%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP104^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:35 AM UTC 2026
% Result : Theorem 2.94s 1.64s
% Output : Refutation 2.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP104^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n016.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:26:21 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : ITP104^2 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.46 % Domain : Interactive Theorem Proving
% 0.28/0.46 % Problem : Sledgehammer ListSlice problem prob_199__5618146_1
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Des21]
% 0.28/0.46 % Names : ListSlice/prob_199__5618146_1 [Des21]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.28/0.46 % Syntax : Number of formulae : 327 ( 70 unt; 44 typ; 0 def)
% 0.28/0.46 % Number of atoms : 920 ( 199 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 13 ( 3 avg)
% 0.28/0.46 % Number of connectives : 3495 ( 89 ~; 20 |; 52 &;2809 @)
% 0.28/0.46 % ( 0 <=>; 525 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 21 ( 8 avg)
% 0.28/0.46 % Number of types : 3 ( 2 usr)
% 0.28/0.46 % Number of type conns : 134 ( 134 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 43 ( 42 usr; 3 con; 0-4 aty)
% 0.28/0.46 % Number of variables : 1068 ( 72 ^; 912 !; 47 ?;1068 :)
% 0.28/0.46 % ( 37 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.46 % SPC : TH1_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:17:05.936
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % Could-be-implicit typings (4)
% 0.28/0.46 thf(ty_t_List_Olist,type,
% 0.28/0.46 list: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Set_Oset,type,
% 0.28/0.46 set: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Nat_Onat,type,
% 0.28/0.46 nat: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_tf_a,type,
% 0.28/0.46 a: $tType ).
% 0.28/0.46
% 0.28/0.46 % Explicit typings (40)
% 0.28/0.46 thf(sy_cl_HOL_Otype,type,
% 0.28/0.46 type:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Nat_Osize,type,
% 0.28/0.46 size:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.28/0.46 ord:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.28/0.46 top:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.28/0.46 order:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.28/0.46 no_bot:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.28/0.46 no_top:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.28/0.46 linorder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.28/0.46 preorder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.28/0.46 order_top:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.28/0.46 wellorder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.28/0.46 dense_order:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.28/0.46 dense_linorder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.28/0.46 condit1656338222tinuum:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.28/0.46 condit1037483654norder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List2_Of__image,type,
% 0.28/0.47 f_image:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List2_Olist__asc,type,
% 0.28/0.47 list_asc:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List2_Olist__desc,type,
% 0.28/0.47 list_desc:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List2_Olist__strict__asc,type,
% 0.28/0.47 list_strict_asc:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List2_Olist__strict__desc,type,
% 0.28/0.47 list_strict_desc:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice,type,
% 0.28/0.47 listSl1903280966_slice:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > nat > ( list @ ( list @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice2,type,
% 0.28/0.47 listSl776277612slice2:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > nat > ( list @ ( list @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice__aux,type,
% 0.28/0.47 listSl1123830174ce_aux:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > nat > nat > ( list @ ( list @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List_Obutlast,type,
% 0.28/0.47 butlast:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List_Odistinct,type,
% 0.28/0.47 distinct:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List_Olist_Oset,type,
% 0.28/0.47 set2:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List_Olist__ex,type,
% 0.28/0.47 list_ex:
% 0.28/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List_Olist__update,type,
% 0.28/0.47 list_update:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List_Onth,type,
% 0.28/0.47 nth:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_List_Ounion,type,
% 0.28/0.47 union:
% 0.28/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.28/0.47 size_size:
% 0.28/0.47 !>[A: $tType] : ( A > nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.28/0.47 ord_less:
% 0.28/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/0.47 ord_less_eq:
% 0.28/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.28/0.47 top_top:
% 0.28/0.47 !>[A: $tType] : A ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.28/0.47 divide_divide:
% 0.28/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_OCollect,type,
% 0.28/0.47 collect:
% 0.28/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_member,type,
% 0.28/0.47 member:
% 0.28/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_k,type,
% 0.28/0.47 k: nat ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_m,type,
% 0.28/0.47 m: nat ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_xs,type,
% 0.28/0.47 xs: list @ a ).
