ) ) ).
% 0.17/0.44
% 0.17/0.44 % prod_decode_aux.induct
% 0.17/0.44 thf(fact_246_remdups__adj__adjacent,axiom,
% 0.17/0.44 ! [A: $tType,I3: nat,Xs: list @ A] :
% 0.17/0.44 ( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
% 0.17/0.44 => ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I3 )
% 0.17/0.44 != ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % remdups_adj_adjacent
% 0.17/0.44 thf(fact_247_remdups__adj__rev,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A] :
% 0.17/0.44 ( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
% 0.17/0.44 = ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % remdups_adj_rev
% 0.17/0.44 thf(fact_248_sorted__remdups__adj,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [Xs: list @ A] :
% 0.17/0.44 ( ( linorder_sorted @ A @ Xs )
% 0.17/0.44 => ( linorder_sorted @ A @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % sorted_remdups_adj
% 0.17/0.44 thf(fact_249_remdups__adj__length,axiom,
% 0.17/0.44 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.17/0.44
% 0.17/0.44 % remdups_adj_length
% 0.17/0.44 thf(fact_250_nth__tl,axiom,
% 0.17/0.44 ! [A: $tType,N: nat,X3: list @ A] :
% 0.17/0.44 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ X3 ) ) )
% 0.17/0.44 => ( ( nth @ A @ ( tl @ A @ X3 ) @ N )
% 0.17/0.44 = ( nth @ A @ X3 @ ( suc @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % nth_tl
% 0.17/0.44 thf(fact_251_Suc__pred,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.17/0.44 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.17/0.44 = N ) ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_pred
% 0.17/0.44 thf(fact_252_le__zero__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [N: A] :
% 0.17/0.44 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.17/0.44 = ( N
% 0.17/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_zero_eq
% 0.17/0.44 thf(fact_253_not__gr__zero,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.17/0.44 => ! [N: A] :
% 0.17/0.44 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.17/0.44 = ( N
% 0.17/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_gr_zero
% 0.17/0.44
% 0.17/0.44 %----Type constructors (20)
% 0.17/0.44 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.17/0.44 ! [A6: $tType,A7: $tType] :
% 0.17/0.44 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.17/0.44 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.17/0.44 ! [A6: $tType,A7: $tType] :
% 0.17/0.44 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.17/0.44 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_fun___Orderings_Oord,axiom,
% 0.17/0.44 ! [A6: $tType,A7: $tType] :
% 0.17/0.44 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.17/0.44 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_fun___Groups_Ominus,axiom,
% 0.26/0.44 ! [A6: $tType,A7: $tType] :
% 0.26/0.44 ( ( minus @ A7 @ ( type2 @ A7 ) )
% 0.26/0.44 => ( minus @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.26/0.44 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.26/0.44 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.26/0.44 cancel146912293up_add @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.26/0.44 wellorder @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.26/0.44 preorder @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.26/0.44 linorder @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.26/0.44 no_top @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.26/0.44 order @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.26/0.44 ord @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
% 0.26/0.44 minus @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.26/0.44 zero @ nat @ ( type2 @ nat ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
% 0.26/0.44 preorder @ $o @ ( type2 @ $o ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
% 0.26/0.44 linorder @ $o @ ( type2 @ $o ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
% 0.26/0.44 order @ $o @ ( type2 @ $o ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 0.26/0.44 ord @ $o @ ( type2 @ $o ) ).
% 0.26/0.44
% 0.26/0.44 thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
% 0.26/0.44 minus @ $o @ ( type2 @ $o ) ).
% 0.26/0.44
% 0.26/0.44 %----Conjectures (1)
% 0.26/0.44 thf(conj_0,conjecture,
% 0.26/0.44 ( ( ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
% 0.26/0.44 => ( ( stream370371455e_snth @ a @ ( stream1035003186_shift @ a @ xs @ s ) @ n )
% 0.26/0.44 = ( nth @ a @ xs @ n ) ) )
% 0.26/0.44 & ( ~ ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
% 0.26/0.44 => ( ( stream370371455e_snth @ a @ ( stream1035003186_shift @ a @ xs @ s ) @ n )
% 0.26/0.44 = ( stream370371455e_snth @ a @ s @ ( minus_minus @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ) ).
% 0.26/0.44
% 0.26/0.44 %------------------------------------------------------------------------------
% 0.26/0.44 ------------------------
% 2.18/1.37 ---- Embedded in THF ---
% 2.18/1.37 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.18/1.37 %%% Generated on Tue Feb 24 06:11:33 EST 2026
% 2.18/1.37 %%% using '$$dhol' embedding, version 1.3.0.
% 2.18/1.37 %%% Logic specification used:
% 2.18/1.37 %%% thf(spec, logic, $$dhol).
% 2.18/1.37
% 2.18/1.37 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.gN7przDacx/DTF2DTF23201.p
% 2.18/1.37 thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream, type, stream170649215stream: ($tType > $tType)).
% 2.18/1.37 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.18/1.37 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.18/1.37 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.18/1.37 thf(ty_tf_a, type, a: $tType).
% 2.18/1.37 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.18/1.37 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.18/1.37 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.18/1.37 thf(sy_c_Hilbert__Choice_OGreatestM, type, hilbert_GreatestM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 2.18/1.37 thf(sy_c_List_Ocount__list, type, count_list: (!> [A:$tType]: (((list @ A) > (A > nat))))).
% 2.18/1.37 thf(sy_c_List_Olinorder__class_Osorted, type, linorder_sorted: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.18/1.37 thf(sy_c_List_Olist_Otl, type, tl: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.18/1.37 thf(sy_c_List_Olist__ex, type, list_ex: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 2.18/1.37 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 2.18/1.37 thf(sy_c_List_Oremdups__adj, type, remdups_adj: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.18/1.37 thf(sy_c_List_Orev, type, rev: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.18/1.37 thf(sy_c_List_OtakeWhile, type, takeWhile: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))).
% 2.18/1.37 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.18/1.37 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.18/1.37 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.18/1.37 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.18/1.37 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.18/1.37 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Oshift, type, stream1035003186_shift: (!> [A:$tType]: (((list @ A) > ((stream170649215stream @ A) > (stream170649215stream @ A)))))).
% 2.18/1.37 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osnth, type, stream370371455e_snth: (!> [A:$tType]: (((stream170649215stream @ A) > (nat > A))))).
% 2.18/1.37 thf(sy_v_n, type, n: nat).
% 2.18/1.37 thf(sy_v_s, type, s: (stream170649215stream @ a)).
% 2.18/1.37 thf(sy_v_xs, type, xs: (list @ a)).
% 2.18/1.37 thf(fact_0_shift__left__inj, axiom, (! [A:$tType,Xs:(list @ A),S1:(stream170649215stream @ A),S2:(stream170649215stream @ A)]: ((((((stream1035003186_shift @ A) @ Xs) @ S1) = (((stream1035003186_shift @ A) @ Xs) @ S2)) = (S1 = S2))))).
% 2.18/1.37 thf(fact_1_shift__snth__less, axiom, (! [A:$tType,P:nat,Xs:(list @ A),S:(stream170649215stream @ A)]: (((((ord_less @ nat) @ P) @ ((size_size @ (list @ A)) @ Xs)) => ((((stream370371455e_snth @ A) @ (((stream1035003186_shift @ A) @ Xs) @ S)) @ P) = (((nth @ A) @ Xs) @ P)))))).
% 2.18/1.37 thf(fact_2_nth__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => ((! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I) = (((nth @ A) @ Ys) @ I))))) => (Xs = Ys)))))).
% 2.18/1.37 thf(fact_3_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P2:(nat > (A > $o))]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => ((^ [P3:(A > $o)]: ((? [X:A]: ((P3 @ X))))) @ (P2 @ I2))))) = (? [Xs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = K) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => ((P2 @ I2) @ (((nth @ A) @ Xs2) @ I2)))))))))))).
% 2.18/1.37 thf(fact_4_list__eq__iff__nth__eq, axiom, (! [A:$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)))))))))))).
% 2.18/1.37 thf(fact_5_minus__apply, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A2:(A > B),B2:(A > B),X2:A]: ((((minus_minus @ B) @ (A2 @ X2)) @ (B2 @ X2))))))))).
% 2.18/1.37 thf(fact_6_diff__less__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less @ nat) @ M) @ N) => ((((ord_less @ nat) @ M) @ L) => (((ord_less @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M))))))).
% 2.18/1.37 thf(fact_7_less__imp__diff__less, axiom, (! [J:nat,K:nat,N:nat]: (((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (((minus_minus @ nat) @ J) @ N)) @ K))))).
% 2.18/1.37 thf(fact_8_length__induct, axiom, (! [A:$tType,P2:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs3:(list @ A)]: (((! [Ys3:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys3)) @ ((size_size @ (list @ A)) @ Xs3)) => (P2 @ Ys3)))) => (P2 @ Xs3)))) => (P2 @ Xs))))).
% 2.18/1.37 thf(fact_9_diff__strict__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,D:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ A) @ D) @ C) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.18/1.37 thf(fact_10_diff__eq__diff__less, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less @ A) @ A3) @ B3) = (((ord_less @ A) @ C) @ D))))))))).
% 2.18/1.37 thf(fact_11_diff__strict__left__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C:A]: (((((ord_less @ A) @ B3) @ A3) => (((ord_less @ A) @ (((minus_minus @ A) @ C) @ A3)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.18/1.37 thf(fact_12_diff__strict__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.18/1.37 thf(fact_13_shift__snth__ge, axiom, (! [A:$tType,Xs:(list @ A),P:nat,S:(stream170649215stream @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ P) => ((((stream370371455e_snth @ A) @ (((stream1035003186_shift @ A) @ Xs) @ S)) @ P) = (((stream370371455e_snth @ A) @ S) @ (((minus_minus @ nat) @ P) @ ((size_size @ (list @ A)) @ Xs)))))))).
% 2.18/1.37 thf(fact_14_diff__diff__cancel, axiom, (! [I3:nat,N:nat]: (((((ord_less_eq @ nat) @ I3) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I3)) = I3))))).
% 2.18/1.37 thf(fact_15_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.18/1.37 thf(fact_16_le__trans, axiom, (! [I3:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I3) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I3) @ K)))))).
% 2.18/1.37 thf(fact_17_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_18_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.18/1.37 thf(fact_19_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.18/1.37 thf(fact_20_diff__eq__diff__less__eq, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less_eq @ A) @ A3) @ B3) = (((ord_less_eq @ A) @ C) @ D))))))))).
% 2.18/1.37 thf(fact_21_diff__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.18/1.37 thf(fact_22_diff__left__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A3)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.18/1.37 thf(fact_23_diff__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,D:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ D) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.18/1.37 thf(fact_24_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I3:nat,J:nat]: (((! [I:nat,J2:nat]: (((((ord_less @ nat) @ I) @ J2) => (((ord_less @ nat) @ (F @ I)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I3) @ J) => (((ord_less_eq @ nat) @ (F @ I3)) @ (F @ J))))))).
% 2.18/1.37 thf(fact_25_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.18/1.37 thf(fact_26_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_27_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 2.18/1.37 thf(fact_28_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_29_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 2.18/1.37 thf(fact_30_diff__le__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M)))))).
% 2.18/1.37 thf(fact_31_le__diff__iff_H, axiom, (! [A3:nat,C:nat,B3:nat]: (((((ord_less_eq @ nat) @ A3) @ C) => ((((ord_less_eq @ nat) @ B3) @ C) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C) @ A3)) @ (((minus_minus @ nat) @ C) @ B3)) = (((ord_less_eq @ nat) @ B3) @ A3))))))).
% 2.18/1.37 thf(fact_32_diff__le__self, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))).
% 2.18/1.37 thf(fact_33_diff__le__mono, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N) @ L)))))).
% 2.18/1.37 thf(fact_34_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))))).
% 2.18/1.37 thf(fact_35_le__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less_eq @ nat) @ M) @ N))))))).
% 2.18/1.37 thf(fact_36_eq__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) = (M = N))))))).
% 2.18/1.37 thf(fact_37_diff__less__mono, axiom, (! [A3:nat,B3:nat,C:nat]: (((((ord_less @ nat) @ A3) @ B3) => ((((ord_less_eq @ nat) @ C) @ A3) => (((ord_less @ nat) @ (((minus_minus @ nat) @ A3) @ C)) @ (((minus_minus @ nat) @ B3) @ C))))))).
% 2.18/1.37 thf(fact_38_less__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less @ nat) @ M) @ N))))))).
% 2.18/1.37 thf(fact_39_diff__right__commute, axiom, (! [A:$tType]: ((((cancel146912293up_add @ A) @ (type2 @ A)) => (! [A3:A,C:A,B3:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ C)) @ B3) = (((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ B3)) @ C)))))))).
% 2.18/1.37 thf(fact_40_diff__eq__diff__eq, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((A3 = B3) = (C = D))))))))).
% 2.18/1.37 thf(fact_41_fun__diff__def, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A2:(A > B),B2:(A > B),X2:A]: ((((minus_minus @ B) @ (A2 @ X2)) @ (B2 @ X2))))))))).
% 2.18/1.37 thf(fact_42_infinite__descent__measure, axiom, (! [A:$tType,P2:(A > $o),V:(A > nat),X3:A]: (((! [X4:A]: (((~ ((P2 @ X4))) => (? [Y2:A]: (((((ord_less @ nat) @ (V @ Y2)) @ (V @ X4)) & (~ ((P2 @ Y2))))))))) => (P2 @ X3))))).
% 2.18/1.37 thf(fact_43_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P2:(A > $o),A3:A]: (((! [X4:A]: (((! [Y2:A]: (((((ord_less @ nat) @ (F @ Y2)) @ (F @ X4)) => (P2 @ Y2)))) => (P2 @ X4)))) => (P2 @ A3))))).
% 2.18/1.37 thf(fact_44_linorder__neqE__nat, axiom, (! [X3:nat,Y3:nat]: (((X3 != Y3) => ((~ ((((ord_less @ nat) @ X3) @ Y3))) => (((ord_less @ nat) @ Y3) @ X3)))))).
% 2.18/1.37 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.18/1.37 thf(fact_46_infinite__descent, axiom, (! [P2:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P2 @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P2 @ M3))))))))) => (P2 @ N))))).
% 2.18/1.37 thf(fact_47_nat__less__induct, axiom, (! [P2:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P2 @ M3)))) => (P2 @ N3)))) => (P2 @ N))))).
% 2.18/1.37 thf(fact_48_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.18/1.37 thf(fact_49_measure__induct, axiom, (! [A:$tType,F:(A > nat),P2:(A > $o),A3:A]: (((! [X4:A]: (((! [Y2:A]: (((((ord_less @ nat) @ (F @ Y2)) @ (F @ X4)) => (P2 @ Y2)))) => (P2 @ X4)))) => (P2 @ A3))))).
% 2.18/1.37 thf(fact_50_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.18/1.37 thf(fact_51_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.18/1.37 thf(fact_52_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.18/1.37 thf(fact_53_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.18/1.37 thf(fact_54_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys)) => (Xs != Ys))))).
% 2.18/1.37 thf(fact_55_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs3:(list @ A)]: ((((size_size @ (list @ A)) @ Xs3) = N)))))).
% 2.18/1.37 thf(fact_56_diff__commute, axiom, (! [I3:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I3) @ J)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I3) @ K)) @ J))))).
% 2.18/1.37 thf(fact_57_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.18/1.37 thf(fact_58_ex__has__greatest__nat, axiom, (! [A:$tType,P2:(A > $o),K:A,M:(A > nat),B3:nat]: (((P2 @ K) => ((! [Y4:A]: (((P2 @ Y4) => (((ord_less @ nat) @ (M @ Y4)) @ B3)))) => (? [X4:A]: (((P2 @ X4) & (! [Y2:A]: (((P2 @ Y2) => (((ord_less_eq @ nat) @ (M @ Y2)) @ (M @ X4))))))))))))).
% 2.18/1.37 thf(fact_59_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A3:A,B3:A,P2:(A > $o)]: (((((ord_less @ A) @ A3) @ B3) => ((P2 @ A3) => ((~ ((P2 @ B3))) => (? [C2:A]: (((((ord_less_eq @ A) @ A3) @ C2) & ((((ord_less_eq @ A) @ C2) @ B3) & ((! [X5:A]: ((((((ord_less_eq @ A) @ A3) @ X5) & (((ord_less @ A) @ X5) @ C2)) => (P2 @ X5)))) & (! [D2:A]: (((! [X4:A]: ((((((ord_less_eq @ A) @ A3) @ X4) & (((ord_less @ A) @ X4) @ D2)) => (P2 @ X4)))) => (((ord_less_eq @ A) @ D2) @ C2)))))))))))))))))).
% 2.18/1.37 thf(fact_60_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((A3 != B3) => ((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less @ A) @ A3) @ B3))))))))).
% 2.18/1.37 thf(fact_61_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 2.18/1.37 thf(fact_62_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (A4 != B4))))))))).
% 2.18/1.37 thf(fact_63_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) | (A4 = B4))))))))).
% 2.18/1.37 thf(fact_64_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.18/1.37 thf(fact_65_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y3) => ((! [W:A]: (((((ord_less @ A) @ X3) @ W) => ((((ord_less @ A) @ W) @ Y3) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y3) @ Z2))))))))).
% 2.18/1.37 thf(fact_66_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,X3:A,Y3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X3) => (((ord_less_eq @ A) @ Y3) @ W))))) => (((ord_less_eq @ A) @ Y3) @ Z2))))))))).
% 2.18/1.37 thf(fact_67_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C:A]: (((((ord_less @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A3))))))))).
% 2.18/1.37 thf(fact_68_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.18/1.37 thf(fact_69_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less_eq @ A) @ C) @ A3))))))))).
% 2.18/1.37 thf(fact_70_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P2 @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P2 @ B5) @ A5) => ((P2 @ A5) @ B5)))) => ((P2 @ A3) @ B3))))))))).
% 2.18/1.37 thf(fact_71_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.18/1.37 thf(fact_72_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X3) @ Z2))))))))).
% 2.18/1.37 thf(fact_73_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.18/1.37 thf(fact_74_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.18/1.37 thf(fact_75_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.18/1.37 thf(fact_76_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y3) = (X3 = Y3))))))))).
% 2.18/1.37 thf(fact_77_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X3) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z2)))) => (((((ord_less_eq @ A) @ X3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X3)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y3))))))))))))))))).
% 2.18/1.37 thf(fact_78_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.18/1.37 thf(fact_79_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.18/1.37 thf(fact_80_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((X3 = Y3) => (((ord_less_eq @ A) @ X3) @ Y3)))))))).
% 2.18/1.37 thf(fact_81_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.18/1.37 thf(fact_82_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X3) => (X3 = Y3))))))))).
% 2.18/1.37 thf(fact_83_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [X2:A,Y5:A]: (((((ord_less_eq @ A) @ X2) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X2))))))))).
% 2.18/1.37 thf(fact_84_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C)))))))))).
% 2.18/1.37 thf(fact_85_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 2.18/1.37 thf(fact_86_order__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C3) @ (F @ B3)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C3) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ C3) @ (F @ A3)) @ C)))))))))).
% 2.18/1.37 thf(fact_87_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 2.18/1.37 thf(fact_88_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 2.18/1.37 thf(fact_89_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.18/1.37 thf(fact_90_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.18/1.37 thf(fact_91_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.18/1.37 thf(fact_92_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C:B]: (((A3 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X4:B,Y4:B]: (((((ord_less @ B) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 2.18/1.37 thf(fact_93_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > B),C:B]: (((((ord_less @ A) @ A3) @ B3) => (((F @ B3) = C) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ B) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ B) @ (F @ A3)) @ C)))))))))).
% 2.18/1.37 thf(fact_94_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A3) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X4:B,Y4:B]: (((((ord_less @ B) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 2.18/1.37 thf(fact_95_order__less__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > C3),C:C3]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ C3) @ (F @ B3)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C3) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C3) @ (F @ A3)) @ C)))))))))).
% 2.18/1.37 thf(fact_96_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X3:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X3))))))))).
% 2.18/1.37 thf(fact_97_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X3:A]: ((? [X1:A]: ((((ord_less @ A) @ X3) @ X1))))))))).
% 2.18/1.37 thf(fact_98_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((X3 != Y3) => ((~ ((((ord_less @ A) @ X3) @ Y3))) => (((ord_less @ A) @ Y3) @ X3))))))))).
% 2.18/1.37 thf(fact_99_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((X3 != Y3) = ((((ord_less @ A) @ X3) @ Y3) | (((ord_less @ A) @ Y3) @ X3))))))))).
% 2.18/1.37 thf(fact_100_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (~ ((((ord_less @ A) @ B3) @ A3)))))))))).
% 2.18/1.37 thf(fact_101_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (? [Z3:A]: (((((ord_less @ A) @ X3) @ Z3) & (((ord_less @ A) @ Z3) @ Y3))))))))))).
% 2.18/1.37 thf(fact_102_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (X3 != Y3)))))))).
% 2.18/1.37 thf(fact_103_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X3)))))))))).
% 2.18/1.37 thf(fact_104_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (~ ((((ord_less @ A) @ B3) @ A3)))))))))).
% 2.18/1.37 thf(fact_105_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y3) => ((((ord_less @ A) @ Y3) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 2.18/1.37 thf(fact_106_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) | ((X3 = Y3) | (((ord_less @ A) @ Y3) @ X3))))))))).
% 2.18/1.37 thf(fact_107_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A]: ((~ ((((ord_less @ A) @ X3) @ X3))))))))).
% 2.18/1.37 thf(fact_108_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((A3 = B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.18/1.37 thf(fact_109_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => ((B3 = C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.18/1.37 thf(fact_110_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (~ ((((ord_less @ A) @ A3) @ B3)))))))))).
% 2.18/1.37 thf(fact_111_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (X3 != Y3)))))))).
% 2.18/1.37 thf(fact_112_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X3)))))))))).
% 2.18/1.37 thf(fact_113_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),A3:A]: (((! [X4:A]: (((! [Y2:A]: (((((ord_less @ A) @ Y2) @ X4) => (P2 @ Y2)))) => (P2 @ X4)))) => (P2 @ A3)))))))).
% 2.18/1.37 thf(fact_114_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y3:A,X3:A]: (((~ ((((ord_less @ A) @ Y3) @ X3))) => ((~ ((((ord_less @ A) @ X3) @ Y3))) = (X3 = Y3))))))))).
% 2.18/1.37 thf(fact_115_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (Y3 != X3)))))))).
% 2.18/1.37 thf(fact_116_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,P2:$o]: (((((ord_less @ A) @ X3) @ Y3) => ((((ord_less @ A) @ Y3) @ X3) => P2)))))))).
% 2.18/1.37 thf(fact_117_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) => ((X3 != Y3) => (((ord_less @ A) @ Y3) @ X3))))))))).
% 2.18/1.37 thf(fact_118_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 2.18/1.37 thf(fact_119_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.18/1.37 thf(fact_120_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X3)))))))))).
% 2.18/1.37 thf(fact_121_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C:A]: (((((ord_less @ A) @ B3) @ A3) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A3))))))))).
% 2.18/1.37 thf(fact_122_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) = ((((ord_less @ A) @ Y3) @ X3) | (X3 = Y3))))))))).
% 2.18/1.37 thf(fact_123_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (A3 != B3)))))))).
% 2.18/1.37 thf(fact_124_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (A3 != B3)))))))).
% 2.18/1.37 thf(fact_125_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A3:A]: ((? [B5:A]: (((((ord_less @ A) @ A3) @ B5) | (((ord_less @ A) @ B5) @ A3)))))))))).
% 2.18/1.37 thf(fact_126_ex__has__least__nat, axiom, (! [A:$tType,P2:(A > $o),K:A,M:(A > nat)]: (((P2 @ K) => (? [X4:A]: (((P2 @ X4) & (! [Y2:A]: (((P2 @ Y2) => (((ord_less_eq @ nat) @ (M @ X4)) @ (M @ Y2)))))))))))).
% 2.18/1.37 thf(fact_127_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ X3) => (~ ((((ord_less @ A) @ X3) @ Y3)))))))))).
% 2.18/1.37 thf(fact_128_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.18/1.37 thf(fact_129_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X2:A,Y5:A]: (((((ord_less @ A) @ X2) @ Y5) | (X2 = Y5))))))))).
% 2.18/1.37 thf(fact_130_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X2:A,Y5:A]: (((((ord_less_eq @ A) @ X2) @ Y5) & (X2 != Y5))))))))).
% 2.18/1.37 thf(fact_131_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X4:B,Y4:B]: (((((ord_less @ B) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 2.18/1.37 thf(fact_132_order__le__less__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less @ C3) @ (F @ B3)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C3) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C3) @ (F @ A3)) @ C)))))))))).
% 2.18/1.37 thf(fact_133_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 2.18/1.37 thf(fact_134_order__less__le__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > C3),C:C3]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less_eq @ C3) @ (F @ B3)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C3) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C3) @ (F @ A3)) @ C)))))))))).
% 2.18/1.37 thf(fact_135_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y3))) = (((ord_less @ A) @ Y3) @ X3)))))))).
% 2.18/1.37 thf(fact_136_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) = (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.18/1.37 thf(fact_137_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((A3 != B3) => (((ord_less @ A) @ A3) @ B3))))))))).
% 2.18/1.37 thf(fact_138_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less_eq @ A) @ X3) @ Y3)))))))).
% 2.18/1.37 thf(fact_139_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) => ((((ord_less_eq @ A) @ X3) @ Y3) = (X3 = Y3))))))))).
% 2.18/1.37 thf(fact_140_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((~ ((((ord_less @ A) @ X3) @ Y3))) = (X3 = Y3))))))))).
% 2.18/1.37 thf(fact_141_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less @ A) @ Y3) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 2.18/1.37 thf(fact_142_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 2.18/1.37 thf(fact_143_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,Y3:A]: (((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((ord_less_eq @ A) @ Y3) @ X4)))) => (((ord_less_eq @ A) @ Y3) @ Z2)))))))).
% 2.18/1.37 thf(fact_144_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y3:A,Z2:A]: (((! [X4:A]: (((((ord_less @ A) @ X4) @ Y3) => (((ord_less_eq @ A) @ X4) @ Z2)))) => (((ord_less_eq @ A) @ Y3) @ Z2)))))))).
% 2.18/1.37 thf(fact_145_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) | (((ord_less @ A) @ Y3) @ X3)))))))).
% 2.18/1.37 thf(fact_146_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less @ A) @ X3) @ Y3) | (X3 = Y3))))))))).
% 2.18/1.37 thf(fact_147_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X2:A,Y5:A]: (((((ord_less_eq @ A) @ X2) @ Y5) & (~ ((((ord_less_eq @ A) @ Y5) @ X2))))))))))).
% 2.18/1.37 thf(fact_148_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y3:A,X3:A]: (((~ ((((ord_less_eq @ A) @ Y3) @ X3))) => (((ord_less @ A) @ X3) @ Y3)))))))).
% 2.18/1.37 thf(fact_149_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.18/1.37 thf(fact_150_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 2.18/1.37 thf(fact_151_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) | (A4 = B4))))))))).
% 2.18/1.37 thf(fact_152_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (A4 != B4))))))))).
% 2.18/1.37 thf(fact_153_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A3))))))))).
% 2.18/1.37 thf(fact_154_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X5)))))))))))))).
% 2.18/1.37 thf(fact_155_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less_eq @ A) @ X5) @ T)))))))))))).
% 2.18/1.37 thf(fact_156_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less_eq @ A) @ T) @ X5)))))))))))).
% 2.18/1.37 thf(fact_157_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ T)))))))))))))).
% 2.18/1.37 thf(fact_158_sorted__nth__mono, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A),I3:nat,J:nat]: ((((linorder_sorted @ A) @ Xs) => ((((ord_less_eq @ nat) @ I3) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I3)) @ (((nth @ A) @ Xs) @ J))))))))))).
% 2.18/1.37 thf(fact_159_sorted__nth__monoI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A)]: (((! [I:nat,J2:nat]: (((((ord_less_eq @ nat) @ I) @ J2) => ((((ord_less @ nat) @ J2) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I)) @ (((nth @ A) @ Xs) @ J2)))))) => ((linorder_sorted @ A) @ Xs)))))))).
% 2.18/1.37 thf(fact_160_sorted__equals__nth__mono, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((linorder_sorted @ A) = (^ [Xs2:(list @ A)]: ((! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less_eq @ nat) @ I2) @ J3) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J3))))))))))))))).
% 2.18/1.37 thf(fact_161_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F2))))))))))).
% 2.18/1.37 thf(fact_162_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((P2 @ X4) = (P4 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P2 @ X5) & (Q @ X5)) = ((P4 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.18/1.37 thf(fact_163_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((P2 @ X4) = (P4 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P2 @ X5) | (Q @ X5)) = ((P4 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.18/1.37 thf(fact_164_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.18/1.37 thf(fact_165_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.18/1.37 thf(fact_166_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less @ A) @ X5) @ T)))))))))))))).
% 2.18/1.37 thf(fact_167_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less @ A) @ T) @ X5)))))))))))).
% 2.18/1.37 thf(fact_168_pinf_I11_J, axiom, (! [C3:$tType,D3:$tType]: ((((ord @ C3) @ (type2 @ C3)) => (! [F3:D3]: ((? [Z3:C3]: ((! [X5:C3]: (((((ord_less @ C3) @ Z3) @ X5) => (F3 = F3)))))))))))).
% 2.18/1.37 thf(fact_169_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((P2 @ X4) = (P4 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P2 @ X5) & (Q @ X5)) = ((P4 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.18/1.37 thf(fact_170_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((P2 @ X4) = (P4 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P2 @ X5) | (Q @ X5)) = ((P4 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.18/1.37 thf(fact_171_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.18/1.37 thf(fact_172_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.18/1.37 thf(fact_173_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less @ A) @ X5) @ T)))))))))))).
% 2.18/1.37 thf(fact_174_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less @ A) @ T) @ X5)))))))))))))).
% 2.18/1.37 thf(fact_175_minf_I11_J, axiom, (! [C3:$tType,D3:$tType]: ((((ord @ C3) @ (type2 @ C3)) => (! [F3:D3]: ((? [Z3:C3]: ((! [X5:C3]: (((((ord_less @ C3) @ X5) @ Z3) => (F3 = F3)))))))))))).
% 2.18/1.37 thf(fact_176_sorted__rev__nth__mono, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A),I3:nat,J:nat]: ((((linorder_sorted @ A) @ ((rev @ A) @ Xs)) => ((((ord_less_eq @ nat) @ I3) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ J)) @ (((nth @ A) @ Xs) @ I3))))))))))).
% 2.18/1.37 thf(fact_177_list__ex__length, axiom, (! [A:$tType]: (((list_ex @ A) = (^ [P5:(A > $o),Xs2:(list @ A)]: ((? [N2:nat]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs2)) & (P5 @ (((nth @ A) @ Xs2) @ N2))))))))))).
% 2.18/1.37 thf(fact_178_length__takeWhile__less__P__nth, axiom, (! [A:$tType,J:nat,P2:(A > $o),Xs:(list @ A)]: (((! [I:nat]: (((((ord_less @ nat) @ I) @ J) => (P2 @ (((nth @ A) @ Xs) @ I))))) => ((((ord_less_eq @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P2) @ Xs)))))))).
% 2.18/1.37 thf(fact_179_rev__rev__ident, axiom, (! [A:$tType,Xs:(list @ A)]: ((((rev @ A) @ ((rev @ A) @ Xs)) = Xs)))).
% 2.18/1.37 thf(fact_180_rev__is__rev__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((rev @ A) @ Xs) = ((rev @ A) @ Ys)) = (Xs = Ys))))).
% 2.18/1.37 thf(fact_181_takeWhile__idem, axiom, (! [A:$tType,P2:(A > $o),Xs:(list @ A)]: (((((takeWhile @ A) @ P2) @ (((takeWhile @ A) @ P2) @ Xs)) = (((takeWhile @ A) @ P2) @ Xs))))).
% 2.18/1.37 thf(fact_182_length__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((rev @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))).
% 2.18/1.37 thf(fact_183_list__ex__rev, axiom, (! [A:$tType,P2:(A > $o),Xs:(list @ A)]: (((((list_ex @ A) @ P2) @ ((rev @ A) @ Xs)) = (((list_ex @ A) @ P2) @ Xs))))).
% 2.18/1.37 thf(fact_184_sorted__takeWhile, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A),P2:(A > $o)]: ((((linorder_sorted @ A) @ Xs) => ((linorder_sorted @ A) @ (((takeWhile @ A) @ P2) @ Xs))))))))).
% 2.18/1.37 thf(fact_185_rev__swap, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((rev @ A) @ Xs) = Ys) = (Xs = ((rev @ A) @ Ys)))))).
% 2.18/1.37 thf(fact_186_length__takeWhile__le, axiom, (! [A:$tType,P2:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P2) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 2.18/1.37 thf(fact_187_takeWhile__nth, axiom, (! [A:$tType,J:nat,P2:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P2) @ Xs))) => ((((nth @ A) @ (((takeWhile @ A) @ P2) @ Xs)) @ J) = (((nth @ A) @ Xs) @ J)))))).
% 2.18/1.37 thf(fact_188_nth__length__takeWhile, axiom, (! [A:$tType,P2:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P2) @ Xs))) @ ((size_size @ (list @ A)) @ Xs)) => (~ ((P2 @ (((nth @ A) @ Xs) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P2) @ Xs)))))))))).
% 2.18/1.37 thf(fact_189_rev__nth, axiom, (! [A:$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)))))))).
% 2.18/1.37 thf(fact_190_count__le__length, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((ord_less_eq @ nat) @ (((count_list @ A) @ Xs) @ X3)) @ ((size_size @ (list @ A)) @ Xs))))).
% 2.18/1.37 thf(fact_191_GreatestM__nat__lemma, axiom, (! [A:$tType,P2:(A > $o),K:A,M:(A > nat),B3:nat]: (((P2 @ K) => ((! [Y4:A]: (((P2 @ Y4) => (((ord_less @ nat) @ (M @ Y4)) @ B3)))) => ((P2 @ ((((hilbert_GreatestM @ A) @ nat) @ M) @ P2)) & (! [Y2:A]: (((P2 @ Y2) => (((ord_less_eq @ nat) @ (M @ Y2)) @ (M @ ((((hilbert_GreatestM @ A) @ nat) @ M) @ P2)))))))))))).