% 0.28/0.47
% 0.28/0.47 % Relevant facts (256)
% 0.28/0.47 thf(fact_0_nth__equalityI,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.28/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.28/0.47 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.28/0.47 => ( ! [I: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( nth @ A @ Xs @ I )
% 0.28/0.47 = ( nth @ A @ Ys @ I ) ) )
% 0.28/0.47 => ( Xs = Ys ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nth_equalityI
% 0.28/0.47 thf(fact_1_Skolem__list__nth,axiom,
% 0.28/0.47 ! [A: $tType,K: nat,P: nat > A > $o] :
% 0.28/0.47 ( ( ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ K )
% 0.28/0.47 => ? [X: A] : ( P @ I2 @ X ) ) )
% 0.28/0.47 = ( ? [Xs2: list @ A] :
% 0.28/0.47 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.28/0.47 = K )
% 0.28/0.47 & ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ K )
% 0.28/0.47 => ( P @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Skolem_list_nth
% 0.28/0.47 thf(fact_2_list__eq__iff__nth__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ^ [Y: list @ A,Z: list @ A] : ( Y = Z ) )
% 0.28/0.47 = ( ^ [Xs2: list @ A,Ys2: list @ A] :
% 0.28/0.47 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.28/0.47 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.28/0.47 & ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ( ( nth @ A @ Xs2 @ I2 )
% 0.28/0.47 = ( nth @ A @ Ys2 @ I2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_eq_iff_nth_eq
% 0.28/0.47 thf(fact_3_list__slice2__list__slice__nth,axiom,
% 0.28/0.47 ! [A: $tType,M: nat,Xs: list @ A,K: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) )
% 0.28/0.47 => ( ( nth @ ( list @ A ) @ ( listSl776277612slice2 @ A @ Xs @ K ) @ M )
% 0.28/0.47 = ( nth @ ( list @ A ) @ ( listSl1903280966_slice @ A @ Xs @ K ) @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice2_list_slice_nth
% 0.28/0.47 thf(fact_4_length__induct,axiom,
% 0.28/0.47 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.28/0.47 ( ! [Xs3: list @ A] :
% 0.28/0.47 ( ! [Ys3: list @ A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.28/0.47 => ( P @ Ys3 ) )
% 0.28/0.47 => ( P @ Xs3 ) )
% 0.28/0.47 => ( P @ Xs ) ) ).
% 0.28/0.47
% 0.28/0.47 % length_induct
% 0.28/0.47 thf(fact_5_list__strict__desc__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ( ( list_strict_desc @ A )
% 0.28/0.47 = ( ^ [Xs2: list @ A] :
% 0.28/0.47 ! [J: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_strict_desc_trans
% 0.28/0.47 thf(fact_6_list__slice__nth__length,axiom,
% 0.28/0.47 ! [A: $tType,M: nat,Xs: list @ A,K: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) )
% 0.28/0.47 => ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ ( listSl1903280966_slice @ A @ Xs @ K ) @ M ) )
% 0.28/0.47 = K ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice_nth_length
% 0.28/0.47 thf(fact_7_list__strict__asc__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ( ( list_strict_asc @ A )
% 0.28/0.47 = ( ^ [Xs2: list @ A] :
% 0.28/0.47 ! [J: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_strict_asc_trans
% 0.28/0.47 thf(fact_8_Ex__list__of__length,axiom,
% 0.28/0.47 ! [A: $tType,N: nat] :
% 0.28/0.47 ? [Xs3: list @ A] :
% 0.28/0.47 ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.28/0.47 = N ) ).
% 0.28/0.47
% 0.28/0.47 % Ex_list_of_length
% 0.28/0.47 thf(fact_9_neq__if__length__neq,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.28/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.28/0.47 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.28/0.47 => ( Xs != Ys ) ) ).