% 2.18/1.37 thf(fact_192_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.18/1.37 thf(fact_193_nat_Oinject, axiom, (! [X22:nat,Y22:nat]: ((((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))).
% 2.18/1.37 thf(fact_194_Suc__less__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_195_Suc__mono, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ (suc @ M)) @ (suc @ N)))))).
% 2.18/1.37 thf(fact_196_lessI, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (suc @ N))))).
% 2.18/1.37 thf(fact_197_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 2.18/1.37 thf(fact_198_diff__Suc__Suc, axiom, (! [M:nat,N:nat]: (((((minus_minus @ nat) @ (suc @ M)) @ (suc @ N)) = (((minus_minus @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_199_Suc__diff__diff, axiom, (! [M:nat,N:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ (suc @ M)) @ N)) @ (suc @ K)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ K))))).
% 2.18/1.37 thf(fact_200_not__less__less__Suc__eq, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 2.18/1.37 thf(fact_201_strict__inc__induct, axiom, (! [I3:nat,J:nat,P2:(nat > $o)]: (((((ord_less @ nat) @ I3) @ J) => ((! [I:nat]: (((J = (suc @ I)) => (P2 @ I)))) => ((! [I:nat]: (((((ord_less @ nat) @ I) @ J) => ((P2 @ (suc @ I)) => (P2 @ I))))) => (P2 @ I3))))))).
% 2.18/1.37 thf(fact_202_less__Suc__induct, axiom, (! [I3:nat,J:nat,P2:(nat > (nat > $o))]: (((((ord_less @ nat) @ I3) @ J) => ((! [I:nat]: (((P2 @ I) @ (suc @ I)))) => ((! [I:nat,J2:nat,K2:nat]: (((((ord_less @ nat) @ I) @ J2) => ((((ord_less @ nat) @ J2) @ K2) => (((P2 @ I) @ J2) => (((P2 @ J2) @ K2) => ((P2 @ I) @ K2))))))) => ((P2 @ I3) @ J))))))).
% 2.18/1.37 thf(fact_203_less__trans__Suc, axiom, (! [I3:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I3) @ J) => ((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (suc @ I3)) @ K)))))).
% 2.18/1.37 thf(fact_204_Suc__less__SucD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) => (((ord_less @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_205_less__antisym, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) => (M = N)))))).
% 2.18/1.37 thf(fact_206_Suc__less__eq2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (suc @ N)) @ M) = (? [M4:nat]: (((M = (suc @ M4)) & (((ord_less @ nat) @ N) @ M4)))))))).
% 2.18/1.37 thf(fact_207_not__less__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) = (((ord_less @ nat) @ N) @ (suc @ M)))))).
% 2.18/1.37 thf(fact_208_less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 2.18/1.37 thf(fact_209_less__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 2.18/1.37 thf(fact_210_less__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less @ nat) @ M) @ N))) => (M = N)))))).
% 2.18/1.37 thf(fact_211_Suc__lessI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((suc @ M) != N) => (((ord_less @ nat) @ (suc @ M)) @ N)))))).
% 2.18/1.37 thf(fact_212_Suc__lessE, axiom, (! [I3:nat,K:nat]: (((((ord_less @ nat) @ (suc @ I3)) @ K) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I3) @ J2) => (K != (suc @ J2))))))))))).
% 2.18/1.37 thf(fact_213_Suc__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_214_lessE, axiom, (! [I3:nat,K:nat]: (((((ord_less @ nat) @ I3) @ K) => ((K != (suc @ I3)) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I3) @ J2) => (K != (suc @ J2)))))))))))).
% 2.18/1.37 thf(fact_215_lift__Suc__mono__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ 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)))))))))).
% 2.18/1.37 thf(fact_216_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N:nat,M:nat]: (((! [N3:nat]: ((((ord_less @ A) @ (F @ N3)) @ (F @ (suc @ N3))))) => ((((ord_less @ A) @ (F @ N)) @ (F @ M)) = (((ord_less @ nat) @ N) @ M))))))))).
% 2.18/1.37 thf(fact_217_lift__Suc__antimono__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ 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)))))))))).
% 2.18/1.37 thf(fact_218_lift__Suc__mono__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ 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)))))))))).
% 2.18/1.37 thf(fact_219_le__imp__less__Suc, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 2.18/1.37 thf(fact_220_less__eq__Suc__le, axiom, ((ord_less @ nat) = (^ [N2:nat]: (((ord_less_eq @ nat) @ (suc @ N2)))))).
% 2.18/1.37 thf(fact_221_less__Suc__eq__le, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_222_le__less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 2.18/1.37 thf(fact_223_Suc__le__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_224_inc__induct, axiom, (! [I3:nat,J:nat,P2:(nat > $o)]: (((((ord_less_eq @ nat) @ I3) @ J) => ((P2 @ J) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ I3) @ N3) => ((((ord_less @ nat) @ N3) @ J) => ((P2 @ (suc @ N3)) => (P2 @ N3)))))) => (P2 @ I3))))))).
% 2.18/1.37 thf(fact_225_dec__induct, axiom, (! [I3:nat,J:nat,P2:(nat > $o)]: (((((ord_less_eq @ nat) @ I3) @ J) => ((P2 @ I3) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ I3) @ N3) => ((((ord_less @ nat) @ N3) @ J) => ((P2 @ N3) => (P2 @ (suc @ N3))))))) => (P2 @ J))))))).
% 2.18/1.37 thf(fact_226_Suc__le__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) = (((ord_less @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_227_Suc__leI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (suc @ M)) @ N))))).
% 2.18/1.37 thf(fact_228_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.18/1.37 thf(fact_229_Suc__inject, axiom, (! [X3:nat,Y3:nat]: ((((suc @ X3) = (suc @ Y3)) => (X3 = Y3))))).
% 2.18/1.37 thf(fact_230_zero__induct__lemma, axiom, (! [P2:(nat > $o),K:nat,I3:nat]: (((P2 @ K) => ((! [N3:nat]: (((P2 @ (suc @ N3)) => (P2 @ N3)))) => (P2 @ (((minus_minus @ nat) @ K) @ I3))))))).
% 2.18/1.37 thf(fact_231_full__nat__induct, axiom, (! [P2:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N3) => (P2 @ M3)))) => (P2 @ N3)))) => (P2 @ N))))).
% 2.18/1.37 thf(fact_232_not__less__eq__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M))))).
% 2.18/1.37 thf(fact_233_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 2.18/1.37 thf(fact_234_le__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M) @ N) | (M = (suc @ N))))))).
% 2.18/1.37 thf(fact_235_Suc__le__D, axiom, (! [N:nat,M5:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M5) => (? [M6:nat]: ((M5 = (suc @ M6)))))))).
% 2.18/1.37 thf(fact_236_le__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ (suc @ N)))))).
% 2.18/1.37 thf(fact_237_le__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (M = (suc @ N))))))).
% 2.18/1.37 thf(fact_238_Suc__leD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.18/1.37 thf(fact_239_diff__less__Suc, axiom, (! [M:nat,N:nat]: ((((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ (suc @ M))))).
% 2.18/1.37 thf(fact_240_Suc__diff__Suc, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => ((suc @ (((minus_minus @ nat) @ M) @ (suc @ N))) = (((minus_minus @ nat) @ M) @ N)))))).
% 2.18/1.37 thf(fact_241_Suc__diff__le, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => ((((minus_minus @ nat) @ (suc @ M)) @ N) = (suc @ (((minus_minus @ nat) @ M) @ N))))))).
% 2.18/1.37 thf(fact_242_GreatestM__natI, axiom, (! [A:$tType,P2:(A > $o),K:A,M:(A > nat),B3:nat]: (((P2 @ K) => ((! [Y4:A]: (((P2 @ Y4) => (((ord_less @ nat) @ (M @ Y4)) @ B3)))) => (P2 @ ((((hilbert_GreatestM @ A) @ nat) @ M) @ P2))))))).
% 2.18/1.37 thf(fact_243_GreatestMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P2:(A > $o),X3:A,M:(A > B),Q:(A > $o)]: (((P2 @ X3) => ((! [Y4:A]: (((P2 @ Y4) => (((ord_less_eq @ B) @ (M @ Y4)) @ (M @ X3))))) => ((! [X4:A]: (((P2 @ X4) => ((! [Y2:A]: (((P2 @ Y2) => (((ord_less_eq @ B) @ (M @ Y2)) @ (M @ X4))))) => (Q @ X4))))) => (Q @ ((((hilbert_GreatestM @ A) @ B) @ M) @ P2))))))))))).
% 2.18/1.37 thf(fact_244_GreatestM__nat__le, axiom, (! [A:$tType,P2:(A > $o),X3:A,M:(A > nat),B3:nat]: (((P2 @ X3) => ((! [Y4:A]: (((P2 @ Y4) => (((ord_less @ nat) @ (M @ Y4)) @ B3)))) => (((ord_less_eq @ nat) @ (M @ X3)) @ (M @ ((((hilbert_GreatestM @ A) @ nat) @ M) @ P2)))))))).
% 2.18/1.37 thf(fact_245_prod__decode__aux_Oinduct, axiom, (! [P2:(nat > (nat > $o)),A0:nat,A1:nat]: (((! [K2:nat,M6:nat]: ((((~ ((((ord_less_eq @ nat) @ M6) @ K2))) => ((P2 @ (suc @ K2)) @ (((minus_minus @ nat) @ M6) @ (suc @ K2)))) => ((P2 @ K2) @ M6)))) => ((P2 @ A0) @ A1))))).
% 2.18/1.37 thf(fact_246_remdups__adj__adjacent, axiom, (! [A:$tType,I3:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ (suc @ I3)) @ ((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs))) => ((((nth @ A) @ ((remdups_adj @ A) @ Xs)) @ I3) != (((nth @ A) @ ((remdups_adj @ A) @ Xs)) @ (suc @ I3))))))).
% 2.18/1.37 thf(fact_247_remdups__adj__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((remdups_adj @ A) @ ((rev @ A) @ Xs)) = ((rev @ A) @ ((remdups_adj @ A) @ Xs)))))).
% 2.18/1.37 thf(fact_248_sorted__remdups__adj, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A)]: ((((linorder_sorted @ A) @ Xs) => ((linorder_sorted @ A) @ ((remdups_adj @ A) @ Xs))))))))).
% 2.18/1.37 thf(fact_249_remdups__adj__length, axiom, (! [A:$tType,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 2.18/1.37 thf(fact_250_nth__tl, axiom, (! [A:$tType,N:nat,X3:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ ((tl @ A) @ X3))) => ((((nth @ A) @ ((tl @ A) @ X3)) @ N) = (((nth @ A) @ X3) @ (suc @ N))))))).
% 2.18/1.37 thf(fact_251_Suc__pred, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((suc @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) = N))))).
% 2.18/1.37 thf(fact_252_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.18/1.37 thf(fact_253_not__gr__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.18/1.37 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.18/1.37 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.18/1.37 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.18/1.37 thf(tcon_fun___Groups_Ominus, axiom, (! [A6:$tType,A7:$tType]: ((((minus @ A7) @ (type2 @ A7)) => ((minus @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.18/1.37 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, ((cancel146912293up_add @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Groups_Ominus_4, axiom, ((minus @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_Nat_Onat___Groups_Ozero, axiom, ((zero @ nat) @ (type2 @ nat))).
% 2.18/1.37 thf(tcon_HOL_Obool___Orderings_Opreorder_5, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.18/1.37 thf(tcon_HOL_Obool___Orderings_Olinorder_6, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.18/1.37 thf(tcon_HOL_Obool___Orderings_Oorder_7, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.18/1.37 thf(tcon_HOL_Obool___Orderings_Oord_8, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.18/1.37 thf(tcon_HOL_Obool___Groups_Ominus_9, axiom, ((minus @ $o) @ (type2 @ $o))).
% 2.18/1.37 thf(conj_0, conjecture, (((((ord_less @ nat) @ n) @ ((size_size @ (list @ a)) @ xs)) => ((((stream370371455e_snth @ a) @ (((stream1035003186_shift @ a) @ xs) @ s)) @ n) = (((nth @ a) @ xs) @ n))) & ((~ ((((ord_less @ nat) @ n) @ ((size_size @ (list @ a)) @ xs)))) => ((((stream370371455e_snth @ a) @ (((stream1035003186_shift @ a) @ xs) @ s)) @ n) = (((stream370371455e_snth @ a) @ s) @ (((minus_minus @ nat) @ n) @ ((size_size @ (list @ a)) @ xs))))))).
% 2.18/1.37 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.gN7przDacx/DTF2DTF23201.p
% 2.18/1.37 ------------------------
% 2.40/1.41 ---- Cleaned THF ---
% 2.40/1.41 thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream,type,
% 2.40/1.41 stream170649215stream: $tType > $tType ).
% 2.40/1.41
% 2.40/1.41 thf(ty_t_List_Olist,type,
% 2.40/1.41 list: $tType > $tType ).
% 2.40/1.41
% 2.40/1.41 thf(ty_t_Nat_Onat,type,
% 2.40/1.41 nat: $tType ).
% 2.40/1.41
% 2.40/1.41 thf(ty_t_itself,type,
% 2.40/1.41 itself: $tType > $tType ).
% 2.40/1.41
% 2.40/1.41 thf(ty_tf_a,type,
% 2.40/1.41 a: $tType ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_HOL_Otype,type,
% 2.40/1.41 type:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Groups_Ozero,type,
% 2.40/1.41 zero:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Groups_Ominus,type,
% 2.40/1.41 minus:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Oord,type,
% 2.40/1.41 ord:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Oorder,type,
% 2.40/1.41 order:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Groups_Ogroup__add,type,
% 2.40/1.41 group_add:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Ono__bot,type,
% 2.40/1.41 no_bot:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Ono__top,type,
% 2.40/1.41 no_top:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Olinorder,type,
% 2.40/1.41 linorder:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Opreorder,type,
% 2.40/1.41 preorder:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Owellorder,type,
% 2.40/1.41 wellorder:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Odense__order,type,
% 2.40/1.41 dense_order:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.40/1.41 dense_linorder:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.40/1.41 ordered_ab_group_add:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.40/1.41 cancel146912293up_add:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.40/1.41 canoni770627133id_add:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.40/1.41 condit1656338222tinuum:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.40/1.41 condit1037483654norder:
% 2.40/1.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.40/1.41 minus_minus:
% 2.40/1.41 !>[A: $tType] : ( A > A > A ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.40/1.41 zero_zero:
% 2.40/1.41 !>[A: $tType] : A ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 2.40/1.41 hilbert_GreatestM:
% 2.40/1.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_List_Ocount__list,type,
% 2.40/1.41 count_list:
% 2.40/1.41 !>[A: $tType] : ( ( list @ A ) > A > nat ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_List_Olinorder__class_Osorted,type,
% 2.40/1.41 linorder_sorted:
% 2.40/1.41 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_List_Olist_Otl,type,
% 2.40/1.41 tl:
% 2.40/1.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_List_Olist__ex,type,
% 2.40/1.41 list_ex:
% 2.40/1.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_List_Onth,type,
% 2.40/1.41 nth:
% 2.40/1.41 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_List_Oremdups__adj,type,
% 2.40/1.41 remdups_adj:
% 2.40/1.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_List_Orev,type,
% 2.40/1.41 rev:
% 2.40/1.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_List_OtakeWhile,type,
% 2.40/1.41 takeWhile:
% 2.40/1.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Nat_OSuc,type,
% 2.40/1.41 suc: nat > nat ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.40/1.41 size_size:
% 2.40/1.41 !>[A: $tType] : ( A > nat ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.40/1.41 ord_less:
% 2.40/1.41 !>[A: $tType] : ( A > A > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.40/1.41 ord_less_eq:
% 2.40/1.41 !>[A: $tType] : ( A > A > $o ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Pure_Otype,type,
% 2.40/1.41 type2:
% 2.40/1.41 !>[A: $tType] : ( itself @ A ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Oshift,type,
% 2.40/1.41 stream1035003186_shift:
% 2.40/1.41 !>[A: $tType] : ( ( list @ A ) > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osnth,type,
% 2.40/1.41 stream370371455e_snth:
% 2.40/1.41 !>[A: $tType] : ( ( stream170649215stream @ A ) > nat > A ) ).
% 2.40/1.41
% 2.40/1.41 thf(sy_v_n,type,
% 2.40/1.41 n: nat ).
% 2.40/1.41
% 2.40/1.41 thf(sy_v_s,type,
% 2.40/1.41 s: stream170649215stream @ a ).
% 2.40/1.41
% 2.40/1.41 thf(sy_v_xs,type,
% 2.40/1.41 xs: list @ a ).
% 2.40/1.41
% 2.40/1.41 thf(fact_0_shift__left__inj,axiom,
% 2.40/1.41 ! [A: $tType,Xs: list @ A,S1: stream170649215stream @ A,S2: stream170649215stream @ A] :
% 2.40/1.41 ( ( ( stream1035003186_shift @ A @ Xs @ S1 )
% 2.40/1.41 = ( stream1035003186_shift @ A @ Xs @ S2 ) )
% 2.40/1.41 = ( S1 = S2 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_1_shift__snth__less,axiom,
% 2.40/1.41 ! [A: $tType,P: nat,Xs: list @ A,S: stream170649215stream @ A] :
% 2.40/1.41 ( ( ord_less @ nat @ P @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.40/1.41 => ( ( stream370371455e_snth @ A @ ( stream1035003186_shift @ A @ Xs @ S ) @ P )
% 2.40/1.41 = ( nth @ A @ Xs @ P ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_2_nth__equalityI,axiom,
% 2.40/1.41 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.40/1.41 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.40/1.41 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.40/1.41 => ( ! [I: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.40/1.41 => ( ( nth @ A @ Xs @ I )
% 2.40/1.41 = ( nth @ A @ Ys @ I ) ) )
% 2.40/1.41 => ( Xs = Ys ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_3_Skolem__list__nth,axiom,
% 2.40/1.41 ! [A: $tType,K: nat,P2: nat > A > $o] :
% 2.40/1.41 ( ( ! [I2: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ I2 @ K )
% 2.40/1.41 => ( ^ [P3: A > $o] :
% 2.40/1.41 ? [X: A] : ( P3 @ X )
% 2.40/1.41 @ ( P2 @ I2 ) ) ) )
% 2.40/1.41 = ( ? [Xs2: list @ A] :
% 2.40/1.41 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.40/1.41 = K )
% 2.40/1.41 & ! [I2: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ I2 @ K )
% 2.40/1.41 => ( P2 @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_4_list__eq__iff__nth__eq,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ^ [Y: list @ A,Z: list @ A] : ( Y = Z ) )
% 2.40/1.41 = ( ^ [Xs2: list @ A,Ys2: list @ A] :
% 2.40/1.41 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.40/1.41 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 2.40/1.41 & ! [I2: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.40/1.41 => ( ( nth @ A @ Xs2 @ I2 )
% 2.40/1.41 = ( nth @ A @ Ys2 @ I2 ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_5_minus__apply,axiom,
% 2.40/1.41 ! [B: $tType,A: $tType] :
% 2.40/1.41 ( ( minus @ B @ ( type2 @ B ) )
% 2.40/1.41 => ( ( minus_minus @ ( A > B ) )
% 2.40/1.41 = ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_6_diff__less__mono2,axiom,
% 2.40/1.41 ! [M: nat,N: nat,L: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ M @ N )
% 2.40/1.41 => ( ( ord_less @ nat @ M @ L )
% 2.40/1.41 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_7_less__imp__diff__less,axiom,
% 2.40/1.41 ! [J: nat,K: nat,N: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ J @ K )
% 2.40/1.41 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_8_length__induct,axiom,
% 2.40/1.41 ! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
% 2.40/1.41 ( ! [Xs3: list @ A] :
% 2.40/1.41 ( ! [Ys3: list @ A] :
% 2.40/1.41 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.40/1.41 => ( P2 @ Ys3 ) )
% 2.40/1.41 => ( P2 @ Xs3 ) )
% 2.40/1.41 => ( P2 @ Xs ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_9_diff__strict__mono,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,D: A,C: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less @ A @ D @ C )
% 2.40/1.41 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_10_diff__eq__diff__less,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A,D: A] :
% 2.40/1.41 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.40/1.41 = ( minus_minus @ A @ C @ D ) )
% 2.40/1.41 => ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_11_diff__strict__left__mono,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A,C: A] :
% 2.40/1.41 ( ( ord_less @ A @ B3 @ A3 )
% 2.40/1.41 => ( ord_less @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_12_diff__strict__right__mono,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_13_shift__snth__ge,axiom,
% 2.40/1.41 ! [A: $tType,Xs: list @ A,P: nat,S: stream170649215stream @ A] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ P )
% 2.40/1.41 => ( ( stream370371455e_snth @ A @ ( stream1035003186_shift @ A @ Xs @ S ) @ P )
% 2.40/1.41 = ( stream370371455e_snth @ A @ S @ ( minus_minus @ nat @ P @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_14_diff__diff__cancel,axiom,
% 2.40/1.41 ! [I3: nat,N: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ I3 @ N )
% 2.40/1.41 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I3 ) )
% 2.40/1.41 = I3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_15_le__refl,axiom,
% 2.40/1.41 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_16_le__trans,axiom,
% 2.40/1.41 ! [I3: nat,J: nat,K: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ I3 @ J )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ J @ K )
% 2.40/1.41 => ( ord_less_eq @ nat @ I3 @ K ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_17_eq__imp__le,axiom,
% 2.40/1.41 ! [M: nat,N: nat] :
% 2.40/1.41 ( ( M = N )
% 2.40/1.41 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_18_le__antisym,axiom,
% 2.40/1.41 ! [M: nat,N: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ N @ M )
% 2.40/1.41 => ( M = N ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_19_nat__le__linear,axiom,
% 2.40/1.41 ! [M: nat,N: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.41 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_20_diff__eq__diff__less__eq,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A,D: A] :
% 2.40/1.41 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.40/1.41 = ( minus_minus @ A @ C @ D ) )
% 2.40/1.41 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_21_diff__right__mono,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_22_diff__left__mono,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A,C: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.40/1.41 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_23_diff__mono,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,D: A,C: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ D @ C )
% 2.40/1.41 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_24_less__mono__imp__le__mono,axiom,
% 2.40/1.41 ! [F: nat > nat,I3: nat,J: nat] :
% 2.40/1.41 ( ! [I: nat,J2: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ I @ J2 )
% 2.40/1.41 => ( ord_less @ nat @ ( F @ I ) @ ( F @ J2 ) ) )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ I3 @ J )
% 2.40/1.41 => ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_25_le__neq__implies__less,axiom,
% 2.40/1.41 ! [M: nat,N: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.41 => ( ( M != N )
% 2.40/1.41 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_26_less__or__eq__imp__le,axiom,
% 2.40/1.41 ! [M: nat,N: nat] :
% 2.40/1.41 ( ( ( ord_less @ nat @ M @ N )
% 2.40/1.41 | ( M = N ) )
% 2.40/1.41 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_27_le__eq__less__or__eq,axiom,
% 2.40/1.41 ( ( ord_less_eq @ nat )
% 2.40/1.41 = ( ^ [M2: nat,N2: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ M2 @ N2 )
% 2.40/1.41 | ( M2 = N2 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_28_less__imp__le__nat,axiom,
% 2.40/1.41 ! [M: nat,N: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ M @ N )
% 2.40/1.41 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_29_nat__less__le,axiom,
% 2.40/1.41 ( ( ord_less @ nat )
% 2.40/1.41 = ( ^ [M2: nat,N2: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.40/1.41 & ( M2 != N2 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_30_diff__le__mono2,axiom,
% 2.40/1.41 ! [M: nat,N: nat,L: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.41 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_31_le__diff__iff_H,axiom,
% 2.40/1.41 ! [A3: nat,C: nat,B3: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ A3 @ C )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ B3 @ C )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A3 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 2.40/1.41 = ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_32_diff__le__self,axiom,
% 2.40/1.41 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_33_diff__le__mono,axiom,
% 2.40/1.41 ! [M: nat,N: nat,L: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.41 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_34_Nat_Odiff__diff__eq,axiom,
% 2.40/1.41 ! [K: nat,M: nat,N: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ K @ M )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ K @ N )
% 2.40/1.41 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.40/1.41 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_35_le__diff__iff,axiom,
% 2.40/1.41 ! [K: nat,M: nat,N: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ K @ M )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ K @ N )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.40/1.41 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_36_eq__diff__iff,axiom,
% 2.40/1.41 ! [K: nat,M: nat,N: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ K @ M )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ K @ N )
% 2.40/1.41 => ( ( ( minus_minus @ nat @ M @ K )
% 2.40/1.41 = ( minus_minus @ nat @ N @ K ) )
% 2.40/1.41 = ( M = N ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_37_diff__less__mono,axiom,
% 2.40/1.41 ! [A3: nat,B3: nat,C: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ C @ A3 )
% 2.40/1.41 => ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_38_less__diff__iff,axiom,
% 2.40/1.41 ! [K: nat,M: nat,N: nat] :
% 2.40/1.41 ( ( ord_less_eq @ nat @ K @ M )
% 2.40/1.41 => ( ( ord_less_eq @ nat @ K @ N )
% 2.40/1.41 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.40/1.41 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_39_diff__right__commute,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,C: A,B3: A] :
% 2.40/1.41 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B3 )
% 2.40/1.41 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_40_diff__eq__diff__eq,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( group_add @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A,D: A] :
% 2.40/1.41 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.40/1.41 = ( minus_minus @ A @ C @ D ) )
% 2.40/1.41 => ( ( A3 = B3 )
% 2.40/1.41 = ( C = D ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_41_fun__diff__def,axiom,
% 2.40/1.41 ! [B: $tType,A: $tType] :
% 2.40/1.41 ( ( minus @ B @ ( type2 @ B ) )
% 2.40/1.41 => ( ( minus_minus @ ( A > B ) )
% 2.40/1.41 = ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_42_infinite__descent__measure,axiom,
% 2.40/1.41 ! [A: $tType,P2: A > $o,V: A > nat,X3: A] :
% 2.40/1.41 ( ! [X4: A] :
% 2.40/1.41 ( ~ ( P2 @ X4 )
% 2.40/1.41 => ? [Y2: A] :
% 2.40/1.41 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X4 ) )
% 2.40/1.41 & ~ ( P2 @ Y2 ) ) )
% 2.40/1.41 => ( P2 @ X3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_43_measure__induct__rule,axiom,
% 2.40/1.41 ! [A: $tType,F: A > nat,P2: A > $o,A3: A] :
% 2.40/1.41 ( ! [X4: A] :
% 2.40/1.41 ( ! [Y2: A] :
% 2.40/1.41 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X4 ) )
% 2.40/1.41 => ( P2 @ Y2 ) )
% 2.40/1.41 => ( P2 @ X4 ) )
% 2.40/1.41 => ( P2 @ A3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_44_linorder__neqE__nat,axiom,
% 2.40/1.41 ! [X3: nat,Y3: nat] :
% 2.40/1.41 ( ( X3 != Y3 )
% 2.40/1.41 => ( ~ ( ord_less @ nat @ X3 @ Y3 )
% 2.40/1.41 => ( ord_less @ nat @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_45_ext,axiom,
% 2.40/1.41 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.40/1.41 ( ! [X4: A] :
% 2.40/1.41 ( ( F @ X4 )
% 2.40/1.41 = ( G @ X4 ) )
% 2.40/1.41 => ( F = G ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_46_infinite__descent,axiom,
% 2.40/1.41 ! [P2: nat > $o,N: nat] :
% 2.40/1.41 ( ! [N3: nat] :
% 2.40/1.41 ( ~ ( P2 @ N3 )
% 2.40/1.41 => ? [M3: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ M3 @ N3 )
% 2.40/1.41 & ~ ( P2 @ M3 ) ) )
% 2.40/1.41 => ( P2 @ N ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_47_nat__less__induct,axiom,
% 2.40/1.41 ! [P2: nat > $o,N: nat] :
% 2.40/1.41 ( ! [N3: nat] :
% 2.40/1.41 ( ! [M3: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ M3 @ N3 )
% 2.40/1.41 => ( P2 @ M3 ) )
% 2.40/1.41 => ( P2 @ N3 ) )
% 2.40/1.41 => ( P2 @ N ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_48_less__irrefl__nat,axiom,
% 2.40/1.41 ! [N: nat] :
% 2.40/1.41 ~ ( ord_less @ nat @ N @ N ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_49_measure__induct,axiom,
% 2.40/1.41 ! [A: $tType,F: A > nat,P2: A > $o,A3: A] :
% 2.40/1.41 ( ! [X4: A] :
% 2.40/1.41 ( ! [Y2: A] :
% 2.40/1.41 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X4 ) )
% 2.40/1.41 => ( P2 @ Y2 ) )
% 2.40/1.41 => ( P2 @ X4 ) )
% 2.40/1.41 => ( P2 @ A3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_50_less__not__refl3,axiom,
% 2.40/1.41 ! [S: nat,T: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ S @ T )
% 2.40/1.41 => ( S != T ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_51_less__not__refl2,axiom,
% 2.40/1.41 ! [N: nat,M: nat] :
% 2.40/1.41 ( ( ord_less @ nat @ N @ M )
% 2.40/1.41 => ( M != N ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_52_less__not__refl,axiom,
% 2.40/1.41 ! [N: nat] :
% 2.40/1.41 ~ ( ord_less @ nat @ N @ N ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_53_nat__neq__iff,axiom,
% 2.40/1.41 ! [M: nat,N: nat] :
% 2.40/1.41 ( ( M != N )
% 2.40/1.41 = ( ( ord_less @ nat @ M @ N )
% 2.40/1.41 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_54_neq__if__length__neq,axiom,
% 2.40/1.41 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.40/1.41 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.40/1.41 != ( size_size @ ( list @ A ) @ Ys ) )
% 2.40/1.41 => ( Xs != Ys ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_55_Ex__list__of__length,axiom,
% 2.40/1.41 ! [A: $tType,N: nat] :
% 2.40/1.41 ? [Xs3: list @ A] :
% 2.40/1.41 ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.40/1.41 = N ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_56_diff__commute,axiom,
% 2.40/1.41 ! [I3: nat,J: nat,K: nat] :
% 2.40/1.41 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J ) @ K )
% 2.40/1.41 = ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ K ) @ J ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_57_order__refl,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_58_ex__has__greatest__nat,axiom,
% 2.40/1.41 ! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
% 2.40/1.41 ( ( P2 @ K )
% 2.40/1.41 => ( ! [Y4: A] :
% 2.40/1.41 ( ( P2 @ Y4 )
% 2.40/1.41 => ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
% 2.40/1.41 => ? [X4: A] :
% 2.40/1.41 ( ( P2 @ X4 )
% 2.40/1.41 & ! [Y2: A] :
% 2.40/1.41 ( ( P2 @ Y2 )
% 2.40/1.41 => ( ord_less_eq @ nat @ ( M @ Y2 ) @ ( M @ X4 ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_59_complete__interval,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,P2: A > $o] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( P2 @ A3 )
% 2.40/1.41 => ( ~ ( P2 @ B3 )
% 2.40/1.41 => ? [C2: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ C2 )
% 2.40/1.41 & ( ord_less_eq @ A @ C2 @ B3 )
% 2.40/1.41 & ! [X5: A] :
% 2.40/1.41 ( ( ( ord_less_eq @ A @ A3 @ X5 )
% 2.40/1.41 & ( ord_less @ A @ X5 @ C2 ) )
% 2.40/1.41 => ( P2 @ X5 ) )
% 2.40/1.41 & ! [D2: A] :
% 2.40/1.41 ( ! [X4: A] :
% 2.40/1.41 ( ( ( ord_less_eq @ A @ A3 @ X4 )
% 2.40/1.41 & ( ord_less @ A @ X4 @ D2 ) )
% 2.40/1.41 => ( P2 @ X4 ) )
% 2.40/1.41 => ( ord_less_eq @ A @ D2 @ C2 ) ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_60_order_Onot__eq__order__implies__strict,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A] :
% 2.40/1.41 ( ( A3 != B3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_61_dual__order_Ostrict__implies__order,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A] :
% 2.40/1.41 ( ( ord_less @ A @ B3 @ A3 )
% 2.40/1.41 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_62_dual__order_Ostrict__iff__order,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ( ( ord_less @ A )
% 2.40/1.41 = ( ^ [B4: A,A4: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.40/1.41 & ( A4 != B4 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_63_dual__order_Oorder__iff__strict,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ( ( ord_less_eq @ A )
% 2.40/1.41 = ( ^ [B4: A,A4: A] :
% 2.40/1.41 ( ( ord_less @ A @ B4 @ A4 )
% 2.40/1.41 | ( A4 = B4 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_64_order_Ostrict__implies__order,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_65_dense__le__bounded,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A,Z2: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ! [W: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ W )
% 2.40/1.41 => ( ( ord_less @ A @ W @ Y3 )
% 2.40/1.41 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.40/1.41 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_66_dense__ge__bounded,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [Z2: A,X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ Z2 @ X3 )
% 2.40/1.41 => ( ! [W: A] :
% 2.40/1.41 ( ( ord_less @ A @ Z2 @ W )
% 2.40/1.41 => ( ( ord_less @ A @ W @ X3 )
% 2.40/1.41 => ( ord_less_eq @ A @ Y3 @ W ) ) )
% 2.40/1.41 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_67_dual__order_Ostrict__trans2,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A,C: A] :
% 2.40/1.41 ( ( ord_less @ A @ B3 @ A3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.40/1.41 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_68_dual__order_Oantisym,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( A3 = B3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_69_dual__order_Otrans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A,C: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.40/1.41 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_70_linorder__wlog,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [P2: A > A > $o,A3: A,B3: A] :
% 2.40/1.41 ( ! [A5: A,B5: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.40/1.41 => ( P2 @ A5 @ B5 ) )
% 2.40/1.41 => ( ! [A5: A,B5: A] :
% 2.40/1.41 ( ( P2 @ B5 @ A5 )
% 2.40/1.41 => ( P2 @ A5 @ B5 ) )
% 2.40/1.41 => ( P2 @ A3 @ B3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_71_dual__order_Orefl,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_72_order__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A,Z2: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.40/1.41 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_73_order__class_Oorder_Oantisym,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.40/1.41 => ( A3 = B3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_74_ord__le__eq__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ord @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( B3 = C )