% 0.28/0.47
% 0.28/0.47 % neq_if_length_neq
% 0.28/0.47 thf(fact_10_size__neq__size__imp__neq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( size @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ( size_size @ A @ X2 )
% 0.28/0.47 != ( size_size @ A @ Y2 ) )
% 0.28/0.47 => ( X2 != Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % size_neq_size_imp_neq
% 0.28/0.47 thf(fact_11_nat__neq__iff,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( M != N )
% 0.28/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_neq_iff
% 0.28/0.47 thf(fact_12_list__slice__length,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,K: nat] :
% 0.28/0.47 ( ( size_size @ ( list @ ( list @ A ) ) @ ( listSl1903280966_slice @ A @ Xs @ K ) )
% 0.28/0.47 = ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice_length
% 0.28/0.47 thf(fact_13_measure__induct__rule,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( wellorder @ B )
% 0.28/0.47 => ! [F: A > B,P: A > $o,A2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ! [Y3: A] :
% 0.28/0.47 ( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
% 0.28/0.47 => ( P @ Y3 ) )
% 0.28/0.47 => ( P @ X3 ) )
% 0.28/0.47 => ( P @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % measure_induct_rule
% 0.28/0.47 thf(fact_14_measure__induct,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( wellorder @ B )
% 0.28/0.47 => ! [F: A > B,P: A > $o,A2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ! [Y3: A] :
% 0.28/0.47 ( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
% 0.28/0.47 => ( P @ Y3 ) )
% 0.28/0.47 => ( P @ X3 ) )
% 0.28/0.47 => ( P @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % measure_induct
% 0.28/0.47 thf(fact_15_infinite__descent__measure,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ~ ( P @ X3 )
% 0.28/0.47 => ? [Y3: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X3 ) )
% 0.28/0.47 & ~ ( P @ Y3 ) ) )
% 0.28/0.47 => ( P @ X2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinite_descent_measure
% 0.28/0.47 thf(fact_16_linorder__neqE__nat,axiom,
% 0.28/0.47 ! [X2: nat,Y2: nat] :
% 0.28/0.47 ( ( X2 != Y2 )
% 0.28/0.47 => ( ~ ( ord_less @ nat @ X2 @ Y2 )
% 0.28/0.47 => ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linorder_neqE_nat
% 0.28/0.47 thf(fact_17_infinite__descent,axiom,
% 0.28/0.47 ! [P: nat > $o,N: nat] :
% 0.28/0.47 ( ! [N2: nat] :
% 0.28/0.47 ( ~ ( P @ N2 )
% 0.28/0.47 => ? [M2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/0.47 & ~ ( P @ M2 ) ) )
% 0.28/0.47 => ( P @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinite_descent
% 0.28/0.47 thf(fact_18_nat__less__induct,axiom,
% 0.28/0.47 ! [P: nat > $o,N: nat] :
% 0.28/0.47 ( ! [N2: nat] :
% 0.28/0.47 ( ! [M2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/0.47 => ( P @ M2 ) )
% 0.28/0.47 => ( P @ N2 ) )
% 0.28/0.47 => ( P @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_less_induct
% 0.28/0.47 thf(fact_19_less__irrefl__nat,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.28/0.47
% 0.28/0.47 % less_irrefl_nat
% 0.28/0.47 thf(fact_20_less__not__refl3,axiom,
% 0.28/0.47 ! [S: nat,T: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ S @ T )
% 0.28/0.47 => ( S != T ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_not_refl3
% 0.28/0.47 thf(fact_21_less__not__refl2,axiom,
% 0.28/0.47 ! [N: nat,M: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ N @ M )
% 0.28/0.47 => ( M != N ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_not_refl2
% 0.28/0.47 thf(fact_22_less__not__refl,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.28/0.47
% 0.28/0.47 % less_not_refl
% 0.28/0.47 thf(fact_23_list__slice__def,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( listSl1903280966_slice @ A )
% 0.28/0.47 = ( ^ [Xs2: list @ A,K2: nat] : ( listSl1123830174ce_aux @ A @ Xs2 @ K2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ K2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice_def
% 0.28/0.47 thf(fact_24_list__strict__desc__imp__list__desc,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [Xs: list @ A] :
% 0.28/0.47 ( ( list_strict_desc @ A @ Xs )
% 0.28/0.47 => ( list_desc @ A @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_strict_desc_imp_list_desc
% 0.28/0.47 thf(fact_25_list__ex__length,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( list_ex @ A )
% 0.28/0.47 = ( ^ [P2: A > $o,Xs2: list @ A] :
% 0.28/0.47 ? [N3: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 & ( P2 @ ( nth @ A @ Xs2 @ N3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_ex_length