% 2.40/1.41 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_75_ord__eq__le__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ord @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A] :
% 2.40/1.41 ( ( A3 = B3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.40/1.41 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_76_antisym__conv,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [Y3: A,X3: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 = ( X3 = Y3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_77_le__cases3,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A,Z2: A] :
% 2.40/1.41 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 2.40/1.41 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.40/1.41 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.40/1.41 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 2.40/1.41 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 2.40/1.41 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 2.40/1.41 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 2.40/1.41 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.40/1.41 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 2.40/1.41 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 2.40/1.41 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_78_order_Otrans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.40/1.41 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_79_le__cases,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_80_eq__refl,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( X3 = Y3 )
% 2.40/1.41 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_81_linear,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_82_antisym,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.40/1.41 => ( X3 = Y3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_83_eq__iff,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.40/1.41 = ( ^ [X2: A,Y5: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 2.40/1.41 & ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_84_ord__le__eq__subst,axiom,
% 2.40/1.41 ! [A: $tType,B: $tType] :
% 2.40/1.41 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.41 & ( ord @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,B3: A,F: A > B,C: B] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ( F @ B3 )
% 2.40/1.41 = C )
% 2.40/1.41 => ( ! [X4: A,Y4: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_85_ord__eq__le__subst,axiom,
% 2.40/1.41 ! [A: $tType,B: $tType] :
% 2.40/1.41 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.41 & ( ord @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.40/1.41 ( ( A3
% 2.40/1.41 = ( F @ B3 ) )
% 2.40/1.41 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.40/1.41 => ( ! [X4: B,Y4: B] :
% 2.40/1.41 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_86_order__subst2,axiom,
% 2.40/1.41 ! [A: $tType,C3: $tType] :
% 2.40/1.41 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.40/1.41 & ( order @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
% 2.40/1.41 => ( ! [X4: A,Y4: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less_eq @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_87_order__subst1,axiom,
% 2.40/1.41 ! [A: $tType,B: $tType] :
% 2.40/1.41 ( ( ( order @ B @ ( type2 @ B ) )
% 2.40/1.41 & ( order @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.40/1.41 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.40/1.41 => ( ! [X4: B,Y4: B] :
% 2.40/1.41 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_88_le__fun__def,axiom,
% 2.40/1.41 ! [B: $tType,A: $tType] :
% 2.40/1.41 ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.41 => ( ( ord_less_eq @ ( A > B ) )
% 2.40/1.41 = ( ^ [F2: A > B,G2: A > B] :
% 2.40/1.41 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_89_le__funI,axiom,
% 2.40/1.41 ! [B: $tType,A: $tType] :
% 2.40/1.41 ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.41 => ! [F: A > B,G: A > B] :
% 2.40/1.41 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.40/1.41 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_90_le__funE,axiom,
% 2.40/1.41 ! [B: $tType,A: $tType] :
% 2.40/1.41 ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.41 => ! [F: A > B,G: A > B,X3: A] :
% 2.40/1.41 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.40/1.41 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_91_le__funD,axiom,
% 2.40/1.41 ! [B: $tType,A: $tType] :
% 2.40/1.41 ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.41 => ! [F: A > B,G: A > B,X3: A] :
% 2.40/1.41 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.40/1.41 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_92_ord__eq__less__subst,axiom,
% 2.40/1.41 ! [A: $tType,B: $tType] :
% 2.40/1.41 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.41 & ( ord @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.40/1.41 ( ( A3
% 2.40/1.41 = ( F @ B3 ) )
% 2.40/1.41 => ( ( ord_less @ B @ B3 @ C )
% 2.40/1.41 => ( ! [X4: B,Y4: B] :
% 2.40/1.41 ( ( ord_less @ B @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_93_ord__less__eq__subst,axiom,
% 2.40/1.41 ! [A: $tType,B: $tType] :
% 2.40/1.41 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.41 & ( ord @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,B3: A,F: A > B,C: B] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ( F @ B3 )
% 2.40/1.41 = C )
% 2.40/1.41 => ( ! [X4: A,Y4: A] :
% 2.40/1.41 ( ( ord_less @ A @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_94_order__less__subst1,axiom,
% 2.40/1.41 ! [A: $tType,B: $tType] :
% 2.40/1.41 ( ( ( order @ B @ ( type2 @ B ) )
% 2.40/1.41 & ( order @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 2.40/1.41 => ( ( ord_less @ B @ B3 @ C )
% 2.40/1.41 => ( ! [X4: B,Y4: B] :
% 2.40/1.41 ( ( ord_less @ B @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_95_order__less__subst2,axiom,
% 2.40/1.41 ! [A: $tType,C3: $tType] :
% 2.40/1.41 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.40/1.41 & ( order @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less @ C3 @ ( F @ B3 ) @ C )
% 2.40/1.41 => ( ! [X4: A,Y4: A] :
% 2.40/1.41 ( ( ord_less @ A @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_96_lt__ex,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A] :
% 2.40/1.41 ? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_97_gt__ex,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( no_top @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A] :
% 2.40/1.41 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_98_neqE,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( X3 != Y3 )
% 2.40/1.41 => ( ~ ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_99_neq__iff,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( X3 != Y3 )
% 2.40/1.41 = ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 | ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_100_order_Oasym,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_101_dense,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ? [Z3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Z3 )
% 2.40/1.41 & ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_102_less__imp__neq,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( X3 != Y3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_103_less__asym,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_104_less__asym_H,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_105_less__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A,Z2: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( ord_less @ A @ Y3 @ Z2 )
% 2.40/1.41 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_106_less__linear,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 | ( X3 = Y3 )
% 2.40/1.41 | ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_107_less__irrefl,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A] :
% 2.40/1.41 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_108_ord__eq__less__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ord @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A] :
% 2.40/1.41 ( ( A3 = B3 )
% 2.40/1.41 => ( ( ord_less @ A @ B3 @ C )
% 2.40/1.41 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_109_ord__less__eq__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( ord @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( B3 = C )
% 2.40/1.41 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_110_dual__order_Oasym,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A] :
% 2.40/1.41 ( ( ord_less @ A @ B3 @ A3 )
% 2.40/1.41 => ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_111_less__imp__not__eq,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( X3 != Y3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_112_less__not__sym,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_113_less__induct,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [P2: A > $o,A3: A] :
% 2.40/1.41 ( ! [X4: A] :
% 2.40/1.41 ( ! [Y2: A] :
% 2.40/1.41 ( ( ord_less @ A @ Y2 @ X4 )
% 2.40/1.41 => ( P2 @ Y2 ) )
% 2.40/1.41 => ( P2 @ X4 ) )
% 2.40/1.41 => ( P2 @ A3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_114_antisym__conv3,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [Y3: A,X3: A] :
% 2.40/1.41 ( ~ ( ord_less @ A @ Y3 @ X3 )
% 2.40/1.41 => ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 2.40/1.41 = ( X3 = Y3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_115_less__imp__not__eq2,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( Y3 != X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_116_less__imp__triv,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A,P2: $o] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( ord_less @ A @ Y3 @ X3 )
% 2.40/1.41 => P2 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_117_linorder__cases,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( X3 != Y3 )
% 2.40/1.41 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_118_dual__order_Oirrefl,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A] :
% 2.40/1.41 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_119_order_Ostrict__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A,C: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less @ A @ B3 @ C )
% 2.40/1.41 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_120_less__imp__not__less,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_121_dual__order_Ostrict__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A,C: A] :
% 2.40/1.41 ( ( ord_less @ A @ B3 @ A3 )
% 2.40/1.41 => ( ( ord_less @ A @ C @ B3 )
% 2.40/1.41 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_122_not__less__iff__gr__or__eq,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 2.40/1.41 = ( ( ord_less @ A @ Y3 @ X3 )
% 2.40/1.41 | ( X3 = Y3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_123_order_Ostrict__implies__not__eq,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( A3 != B3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_124_dual__order_Ostrict__implies__not__eq,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [B3: A,A3: A] :
% 2.40/1.41 ( ( ord_less @ A @ B3 @ A3 )
% 2.40/1.41 => ( A3 != B3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_125_ex__gt__or__lt,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A] :
% 2.40/1.41 ? [B5: A] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B5 )
% 2.40/1.41 | ( ord_less @ A @ B5 @ A3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_126_ex__has__least__nat,axiom,
% 2.40/1.41 ! [A: $tType,P2: A > $o,K: A,M: A > nat] :
% 2.40/1.41 ( ( P2 @ K )
% 2.40/1.41 => ? [X4: A] :
% 2.40/1.41 ( ( P2 @ X4 )
% 2.40/1.41 & ! [Y2: A] :
% 2.40/1.41 ( ( P2 @ Y2 )
% 2.40/1.41 => ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y2 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_127_leD,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [Y3: A,X3: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.40/1.41 => ~ ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_128_leI,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_129_le__less,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ( ( ord_less_eq @ A )
% 2.40/1.41 = ( ^ [X2: A,Y5: A] :
% 2.40/1.41 ( ( ord_less @ A @ X2 @ Y5 )
% 2.40/1.41 | ( X2 = Y5 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_130_less__le,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ( ( ord_less @ A )
% 2.40/1.41 = ( ^ [X2: A,Y5: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 2.40/1.41 & ( X2 != Y5 ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_131_order__le__less__subst1,axiom,
% 2.40/1.41 ! [A: $tType,B: $tType] :
% 2.40/1.41 ( ( ( order @ B @ ( type2 @ B ) )
% 2.40/1.41 & ( order @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.40/1.41 => ( ( ord_less @ B @ B3 @ C )
% 2.40/1.41 => ( ! [X4: B,Y4: B] :
% 2.40/1.41 ( ( ord_less @ B @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_132_order__le__less__subst2,axiom,
% 2.40/1.41 ! [A: $tType,C3: $tType] :
% 2.40/1.41 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.40/1.41 & ( order @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less @ C3 @ ( F @ B3 ) @ C )
% 2.40/1.41 => ( ! [X4: A,Y4: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_133_order__less__le__subst1,axiom,
% 2.40/1.41 ! [A: $tType,B: $tType] :
% 2.40/1.41 ( ( ( order @ B @ ( type2 @ B ) )
% 2.40/1.41 & ( order @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 2.40/1.41 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.40/1.41 => ( ! [X4: B,Y4: B] :
% 2.40/1.41 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_134_order__less__le__subst2,axiom,
% 2.40/1.41 ! [A: $tType,C3: $tType] :
% 2.40/1.41 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.40/1.41 & ( order @ A @ ( type2 @ A ) ) )
% 2.40/1.41 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 2.40/1.41 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
% 2.40/1.41 => ( ! [X4: A,Y4: A] :
% 2.40/1.41 ( ( ord_less @ A @ X4 @ Y4 )
% 2.40/1.41 => ( ord_less @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.40/1.41 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_135_not__le,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ~ ( ord_less_eq @ A @ X3 @ Y3 ) )
% 2.40/1.41 = ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_136_not__less,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 2.40/1.41 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_137_le__neq__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [A3: A,B3: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.41 => ( ( A3 != B3 )
% 2.40/1.41 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_138_less__imp__le,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_139_antisym__conv1,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 = ( X3 = Y3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_140_antisym__conv2,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 2.40/1.41 = ( X3 = Y3 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_141_le__less__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A,Z2: A] :
% 2.40/1.41 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( ord_less @ A @ Y3 @ Z2 )
% 2.40/1.41 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_142_less__le__trans,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [X3: A,Y3: A,Z2: A] :
% 2.40/1.41 ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.41 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.40/1.41 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_143_dense__ge,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [Z2: A,Y3: A] :
% 2.40/1.41 ( ! [X4: A] :
% 2.40/1.41 ( ( ord_less @ A @ Z2 @ X4 )
% 2.40/1.41 => ( ord_less_eq @ A @ Y3 @ X4 ) )
% 2.40/1.41 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% 2.40/1.41
% 2.40/1.41 thf(fact_144_dense__le,axiom,
% 2.40/1.41 ! [A: $tType] :
% 2.40/1.41 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.40/1.41 => ! [Y3: A,Z2: A] :
% 2.40/1.41 ( ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ X4 @ Y3 )
% 2.40/1.42 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.40/1.42 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_145_le__less__linear,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [X3: A,Y3: A] :
% 2.40/1.42 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.42 | ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_146_le__imp__less__or__eq,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [X3: A,Y3: A] :
% 2.40/1.42 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.40/1.42 => ( ( ord_less @ A @ X3 @ Y3 )
% 2.40/1.42 | ( X3 = Y3 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_147_less__le__not__le,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( preorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ( ( ord_less @ A )
% 2.40/1.42 = ( ^ [X2: A,Y5: A] :
% 2.40/1.42 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 2.40/1.42 & ~ ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_148_not__le__imp__less,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [Y3: A,X3: A] :
% 2.40/1.42 ( ~ ( ord_less_eq @ A @ Y3 @ X3 )
% 2.40/1.42 => ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_149_order_Ostrict__trans1,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [A3: A,B3: A,C: A] :
% 2.40/1.42 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.40/1.42 => ( ( ord_less @ A @ B3 @ C )
% 2.40/1.42 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_150_order_Ostrict__trans2,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [A3: A,B3: A,C: A] :
% 2.40/1.42 ( ( ord_less @ A @ A3 @ B3 )
% 2.40/1.42 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.40/1.42 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_151_order_Oorder__iff__strict,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ( ( ord_less_eq @ A )
% 2.40/1.42 = ( ^ [A4: A,B4: A] :
% 2.40/1.42 ( ( ord_less @ A @ A4 @ B4 )
% 2.40/1.42 | ( A4 = B4 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_152_order_Ostrict__iff__order,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ( ( ord_less @ A )
% 2.40/1.42 = ( ^ [A4: A,B4: A] :
% 2.40/1.42 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.40/1.42 & ( A4 != B4 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_153_dual__order_Ostrict__trans1,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [B3: A,A3: A,C: A] :
% 2.40/1.42 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.40/1.42 => ( ( ord_less @ A @ C @ B3 )
% 2.40/1.42 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_154_minf_I8_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ X5 @ Z3 )
% 2.40/1.42 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_155_minf_I6_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ X5 @ Z3 )
% 2.40/1.42 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_156_pinf_I8_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z3 @ X5 )
% 2.40/1.42 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_157_pinf_I6_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z3 @ X5 )
% 2.40/1.42 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_158_sorted__nth__mono,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [Xs: list @ A,I3: nat,J: nat] :
% 2.40/1.42 ( ( linorder_sorted @ A @ Xs )
% 2.40/1.42 => ( ( ord_less_eq @ nat @ I3 @ J )
% 2.40/1.42 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.40/1.42 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_159_sorted__nth__monoI,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [Xs: list @ A] :
% 2.40/1.42 ( ! [I: nat,J2: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ I @ J2 )
% 2.40/1.42 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.40/1.42 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J2 ) ) ) )
% 2.40/1.42 => ( linorder_sorted @ A @ Xs ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_160_sorted__equals__nth__mono,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ( ( linorder_sorted @ A )
% 2.40/1.42 = ( ^ [Xs2: list @ A] :
% 2.40/1.42 ! [J3: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.40/1.42 => ! [I2: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 2.40/1.42 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_161_less__fun__def,axiom,
% 2.40/1.42 ! [B: $tType,A: $tType] :
% 2.40/1.42 ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.42 => ( ( ord_less @ ( A > B ) )
% 2.40/1.42 = ( ^ [F2: A > B,G2: A > B] :
% 2.40/1.42 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.40/1.42 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_162_pinf_I1_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.40/1.42 ( ? [Z4: A] :
% 2.40/1.42 ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z4 @ X4 )
% 2.40/1.42 => ( ( P2 @ X4 )
% 2.40/1.42 = ( P4 @ X4 ) ) )
% 2.40/1.42 => ( ? [Z4: A] :
% 2.40/1.42 ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z4 @ X4 )
% 2.40/1.42 => ( ( Q @ X4 )
% 2.40/1.42 = ( Q2 @ X4 ) ) )
% 2.40/1.42 => ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z3 @ X5 )
% 2.40/1.42 => ( ( ( P2 @ X5 )
% 2.40/1.42 & ( Q @ X5 ) )
% 2.40/1.42 = ( ( P4 @ X5 )
% 2.40/1.42 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_163_pinf_I2_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.40/1.42 ( ? [Z4: A] :
% 2.40/1.42 ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z4 @ X4 )
% 2.40/1.42 => ( ( P2 @ X4 )
% 2.40/1.42 = ( P4 @ X4 ) ) )
% 2.40/1.42 => ( ? [Z4: A] :
% 2.40/1.42 ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z4 @ X4 )
% 2.40/1.42 => ( ( Q @ X4 )
% 2.40/1.42 = ( Q2 @ X4 ) ) )
% 2.40/1.42 => ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z3 @ X5 )
% 2.40/1.42 => ( ( ( P2 @ X5 )
% 2.40/1.42 | ( Q @ X5 ) )
% 2.40/1.42 = ( ( P4 @ X5 )
% 2.40/1.42 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_164_pinf_I3_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z3 @ X5 )
% 2.40/1.42 => ( X5 != T ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_165_pinf_I4_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z3 @ X5 )
% 2.40/1.42 => ( X5 != T ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_166_pinf_I5_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z3 @ X5 )
% 2.40/1.42 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_167_pinf_I7_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ Z3 @ X5 )
% 2.40/1.42 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_168_pinf_I11_J,axiom,
% 2.40/1.42 ! [C3: $tType,D3: $tType] :
% 2.40/1.42 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 2.40/1.42 => ! [F3: D3] :
% 2.40/1.42 ? [Z3: C3] :
% 2.40/1.42 ! [X5: C3] :
% 2.40/1.42 ( ( ord_less @ C3 @ Z3 @ X5 )
% 2.40/1.42 => ( F3 = F3 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_169_minf_I1_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.40/1.42 ( ? [Z4: A] :
% 2.40/1.42 ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ X4 @ Z4 )
% 2.40/1.42 => ( ( P2 @ X4 )
% 2.40/1.42 = ( P4 @ X4 ) ) )
% 2.40/1.42 => ( ? [Z4: A] :
% 2.40/1.42 ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ X4 @ Z4 )
% 2.40/1.42 => ( ( Q @ X4 )
% 2.40/1.42 = ( Q2 @ X4 ) ) )
% 2.40/1.42 => ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ X5 @ Z3 )
% 2.40/1.42 => ( ( ( P2 @ X5 )
% 2.40/1.42 & ( Q @ X5 ) )
% 2.40/1.42 = ( ( P4 @ X5 )
% 2.40/1.42 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_170_minf_I2_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.40/1.42 ( ? [Z4: A] :
% 2.40/1.42 ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ X4 @ Z4 )
% 2.40/1.42 => ( ( P2 @ X4 )
% 2.40/1.42 = ( P4 @ X4 ) ) )
% 2.40/1.42 => ( ? [Z4: A] :
% 2.40/1.42 ! [X4: A] :
% 2.40/1.42 ( ( ord_less @ A @ X4 @ Z4 )
% 2.40/1.42 => ( ( Q @ X4 )
% 2.40/1.42 = ( Q2 @ X4 ) ) )
% 2.40/1.42 => ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ X5 @ Z3 )
% 2.40/1.42 => ( ( ( P2 @ X5 )
% 2.40/1.42 | ( Q @ X5 ) )
% 2.40/1.42 = ( ( P4 @ X5 )
% 2.40/1.42 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_171_minf_I3_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ X5 @ Z3 )
% 2.40/1.42 => ( X5 != T ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_172_minf_I4_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ X5 @ Z3 )
% 2.40/1.42 => ( X5 != T ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_173_minf_I5_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ X5 @ Z3 )
% 2.40/1.42 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_174_minf_I7_J,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [T: A] :
% 2.40/1.42 ? [Z3: A] :
% 2.40/1.42 ! [X5: A] :
% 2.40/1.42 ( ( ord_less @ A @ X5 @ Z3 )
% 2.40/1.42 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_175_minf_I11_J,axiom,
% 2.40/1.42 ! [C3: $tType,D3: $tType] :
% 2.40/1.42 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 2.40/1.42 => ! [F3: D3] :
% 2.40/1.42 ? [Z3: C3] :
% 2.40/1.42 ! [X5: C3] :
% 2.40/1.42 ( ( ord_less @ C3 @ X5 @ Z3 )
% 2.40/1.42 => ( F3 = F3 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_176_sorted__rev__nth__mono,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [Xs: list @ A,I3: nat,J: nat] :
% 2.40/1.42 ( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
% 2.40/1.42 => ( ( ord_less_eq @ nat @ I3 @ J )
% 2.40/1.42 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.40/1.42 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_177_list__ex__length,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( list_ex @ A )
% 2.40/1.42 = ( ^ [P5: A > $o,Xs2: list @ A] :
% 2.40/1.42 ? [N2: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.40/1.42 & ( P5 @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_178_length__takeWhile__less__P__nth,axiom,
% 2.40/1.42 ! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
% 2.40/1.42 ( ! [I: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ I @ J )
% 2.40/1.42 => ( P2 @ ( nth @ A @ Xs @ I ) ) )
% 2.40/1.42 => ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.40/1.42 => ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_179_rev__rev__ident,axiom,
% 2.40/1.42 ! [A: $tType,Xs: list @ A] :
% 2.40/1.42 ( ( rev @ A @ ( rev @ A @ Xs ) )
% 2.40/1.42 = Xs ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_180_rev__is__rev__conv,axiom,
% 2.40/1.42 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.40/1.42 ( ( ( rev @ A @ Xs )
% 2.40/1.42 = ( rev @ A @ Ys ) )
% 2.40/1.42 = ( Xs = Ys ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_181_takeWhile__idem,axiom,
% 2.40/1.42 ! [A: $tType,P2: A > $o,Xs: list @ A] :
% 2.40/1.42 ( ( takeWhile @ A @ P2 @ ( takeWhile @ A @ P2 @ Xs ) )
% 2.40/1.42 = ( takeWhile @ A @ P2 @ Xs ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_182_length__rev,axiom,
% 2.40/1.42 ! [A: $tType,Xs: list @ A] :
% 2.40/1.42 ( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
% 2.40/1.42 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_183_list__ex__rev,axiom,
% 2.40/1.42 ! [A: $tType,P2: A > $o,Xs: list @ A] :
% 2.40/1.42 ( ( list_ex @ A @ P2 @ ( rev @ A @ Xs ) )
% 2.40/1.42 = ( list_ex @ A @ P2 @ Xs ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_184_sorted__takeWhile,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [Xs: list @ A,P2: A > $o] :
% 2.40/1.42 ( ( linorder_sorted @ A @ Xs )
% 2.40/1.42 => ( linorder_sorted @ A @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_185_rev__swap,axiom,
% 2.40/1.42 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.40/1.42 ( ( ( rev @ A @ Xs )
% 2.40/1.42 = Ys )
% 2.40/1.42 = ( Xs
% 2.40/1.42 = ( rev @ A @ Ys ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_186_length__takeWhile__le,axiom,
% 2.40/1.42 ! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_187_takeWhile__nth,axiom,
% 2.40/1.42 ! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
% 2.40/1.42 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) )
% 2.40/1.42 => ( ( nth @ A @ ( takeWhile @ A @ P2 @ Xs ) @ J )
% 2.40/1.42 = ( nth @ A @ Xs @ J ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_188_nth__length__takeWhile,axiom,
% 2.40/1.42 ! [A: $tType,P2: A > $o,Xs: list @ A] :
% 2.40/1.42 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.40/1.42 => ~ ( P2 @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_189_rev__nth,axiom,
% 2.40/1.42 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.40/1.42 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.40/1.42 => ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
% 2.40/1.42 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_190_count__le__length,axiom,
% 2.40/1.42 ! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_191_GreatestM__nat__lemma,axiom,
% 2.40/1.42 ! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
% 2.40/1.42 ( ( P2 @ K )
% 2.40/1.42 => ( ! [Y4: A] :
% 2.40/1.42 ( ( P2 @ Y4 )
% 2.40/1.42 => ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
% 2.40/1.42 => ( ( P2 @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) )
% 2.40/1.42 & ! [Y2: A] :
% 2.40/1.42 ( ( P2 @ Y2 )
% 2.40/1.42 => ( ord_less_eq @ nat @ ( M @ Y2 ) @ ( M @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_192_old_Onat_Oinject,axiom,
% 2.40/1.42 ! [Nat: nat,Nat2: nat] :
% 2.40/1.42 ( ( ( suc @ Nat )
% 2.40/1.42 = ( suc @ Nat2 ) )
% 2.40/1.42 = ( Nat = Nat2 ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_193_nat_Oinject,axiom,
% 2.40/1.42 ! [X22: nat,Y22: nat] :
% 2.40/1.42 ( ( ( suc @ X22 )
% 2.40/1.42 = ( suc @ Y22 ) )
% 2.40/1.42 = ( X22 = Y22 ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_194_Suc__less__eq,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.40/1.42 = ( ord_less @ nat @ M @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_195_Suc__mono,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ M @ N )
% 2.40/1.42 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_196_lessI,axiom,
% 2.40/1.42 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_197_Suc__le__mono,axiom,
% 2.40/1.42 ! [N: nat,M: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 2.40/1.42 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_198_diff__Suc__Suc,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.40/1.42 = ( minus_minus @ nat @ M @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_199_Suc__diff__diff,axiom,
% 2.40/1.42 ! [M: nat,N: nat,K: nat] :
% 2.40/1.42 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 2.40/1.42 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_200_not__less__less__Suc__eq,axiom,
% 2.40/1.42 ! [N: nat,M: nat] :
% 2.40/1.42 ( ~ ( ord_less @ nat @ N @ M )
% 2.40/1.42 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 2.40/1.42 = ( N = M ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_201_strict__inc__induct,axiom,
% 2.40/1.42 ! [I3: nat,J: nat,P2: nat > $o] :
% 2.40/1.42 ( ( ord_less @ nat @ I3 @ J )
% 2.40/1.42 => ( ! [I: nat] :
% 2.40/1.42 ( ( J
% 2.40/1.42 = ( suc @ I ) )
% 2.40/1.42 => ( P2 @ I ) )
% 2.40/1.42 => ( ! [I: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ I @ J )
% 2.40/1.42 => ( ( P2 @ ( suc @ I ) )
% 2.40/1.42 => ( P2 @ I ) ) )
% 2.40/1.42 => ( P2 @ I3 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_202_less__Suc__induct,axiom,
% 2.40/1.42 ! [I3: nat,J: nat,P2: nat > nat > $o] :
% 2.40/1.42 ( ( ord_less @ nat @ I3 @ J )
% 2.40/1.42 => ( ! [I: nat] : ( P2 @ I @ ( suc @ I ) )
% 2.40/1.42 => ( ! [I: nat,J2: nat,K2: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ I @ J2 )
% 2.40/1.42 => ( ( ord_less @ nat @ J2 @ K2 )
% 2.40/1.42 => ( ( P2 @ I @ J2 )
% 2.40/1.42 => ( ( P2 @ J2 @ K2 )
% 2.40/1.42 => ( P2 @ I @ K2 ) ) ) ) )
% 2.40/1.42 => ( P2 @ I3 @ J ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_203_less__trans__Suc,axiom,
% 2.40/1.42 ! [I3: nat,J: nat,K: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ I3 @ J )
% 2.40/1.42 => ( ( ord_less @ nat @ J @ K )
% 2.40/1.42 => ( ord_less @ nat @ ( suc @ I3 ) @ K ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_204_Suc__less__SucD,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.40/1.42 => ( ord_less @ nat @ M @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_205_less__antisym,axiom,
% 2.40/1.42 ! [N: nat,M: nat] :
% 2.40/1.42 ( ~ ( ord_less @ nat @ N @ M )
% 2.40/1.42 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 2.40/1.42 => ( M = N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_206_Suc__less__eq2,axiom,
% 2.40/1.42 ! [N: nat,M: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 2.40/1.42 = ( ? [M4: nat] :
% 2.40/1.42 ( ( M
% 2.40/1.42 = ( suc @ M4 ) )
% 2.40/1.42 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_207_not__less__eq,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 2.40/1.42 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_208_less__Suc__eq,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.40/1.42 = ( ( ord_less @ nat @ M @ N )
% 2.40/1.42 | ( M = N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_209_less__SucI,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ M @ N )
% 2.40/1.42 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_210_less__SucE,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.40/1.42 => ( ~ ( ord_less @ nat @ M @ N )
% 2.40/1.42 => ( M = N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_211_Suc__lessI,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ M @ N )
% 2.40/1.42 => ( ( ( suc @ M )
% 2.40/1.42 != N )
% 2.40/1.42 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_212_Suc__lessE,axiom,
% 2.40/1.42 ! [I3: nat,K: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ ( suc @ I3 ) @ K )
% 2.40/1.42 => ~ ! [J2: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ I3 @ J2 )
% 2.40/1.42 => ( K
% 2.40/1.42 != ( suc @ J2 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_213_Suc__lessD,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 2.40/1.42 => ( ord_less @ nat @ M @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_214_lessE,axiom,
% 2.40/1.42 ! [I3: nat,K: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ I3 @ K )
% 2.40/1.42 => ( ( K
% 2.40/1.42 != ( suc @ I3 ) )
% 2.40/1.42 => ~ ! [J2: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ I3 @ J2 )
% 2.40/1.42 => ( K
% 2.40/1.42 != ( suc @ J2 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_215_lift__Suc__mono__less,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [F: nat > A,N: nat,N4: nat] :
% 2.40/1.42 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 2.40/1.42 => ( ( ord_less @ nat @ N @ N4 )
% 2.40/1.42 => ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_216_lift__Suc__mono__less__iff,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [F: nat > A,N: nat,M: nat] :
% 2.40/1.42 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 2.40/1.42 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 2.40/1.42 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_217_lift__Suc__antimono__le,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [F: nat > A,N: nat,N4: nat] :
% 2.40/1.42 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 2.40/1.42 => ( ( ord_less_eq @ nat @ N @ N4 )
% 2.40/1.42 => ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_218_lift__Suc__mono__le,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( order @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [F: nat > A,N: nat,N4: nat] :
% 2.40/1.42 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 2.40/1.42 => ( ( ord_less_eq @ nat @ N @ N4 )
% 2.40/1.42 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_219_le__imp__less__Suc,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.42 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_220_less__eq__Suc__le,axiom,
% 2.40/1.42 ( ( ord_less @ nat )
% 2.40/1.42 = ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_221_less__Suc__eq__le,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.40/1.42 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_222_le__less__Suc__eq,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.42 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 2.40/1.42 = ( N = M ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_223_Suc__le__lessD,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 2.40/1.42 => ( ord_less @ nat @ M @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_224_inc__induct,axiom,
% 2.40/1.42 ! [I3: nat,J: nat,P2: nat > $o] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ I3 @ J )
% 2.40/1.42 => ( ( P2 @ J )
% 2.40/1.42 => ( ! [N3: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ I3 @ N3 )
% 2.40/1.42 => ( ( ord_less @ nat @ N3 @ J )
% 2.40/1.42 => ( ( P2 @ ( suc @ N3 ) )
% 2.40/1.42 => ( P2 @ N3 ) ) ) )
% 2.40/1.42 => ( P2 @ I3 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_225_dec__induct,axiom,
% 2.40/1.42 ! [I3: nat,J: nat,P2: nat > $o] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ I3 @ J )
% 2.40/1.42 => ( ( P2 @ I3 )
% 2.40/1.42 => ( ! [N3: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ I3 @ N3 )
% 2.40/1.42 => ( ( ord_less @ nat @ N3 @ J )
% 2.40/1.42 => ( ( P2 @ N3 )
% 2.40/1.42 => ( P2 @ ( suc @ N3 ) ) ) ) )
% 2.40/1.42 => ( P2 @ J ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_226_Suc__le__eq,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 2.40/1.42 = ( ord_less @ nat @ M @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_227_Suc__leI,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ M @ N )
% 2.40/1.42 => ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_228_n__not__Suc__n,axiom,
% 2.40/1.42 ! [N: nat] :
% 2.40/1.42 ( N
% 2.40/1.42 != ( suc @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_229_Suc__inject,axiom,
% 2.40/1.42 ! [X3: nat,Y3: nat] :
% 2.40/1.42 ( ( ( suc @ X3 )
% 2.40/1.42 = ( suc @ Y3 ) )
% 2.40/1.42 => ( X3 = Y3 ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_230_zero__induct__lemma,axiom,
% 2.40/1.42 ! [P2: nat > $o,K: nat,I3: nat] :
% 2.40/1.42 ( ( P2 @ K )
% 2.40/1.42 => ( ! [N3: nat] :
% 2.40/1.42 ( ( P2 @ ( suc @ N3 ) )
% 2.40/1.42 => ( P2 @ N3 ) )
% 2.40/1.42 => ( P2 @ ( minus_minus @ nat @ K @ I3 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_231_full__nat__induct,axiom,
% 2.40/1.42 ! [P2: nat > $o,N: nat] :
% 2.40/1.42 ( ! [N3: nat] :
% 2.40/1.42 ( ! [M3: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
% 2.40/1.42 => ( P2 @ M3 ) )
% 2.40/1.42 => ( P2 @ N3 ) )
% 2.40/1.42 => ( P2 @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_232_not__less__eq__eq,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 2.40/1.42 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_233_Suc__n__not__le__n,axiom,
% 2.40/1.42 ! [N: nat] :
% 2.40/1.42 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_234_le__Suc__eq,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 2.40/1.42 = ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.42 | ( M
% 2.40/1.42 = ( suc @ N ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_235_Suc__le__D,axiom,
% 2.40/1.42 ! [N: nat,M5: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
% 2.40/1.42 => ? [M6: nat] :
% 2.40/1.42 ( M5
% 2.40/1.42 = ( suc @ M6 ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_236_le__SucI,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ M @ N )
% 2.40/1.42 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_237_le__SucE,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 2.40/1.42 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 2.40/1.42 => ( M
% 2.40/1.42 = ( suc @ N ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_238_Suc__leD,axiom,
% 2.40/1.42 ! [M: nat,N: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 2.40/1.42 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_239_diff__less__Suc,axiom,
% 2.40/1.42 ! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_240_Suc__diff__Suc,axiom,
% 2.40/1.42 ! [N: nat,M: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ N @ M )
% 2.40/1.42 => ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
% 2.40/1.42 = ( minus_minus @ nat @ M @ N ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_241_Suc__diff__le,axiom,
% 2.40/1.42 ! [N: nat,M: nat] :
% 2.40/1.42 ( ( ord_less_eq @ nat @ N @ M )
% 2.40/1.42 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 2.40/1.42 = ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_242_GreatestM__natI,axiom,
% 2.40/1.42 ! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
% 2.40/1.42 ( ( P2 @ K )
% 2.40/1.42 => ( ! [Y4: A] :
% 2.40/1.42 ( ( P2 @ Y4 )
% 2.40/1.42 => ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
% 2.40/1.42 => ( P2 @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_243_GreatestMI2,axiom,
% 2.40/1.42 ! [B: $tType,A: $tType] :
% 2.40/1.42 ( ( ord @ B @ ( type2 @ B ) )
% 2.40/1.42 => ! [P2: A > $o,X3: A,M: A > B,Q: A > $o] :
% 2.40/1.42 ( ( P2 @ X3 )
% 2.40/1.42 => ( ! [Y4: A] :
% 2.40/1.42 ( ( P2 @ Y4 )
% 2.40/1.42 => ( ord_less_eq @ B @ ( M @ Y4 ) @ ( M @ X3 ) ) )
% 2.40/1.42 => ( ! [X4: A] :
% 2.40/1.42 ( ( P2 @ X4 )
% 2.40/1.42 => ( ! [Y2: A] :
% 2.40/1.42 ( ( P2 @ Y2 )
% 2.40/1.42 => ( ord_less_eq @ B @ ( M @ Y2 ) @ ( M @ X4 ) ) )
% 2.40/1.42 => ( Q @ X4 ) ) )
% 2.40/1.42 => ( Q @ ( hilbert_GreatestM @ A @ B @ M @ P2 ) ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_244_GreatestM__nat__le,axiom,
% 2.40/1.42 ! [A: $tType,P2: A > $o,X3: A,M: A > nat,B3: nat] :
% 2.40/1.42 ( ( P2 @ X3 )
% 2.40/1.42 => ( ! [Y4: A] :
% 2.40/1.42 ( ( P2 @ Y4 )
% 2.40/1.42 => ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
% 2.40/1.42 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_245_prod__decode__aux_Oinduct,axiom,
% 2.40/1.42 ! [P2: nat > nat > $o,A0: nat,A1: nat] :
% 2.40/1.42 ( ! [K2: nat,M6: nat] :
% 2.40/1.42 ( ( ~ ( ord_less_eq @ nat @ M6 @ K2 )
% 2.40/1.42 => ( P2 @ ( suc @ K2 ) @ ( minus_minus @ nat @ M6 @ ( suc @ K2 ) ) ) )
% 2.40/1.42 => ( P2 @ K2 @ M6 ) )
% 2.40/1.42 => ( P2 @ A0 @ A1 ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_246_remdups__adj__adjacent,axiom,
% 2.40/1.42 ! [A: $tType,I3: nat,Xs: list @ A] :
% 2.40/1.42 ( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
% 2.40/1.42 => ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I3 )
% 2.40/1.42 != ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_247_remdups__adj__rev,axiom,
% 2.40/1.42 ! [A: $tType,Xs: list @ A] :
% 2.40/1.42 ( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
% 2.40/1.42 = ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_248_sorted__remdups__adj,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( linorder @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [Xs: list @ A] :
% 2.40/1.42 ( ( linorder_sorted @ A @ Xs )
% 2.40/1.42 => ( linorder_sorted @ A @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_249_remdups__adj__length,axiom,
% 2.40/1.42 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_250_nth__tl,axiom,
% 2.40/1.42 ! [A: $tType,N: nat,X3: list @ A] :
% 2.40/1.42 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ X3 ) ) )
% 2.40/1.42 => ( ( nth @ A @ ( tl @ A @ X3 ) @ N )
% 2.40/1.42 = ( nth @ A @ X3 @ ( suc @ N ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_251_Suc__pred,axiom,
% 2.40/1.42 ! [N: nat] :
% 2.40/1.42 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.40/1.42 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 2.40/1.42 = N ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_252_le__zero__eq,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [N: A] :
% 2.40/1.42 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.40/1.42 = ( N
% 2.40/1.42 = ( zero_zero @ A ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(fact_253_not__gr__zero,axiom,
% 2.40/1.42 ! [A: $tType] :
% 2.40/1.42 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.40/1.42 => ! [N: A] :
% 2.40/1.42 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.40/1.42 = ( N
% 2.40/1.42 = ( zero_zero @ A ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.40/1.42 ! [A6: $tType,A7: $tType] :
% 2.40/1.42 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 2.40/1.42 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.40/1.42 ! [A6: $tType,A7: $tType] :
% 2.40/1.42 ( ( order @ A7 @ ( type2 @ A7 ) )
% 2.40/1.42 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_fun___Orderings_Oord,axiom,
% 2.40/1.42 ! [A6: $tType,A7: $tType] :
% 2.40/1.42 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 2.40/1.42 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_fun___Groups_Ominus,axiom,
% 2.40/1.42 ! [A6: $tType,A7: $tType] :
% 2.40/1.42 ( ( minus @ A7 @ ( type2 @ A7 ) )
% 2.40/1.42 => ( minus @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.40/1.42 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.40/1.42 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.40/1.42 cancel146912293up_add @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.40/1.42 wellorder @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.40/1.42 preorder @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.40/1.42 linorder @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.40/1.42 no_top @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.40/1.42 order @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.40/1.42 ord @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
% 2.40/1.42 minus @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.40/1.42 zero @ nat @ ( type2 @ nat ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
% 2.40/1.42 preorder @ $o @ ( type2 @ $o ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
% 2.40/1.42 linorder @ $o @ ( type2 @ $o ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
% 2.40/1.42 order @ $o @ ( type2 @ $o ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 2.40/1.42 ord @ $o @ ( type2 @ $o ) ).
% 2.40/1.42
% 2.40/1.42 thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
% 2.40/1.42 minus @ $o @ ( type2 @ $o ) ).
% 2.40/1.42
% 2.40/1.42 thf(conj_0,conjecture,
% 2.40/1.42 ( ( ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
% 2.40/1.42 => ( ( stream370371455e_snth @ a @ ( stream1035003186_shift @ a @ xs @ s ) @ n )
% 2.40/1.42 = ( nth @ a @ xs @ n ) ) )
% 2.40/1.42 & ( ~ ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
% 2.40/1.42 => ( ( stream370371455e_snth @ a @ ( stream1035003186_shift @ a @ xs @ s ) @ n )
% 2.40/1.42 = ( stream370371455e_snth @ a @ s @ ( minus_minus @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ) ).
% 2.40/1.42 ------------------------
% 2.40/1.43 % (23307)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_23201 for (2999ds/275Mi)
% 2.40/1.44 % (23303)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_23201 for (2999ds/4Mi)
% 2.40/1.44 % (23305)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_23201 for (2999ds/2Mi)
% 2.40/1.44 % (23302)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_23201 for (2999ds/183Mi)
% 2.40/1.44 % (23304)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_23201 for (2999ds/27Mi)
% 2.40/1.44 % (23306)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_23201 for (2999ds/2Mi)
% 2.40/1.44 % (23308)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_23201 for (2999ds/18Mi)
% 2.40/1.44 % (23305)Instruction limit reached!
% 2.40/1.44 % (23305)------------------------------
% 2.40/1.44 % (23305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.44 % (23306)Instruction limit reached!
% 2.40/1.44 % (23306)------------------------------
% 2.40/1.44 % (23306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.44 % (23305)Termination reason: Unknown
% 2.40/1.44 % (23305)Termination phase: shuffling
% 2.40/1.44
% 2.40/1.44 % (23305)Memory used [KB]: 1407
% 2.40/1.44 % (23305)Time elapsed: 0.003 s
% 2.40/1.44 % (23305)Instructions burned: 3 (million)
% 2.40/1.44 % (23305)------------------------------
% 2.40/1.44 % (23305)------------------------------
% 2.40/1.44 % (23306)Termination reason: Unknown
% 2.40/1.44 % (23306)Termination phase: shuffling
% 2.40/1.44
% 2.40/1.44 % (23306)Memory used [KB]: 1407
% 2.40/1.44 % (23306)Time elapsed: 0.003 s
% 2.40/1.44 % (23306)Instructions burned: 3 (million)
% 2.40/1.44 % (23306)------------------------------
% 2.40/1.44 % (23306)------------------------------
% 2.40/1.44 % (23303)Instruction limit reached!
% 2.40/1.44 % (23303)------------------------------
% 2.40/1.44 % (23303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.44 % (23303)Termination reason: Unknown
% 2.40/1.44 % (23303)Termination phase: shuffling
% 2.40/1.44
% 2.40/1.44 % (23303)Memory used [KB]: 1407
% 2.40/1.44 % (23303)Time elapsed: 0.004 s
% 2.40/1.44 % (23303)Instructions burned: 5 (million)
% 2.40/1.44 % (23303)------------------------------
% 2.40/1.44 % (23303)------------------------------
% 2.40/1.45 % (23308)Instruction limit reached!
% 2.40/1.45 % (23308)------------------------------
% 2.40/1.45 % (23308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.45 % (23308)Termination reason: Unknown
% 2.40/1.45 % (23308)Termination phase: shuffling
% 2.40/1.45
% 2.40/1.45 % (23308)Memory used [KB]: 1663
% 2.40/1.45 % (23308)Time elapsed: 0.011 s
% 2.40/1.45 % (23308)Instructions burned: 18 (million)
% 2.40/1.45 % (23308)------------------------------
% 2.40/1.45 % (23308)------------------------------
% 2.40/1.45 % (23304)Instruction limit reached!
% 2.40/1.45 % (23304)------------------------------
% 2.40/1.45 % (23304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.45 % (23304)Termination reason: Unknown
% 2.40/1.45 % (23304)Termination phase: Property scanning
% 2.40/1.45
% 2.40/1.45 % (23304)Memory used [KB]: 1791
% 2.40/1.45 % (23304)Time elapsed: 0.015 s
% 2.40/1.45 % (23304)Instructions burned: 27 (million)
% 2.40/1.45 % (23304)------------------------------
% 2.40/1.45 % (23304)------------------------------
% 2.40/1.46 % (23309)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23201 for (2999ds/3Mi)
% 2.40/1.46 % (23310)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_23201 for (2999ds/37Mi)
% 2.40/1.46 % (23311)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_23201 for (2999ds/15Mi)
% 2.40/1.46 % (23307)First to succeed.
% 2.40/1.46 % (23309)Instruction limit reached!
% 2.40/1.46 % (23309)------------------------------
% 2.40/1.46 % (23309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.46 % (23309)Termination reason: Unknown
% 2.40/1.46 % (23309)Termination phase: shuffling
% 2.40/1.46
% 2.40/1.46 % (23309)Memory used [KB]: 1407
% 2.40/1.46 % (23309)Time elapsed: 0.003 s
% 2.40/1.46 % (23309)Instructions burned: 3 (million)
% 2.40/1.46 % (23309)------------------------------
% 2.40/1.46 % (23309)------------------------------
% 2.40/1.46 % (23307)Refutation found. Thanks to Tanya!
% 2.40/1.46 % SZS status Theorem for DTF2THF_23201
% 2.40/1.46 % SZS output start Proof for DTF2THF_23201
% 2.40/1.46 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.40/1.46 thf(type_def_7, type, list: ($tType) > $tType).
% 2.40/1.46 thf(type_def_8, type, nat: $tType).
% 2.40/1.46 thf(type_def_9, type, stream170649215stream: ($tType) > $tType).
% 2.40/1.46 thf(type_def_10, type, a: $tType).
% 2.40/1.46 thf(func_def_0, type, stream170649215stream: ($tType) > $tType).
% 2.40/1.46 thf(func_def_1, type, list: ($tType) > $tType).
% 2.40/1.46 thf(func_def_2, type, nat: $tType).
% 2.40/1.46 thf(func_def_3, type, itself: ($tType) > $tType).
% 2.40/1.46 thf(func_def_4, type, a: $tType).
% 2.40/1.46 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_6, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_7, type, minus: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_8, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_9, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_10, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_11, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_12, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_13, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_14, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_15, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_16, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_17, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_18, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_19, type, cancel146912293up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_20, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_21, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_22, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.40/1.46 thf(func_def_23, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.40/1.46 thf(func_def_24, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.40/1.46 thf(func_def_25, type, hilbert_GreatestM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 2.40/1.46 thf(func_def_26, type, count_list: !>[X0: $tType]:(list @ X0 > X0 > nat)).
% 2.40/1.46 thf(func_def_27, type, linorder_sorted: !>[X0: $tType]:(list @ X0 > $o)).
% 2.40/1.46 thf(func_def_28, type, tl: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.40/1.46 thf(func_def_29, type, list_ex: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 2.40/1.46 thf(func_def_30, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 2.40/1.46 thf(func_def_31, type, remdups_adj: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.40/1.46 thf(func_def_32, type, rev: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.40/1.46 thf(func_def_33, type, takeWhile: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.40/1.46 thf(func_def_34, type, suc: nat > nat).
% 2.40/1.46 thf(func_def_35, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.40/1.46 thf(func_def_36, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.40/1.46 thf(func_def_37, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.40/1.46 thf(func_def_38, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.40/1.46 thf(func_def_39, type, stream1035003186_shift: !>[X0: $tType]:(list @ X0 > stream170649215stream @ X0 > stream170649215stream @ X0)).
% 2.40/1.46 thf(func_def_40, type, stream370371455e_snth: !>[X0: $tType]:(stream170649215stream @ X0 > nat > X0)).
% 2.40/1.46 thf(func_def_41, type, n: nat).
% 2.40/1.46 thf(func_def_42, type, s: stream170649215stream @ a).
% 2.40/1.46 thf(func_def_43, type, xs: list @ a).
% 2.40/1.46 thf(func_def_58, type, sK0: (nat > $o) > nat).
% 2.40/1.46 thf(func_def_59, type, sK1: !>[X0: $tType]:(nat > list @ X0)).
% 2.40/1.46 thf(func_def_60, type, sK2: !>[X0: $tType]:(list @ X0 > list @ X0 > nat)).
% 2.40/1.46 thf(func_def_61, type, sK3: nat > nat > nat).
% 2.40/1.46 thf(func_def_62, type, sK4: (nat > $o) > nat).
% 2.40/1.46 thf(func_def_63, type, sK5: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.40/1.46 thf(func_def_64, type, sK6: (nat > $o) > nat > nat).
% 2.40/1.46 thf(func_def_65, type, sK7: (nat > $o) > nat > nat).
% 2.40/1.46 thf(func_def_66, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.40/1.46 thf(func_def_67, type, sK9: nat > nat > nat).
% 2.40/1.46 thf(func_def_68, type, sK10: nat > nat > nat).
% 2.40/1.46 thf(func_def_69, type, sK11: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.40/1.46 thf(func_def_70, type, sK12: !>[X0: $tType]:(nat > (X0 > nat) > (X0 > $o) > X0)).
% 2.40/1.46 thf(func_def_71, type, sK13: (nat > nat) > nat).
% 2.40/1.46 thf(func_def_72, type, sK14: (nat > nat) > nat).
% 2.40/1.46 thf(func_def_73, type, sK15: nat > nat).
% 2.40/1.46 thf(func_def_74, type, sK16: (nat > $o) > nat).
% 2.40/1.46 thf(func_def_75, type, sK17: (nat > $o) > nat > nat > nat).
% 2.40/1.46 thf(func_def_76, type, sK18: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.40/1.46 thf(func_def_77, type, sK19: (nat > $o) > nat > nat > nat).
% 2.40/1.46 thf(func_def_78, type, sK20: (nat > $o) > nat).
% 2.40/1.46 thf(func_def_79, type, sK21: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.40/1.46 thf(func_def_80, type, sK22: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.40/1.46 thf(func_def_81, type, sK23: (nat > nat > $o) > nat).
% 2.40/1.46 thf(func_def_82, type, sK24: (nat > nat > $o) > nat).
% 2.40/1.46 thf(func_def_83, type, sK25: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.40/1.46 thf(func_def_84, type, sK26: (nat > nat > $o) > nat).
% 2.40/1.46 thf(func_def_85, type, sK27: (nat > nat > $o) > nat).
% 2.40/1.46 thf(func_def_86, type, sK28: (nat > nat > $o) > nat).
% 2.40/1.46 thf(func_def_87, type, sK29: (nat > nat > $o) > nat).
% 2.40/1.46 thf(func_def_89, type, ph31: !>[X0: $tType]:(X0)).
% 2.40/1.46 thf(f1211,plain,(
% 2.40/1.46 $false),
% 2.40/1.46 inference(avatar_sat_refutation,[status(thm)],[f1146,f1167,f1179,f1209])).
% 2.40/1.46 thf(f1209,plain,(
% 2.40/1.46 spl30_2 | ~spl30_4),
% 2.40/1.46 inference(avatar_contradiction_clause,[status(thm)],[f1208])).
% 2.40/1.46 thf(f1208,plain,(
% 2.40/1.46 $false | (spl30_2 | ~spl30_4)),
% 2.40/1.46 inference(subsumption_resolution,[status(thm)],[f1197,f1119])).
% 2.40/1.46 thf(f1119,plain,(
% 2.40/1.46 ( ! [X1 : nat] : (((ord_less_eq @ nat @ X1 @ X1) = $true)) )),
% 2.40/1.46 inference(equality_resolution,[status(thm)],[f987])).
% 2.40/1.46 thf(f987,plain,(
% 2.40/1.46 ( ! [X0 : nat,X1 : nat] : ((X0 != X1) | ($true = (ord_less_eq @ nat @ X0 @ X1))) )),
% 2.40/1.46 inference(cnf_transformation,[status(thm)],[f825])).
% 2.40/1.46 thf(f825,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : ((((ord_less @ nat @ X0 @ X1) != $true) & (X0 != X1)) | ($true = (ord_less_eq @ nat @ X0 @ X1)))),
% 2.40/1.46 inference(ennf_transformation,[status(thm)],[f537])).
% 2.40/1.46 thf(f537,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : (((X0 = X1) | ((ord_less @ nat @ X0 @ X1) = $true)) => ($true = (ord_less_eq @ nat @ X0 @ X1)))),
% 2.40/1.46 inference(fool_elimination,[status(thm)],[f536])).
% 2.40/1.46 thf(f536,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) | (X0 = X1)) => (ord_less_eq @ nat @ X0 @ X1))),
% 2.40/1.46 inference(rectify,[status(thm)],[f27])).
% 2.40/1.46 thf(f27,axiom,(
% 2.40/1.46 ! [X21 : nat,X22 : nat] : (((ord_less @ nat @ X21 @ X22) | (X21 = X22)) => (ord_less_eq @ nat @ X21 @ X22))),
% 2.40/1.46 file('/export/starexec/sandbox2/tmp/tmp.gN7przDacx/DTF2THF_23201.p',fact_26_less__or__eq__imp__le)).
% 2.40/1.46 thf(f1197,plain,(
% 2.40/1.46 ((ord_less_eq @ nat @ n @ n) != $true) | (spl30_2 | ~spl30_4)),
% 2.40/1.46 inference(superposition,[status(thm)],[f1169,f1160])).
% 2.40/1.46 thf(f1160,plain,(
% 2.40/1.46 (n = (size_size @ list @ a @ xs)) | ~spl30_4),
% 2.40/1.46 inference(avatar_component_clause,[status(thm)],[f1158])).
% 2.40/1.46 thf(f1158,definition,(
% 2.40/1.46 spl30_4 <=> (n = (size_size @ list @ a @ xs))),
% 2.40/1.46 introduced(definition,[new_symbols(naming,[spl30_4])],[avatar_definition])).
% 2.40/1.46 thf(f1169,plain,(
% 2.40/1.46 ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) != $true) | spl30_2),
% 2.40/1.46 inference(trivial_inequality_removal,[status(thm)],[f1168])).
% 2.40/1.46 thf(f1168,plain,(
% 2.40/1.46 ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) != (stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n)) | ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) != $true) | spl30_2),
% 2.40/1.46 inference(superposition,[status(thm)],[f1142,f985])).
% 2.40/1.46 thf(f985,plain,(
% 2.40/1.46 ( ! [X0 : $tType,X2 : nat,X3 : stream170649215stream @ X0,X1 : list @ X0] : (((stream370371455e_snth @ X0 @ X3 @ (minus_minus @ nat @ X2 @ (size_size @ list @ X0 @ X1))) = (stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X1 @ X3) @ X2)) | ((ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X2) != $true)) )),
% 2.40/1.46 inference(cnf_transformation,[status(thm)],[f882])).
% 2.40/1.46 thf(f882,plain,(
% 2.40/1.46 ! [X0 : $tType,X1 : list @ X0,X2 : nat,X3 : stream170649215stream @ X0] : (((stream370371455e_snth @ X0 @ X3 @ (minus_minus @ nat @ X2 @ (size_size @ list @ X0 @ X1))) = (stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X1 @ X3) @ X2)) | ((ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X2) != $true))),
% 2.40/1.46 inference(ennf_transformation,[status(thm)],[f711])).
% 2.40/1.46 thf(f711,plain,(
% 2.40/1.46 ! [X0 : $tType,X1 : list @ X0,X2 : nat,X3 : stream170649215stream @ X0] : (((ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X2) = $true) => ((stream370371455e_snth @ X0 @ X3 @ (minus_minus @ nat @ X2 @ (size_size @ list @ X0 @ X1))) = (stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X1 @ X3) @ X2)))),
% 2.40/1.46 inference(fool_elimination,[status(thm)],[f710])).
% 2.40/1.46 thf(f710,plain,(
% 2.40/1.46 ! [X0 : $tType,X1 : list @ X0,X2 : nat,X3 : stream170649215stream @ X0] : ((ord_less_eq @ nat @ (size_size @ list @ X0 @ X1) @ X2) => ((stream370371455e_snth @ X0 @ X3 @ (minus_minus @ nat @ X2 @ (size_size @ list @ X0 @ X1))) = (stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X1 @ X3) @ X2)))),
% 2.40/1.46 inference(rectify,[status(thm)],[f14])).
% 2.40/1.46 thf(f14,axiom,(
% 2.40/1.46 ! [X0 : $tType,X2 : list @ X0,X5 : nat,X6 : stream170649215stream @ X0] : ((ord_less_eq @ nat @ (size_size @ list @ X0 @ X2) @ X5) => ((stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X2 @ X6) @ X5) = (stream370371455e_snth @ X0 @ X6 @ (minus_minus @ nat @ X5 @ (size_size @ list @ X0 @ X2)))))),
% 2.40/1.46 file('/export/starexec/sandbox2/tmp/tmp.gN7przDacx/DTF2THF_23201.p',fact_13_shift__snth__ge)).
% 2.40/1.46 thf(f1142,plain,(
% 2.40/1.46 ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) != (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)))) | spl30_2),
% 2.40/1.46 inference(avatar_component_clause,[status(thm)],[f1140])).
% 2.40/1.46 thf(f1140,definition,(
% 2.40/1.46 spl30_2 <=> ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs))))),
% 2.40/1.46 introduced(definition,[new_symbols(naming,[spl30_2])],[avatar_definition])).
% 2.40/1.46 thf(f1179,plain,(
% 2.40/1.46 spl30_2 | spl30_5),
% 2.40/1.46 inference(avatar_contradiction_clause,[status(thm)],[f1178])).
% 2.40/1.46 thf(f1178,plain,(
% 2.40/1.46 $false | (spl30_2 | spl30_5)),
% 2.40/1.46 inference(subsumption_resolution,[status(thm)],[f1174,f1169])).
% 2.40/1.46 thf(f1174,plain,(
% 2.40/1.46 ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) = $true) | spl30_5),
% 2.40/1.46 inference(trivial_inequality_removal,[status(thm)],[f1170])).
% 2.40/1.46 thf(f1170,plain,(
% 2.40/1.46 ($true != $true) | ((ord_less_eq @ nat @ (size_size @ list @ a @ xs) @ n) = $true) | spl30_5),
% 2.40/1.46 inference(superposition,[status(thm)],[f1166,f1107])).
% 2.40/1.46 thf(f1107,plain,(
% 2.40/1.46 ( ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less_eq @ nat @ X1 @ X0) = $true)) )),
% 2.40/1.46 inference(cnf_transformation,[status(thm)],[f519])).
% 2.40/1.46 thf(f519,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less_eq @ nat @ X1 @ X0) = $true))),
% 2.40/1.46 inference(fool_elimination,[status(thm)],[f518])).
% 2.40/1.46 thf(f518,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X1 @ X0) | (ord_less_eq @ nat @ X0 @ X1))),
% 2.40/1.46 inference(rectify,[status(thm)],[f20])).
% 2.40/1.46 thf(f20,axiom,(
% 2.40/1.46 ! [X21 : nat,X22 : nat] : ((ord_less_eq @ nat @ X22 @ X21) | (ord_less_eq @ nat @ X21 @ X22))),
% 2.40/1.46 file('/export/starexec/sandbox2/tmp/tmp.gN7przDacx/DTF2THF_23201.p',fact_19_nat__le__linear)).
% 2.40/1.46 thf(f1166,plain,(
% 2.40/1.46 ((ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs)) != $true) | spl30_5),
% 2.40/1.46 inference(avatar_component_clause,[status(thm)],[f1164])).
% 2.40/1.46 thf(f1164,definition,(
% 2.40/1.46 spl30_5 <=> ((ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs)) = $true)),
% 2.40/1.46 introduced(definition,[new_symbols(naming,[spl30_5])],[avatar_definition])).
% 2.40/1.46 thf(f1167,plain,(
% 2.40/1.46 ~spl30_5 | spl30_4),
% 2.40/1.46 inference(avatar_split_clause,[status(thm)],[f1150,f1158,f1164])).
% 2.40/1.46 thf(f1150,plain,(
% 2.40/1.46 ((ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs)) != $true) | (n = (size_size @ list @ a @ xs))),
% 2.40/1.46 inference(trivial_inequality_removal,[status(thm)],[f1148])).
% 2.40/1.46 thf(f1148,plain,(
% 2.40/1.46 (n = (size_size @ list @ a @ xs)) | ($true != $true) | ((ord_less_eq @ nat @ n @ (size_size @ list @ a @ xs)) != $true)),
% 2.40/1.46 inference(superposition,[status(thm)],[f1144,f1015])).
% 2.40/1.46 thf(f1015,plain,(
% 2.40/1.46 ( ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) = $true) | ($true != (ord_less_eq @ nat @ X0 @ X1)) | (X0 = X1)) )),
% 2.40/1.46 inference(cnf_transformation,[status(thm)],[f870])).
% 2.40/1.46 thf(f870,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : (($true != (ord_less_eq @ nat @ X0 @ X1)) | (X0 = X1) | ((ord_less @ nat @ X0 @ X1) = $true))),
% 2.40/1.46 inference(flattening,[status(thm)],[f869])).
% 2.40/1.46 thf(f869,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : ((((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1)) | ($true != (ord_less_eq @ nat @ X0 @ X1)))),
% 2.40/1.46 inference(ennf_transformation,[status(thm)],[f293])).
% 2.40/1.46 thf(f293,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) => ((X0 != X1) => ((ord_less @ nat @ X0 @ X1) = $true)))),
% 2.40/1.46 inference(fool_elimination,[status(thm)],[f292])).
% 2.40/1.46 thf(f292,plain,(
% 2.40/1.46 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) => ((X0 != X1) => (ord_less @ nat @ X0 @ X1)))),
% 2.40/1.46 inference(rectify,[status(thm)],[f26])).
% 2.40/1.46 thf(f26,axiom,(
% 2.40/1.46 ! [X21 : nat,X22 : nat] : ((ord_less_eq @ nat @ X21 @ X22) => ((X21 != X22) => (ord_less @ nat @ X21 @ X22)))),
% 2.40/1.46 file('/export/starexec/sandbox2/tmp/tmp.gN7przDacx/DTF2THF_23201.p',fact_25_le__neq__implies__less)).
% 2.40/1.46 thf(f1144,plain,(
% 2.40/1.46 ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true)),
% 2.40/1.46 inference(subsumption_resolution,[status(thm)],[f1102,f1100])).
% 2.40/1.46 thf(f1100,plain,(
% 2.40/1.46 ( ! [X0 : $tType,X2 : list @ X0,X3 : stream170649215stream @ X0,X1 : nat] : (((stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X2 @ X3) @ X1) = (nth @ X0 @ X2 @ X1)) | ($true != (ord_less @ nat @ X1 @ (size_size @ list @ X0 @ X2)))) )),
% 2.40/1.46 inference(cnf_transformation,[status(thm)],[f868])).
% 2.40/1.46 thf(f868,plain,(
% 2.40/1.46 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : stream170649215stream @ X0] : (($true != (ord_less @ nat @ X1 @ (size_size @ list @ X0 @ X2))) | ((stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X2 @ X3) @ X1) = (nth @ X0 @ X2 @ X1)))),
% 2.40/1.46 inference(ennf_transformation,[status(thm)],[f673])).
% 2.40/1.46 thf(f673,plain,(
% 2.40/1.46 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : stream170649215stream @ X0] : (($true = (ord_less @ nat @ X1 @ (size_size @ list @ X0 @ X2))) => ((stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X2 @ X3) @ X1) = (nth @ X0 @ X2 @ X1)))),
% 2.40/1.46 inference(fool_elimination,[status(thm)],[f672])).
% 2.40/1.46 thf(f672,plain,(
% 2.40/1.46 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : stream170649215stream @ X0] : ((ord_less @ nat @ X1 @ (size_size @ list @ X0 @ X2)) => ((stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X2 @ X3) @ X1) = (nth @ X0 @ X2 @ X1)))),
% 2.40/1.46 inference(rectify,[status(thm)],[f2])).
% 2.40/1.46 thf(f2,axiom,(
% 2.40/1.46 ! [X0 : $tType,X5 : nat,X2 : list @ X0,X6 : stream170649215stream @ X0] : ((ord_less @ nat @ X5 @ (size_size @ list @ X0 @ X2)) => ((stream370371455e_snth @ X0 @ (stream1035003186_shift @ X0 @ X2 @ X6) @ X5) = (nth @ X0 @ X2 @ X5)))),
% 2.40/1.46 file('/export/starexec/sandbox2/tmp/tmp.gN7przDacx/DTF2THF_23201.p',fact_1_shift__snth__less)).
% 2.40/1.46 thf(f1102,plain,(
% 2.40/1.46 ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true) | ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) != (nth @ a @ xs @ n))),
% 2.40/1.46 inference(cnf_transformation,[status(thm)],[f859])).
% 2.40/1.46 thf(f859,plain,(
% 2.40/1.46 (((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) != (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs)))) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true)) | (((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) != (nth @ a @ xs @ n)) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true))),
% 2.40/1.46 inference(ennf_transformation,[status(thm)],[f794])).
% 2.40/1.46 thf(f794,plain,(
% 2.40/1.46 ~((((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (nth @ a @ xs @ n))) & (((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) != $true) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs))))))),
% 2.40/1.46 inference(flattening,[status(thm)],[f301])).
% 2.40/1.46 thf(f301,plain,(
% 2.40/1.46 ~((~((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs))))) & (((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (nth @ a @ xs @ n))))),
% 2.40/1.46 inference(fool_elimination,[status(thm)],[f300])).
% 2.40/1.46 thf(f300,plain,(
% 2.40/1.46 ~((~(ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs))))) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (nth @ a @ xs @ n))))),
% 2.40/1.46 inference(rectify,[status(thm)],[f276])).
% 2.40/1.46 thf(f276,negated_conjecture,(
% 2.40/1.46 ~((~(ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs))))) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (nth @ a @ xs @ n))))),
% 2.40/1.46 inference(negated_conjecture,[status(cth)],[f275])).
% 2.40/1.46 thf(f275,conjecture,(
% 2.40/1.46 (~(ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs))))) & ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) => ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) = (nth @ a @ xs @ n)))),
% 2.40/1.46 file('/export/starexec/sandbox2/tmp/tmp.gN7przDacx/DTF2THF_23201.p',conj_0)).
% 2.40/1.46 thf(f1146,plain,(
% 2.40/1.46 ~spl30_2),
% 2.40/1.46 inference(avatar_split_clause,[status(thm)],[f1145,f1140])).
% 2.40/1.46 thf(f1145,plain,(
% 2.40/1.46 ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) != (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs))))),
% 2.40/1.46 inference(subsumption_resolution,[status(thm)],[f1103,f1144])).
% 2.40/1.46 thf(f1103,plain,(
% 2.40/1.46 ((ord_less @ nat @ n @ (size_size @ list @ a @ xs)) = $true) | ((stream370371455e_snth @ a @ (stream1035003186_shift @ a @ xs @ s) @ n) != (stream370371455e_snth @ a @ s @ (minus_minus @ nat @ n @ (size_size @ list @ a @ xs))))),
% 2.40/1.46 inference(cnf_transformation,[status(thm)],[f859])).
% 2.40/1.46 % SZS output end Proof for DTF2THF_23201
% 2.40/1.46 % (23307)------------------------------
% 2.40/1.46 % (23307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.46 % (23307)Termination reason: Refutation
% 2.40/1.46
% 2.40/1.46 % (23307)Memory used [KB]: 6780
% 2.40/1.46 % (23307)Time elapsed: 0.027 s
% 2.40/1.46 % (23307)Instructions burned: 79 (million)
% 2.40/1.46 % (23307)------------------------------
% 2.40/1.46 % (23307)------------------------------
% 2.40/1.46 % (23301)Success in time 0.043 s
%------------------------------------------------------------------------------