% 0.28/0.47 thf(fact_26_f__image__eqI,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Xs: list @ A,N: nat,A3: set @ nat] :
% 0.28/0.47 ( ( X2
% 0.28/0.47 = ( nth @ A @ Xs @ N ) )
% 0.28/0.47 => ( ( member @ nat @ N @ A3 )
% 0.28/0.47 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( member @ A @ X2 @ ( f_image @ A @ Xs @ A3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_image_eqI
% 0.28/0.47 thf(fact_27_list__strict__asc__imp__list__asc,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [Xs: list @ A] :
% 0.28/0.47 ( ( list_strict_asc @ A @ Xs )
% 0.28/0.47 => ( list_asc @ A @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_strict_asc_imp_list_asc
% 0.28/0.47 thf(fact_28_nth__list__update__eq,axiom,
% 0.28/0.47 ! [A: $tType,I3: nat,Xs: list @ A,X2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ I3 )
% 0.28/0.47 = X2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % nth_list_update_eq
% 0.28/0.47 thf(fact_29_rev__f__imageI,axiom,
% 0.28/0.47 ! [A: $tType,N: nat,A3: set @ nat,Xs: list @ A,X2: A] :
% 0.28/0.47 ( ( member @ nat @ N @ A3 )
% 0.28/0.47 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( X2
% 0.28/0.47 = ( nth @ A @ Xs @ N ) )
% 0.28/0.47 => ( member @ A @ X2 @ ( f_image @ A @ Xs @ A3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_f_imageI
% 0.28/0.47 thf(fact_30_f__image__iff,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Xs: list @ A,A3: set @ nat] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( f_image @ A @ Xs @ A3 ) )
% 0.28/0.47 = ( ? [X4: nat] :
% 0.28/0.47 ( ( member @ nat @ X4 @ A3 )
% 0.28/0.47 & ( ord_less @ nat @ X4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 & ( X2
% 0.28/0.47 = ( nth @ A @ Xs @ X4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_image_iff
% 0.28/0.47 thf(fact_31_nth__butlast,axiom,
% 0.28/0.47 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.28/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
% 0.28/0.47 => ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
% 0.28/0.47 = ( nth @ A @ Xs @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nth_butlast
% 0.28/0.47 thf(fact_32_f__imageI,axiom,
% 0.28/0.47 ! [A: $tType,N: nat,A3: set @ nat,Xs: list @ A] :
% 0.28/0.47 ( ( member @ nat @ N @ A3 )
% 0.28/0.47 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( member @ A @ ( nth @ A @ Xs @ N ) @ ( f_image @ A @ Xs @ A3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_imageI
% 0.28/0.47 thf(fact_33_f__imageE,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Xs: list @ A,A3: set @ nat] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( f_image @ A @ Xs @ A3 ) )
% 0.28/0.47 => ~ ! [N2: nat] :
% 0.28/0.47 ( ( X2
% 0.28/0.47 = ( nth @ A @ Xs @ N2 ) )
% 0.28/0.47 => ( ( member @ nat @ N2 @ A3 )
% 0.28/0.47 => ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_imageE
% 0.28/0.47 thf(fact_34_list__update__overwrite,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,I3: nat,X2: A,Y2: A] :
% 0.28/0.47 ( ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ I3 @ Y2 )
% 0.28/0.47 = ( list_update @ A @ Xs @ I3 @ Y2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_update_overwrite
% 0.28/0.47 thf(fact_35_length__list__update,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,I3: nat,X2: A] :
% 0.28/0.47 ( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I3 @ X2 ) )
% 0.28/0.47 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.28/0.47
% 0.28/0.47 % length_list_update
% 0.28/0.47 thf(fact_36_list__update__id,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,I3: nat] :
% 0.28/0.47 ( ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ I3 ) )
% 0.28/0.47 = Xs ) ).
% 0.28/0.47
% 0.28/0.47 % list_update_id
% 0.28/0.47 thf(fact_37_nth__list__update__neq,axiom,
% 0.28/0.47 ! [A: $tType,I3: nat,J2: nat,Xs: list @ A,X2: A] :
% 0.28/0.47 ( ( I3 != J2 )
% 0.28/0.47 => ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ J2 )
% 0.28/0.47 = ( nth @ A @ Xs @ J2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nth_list_update_neq
% 0.28/0.47 thf(fact_38_list__update__swap,axiom,
% 0.28/0.47 ! [A: $tType,I3: nat,I4: nat,Xs: list @ A,X2: A,X5: A] :
% 0.28/0.47 ( ( I3 != I4 )
% 0.28/0.47 => ( ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ I4 @ X5 )
% 0.28/0.47 = ( list_update @ A @ ( list_update @ A @ Xs @ I4 @ X5 ) @ I3 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_update_swap
% 0.28/0.47 thf(fact_39_list__slice__aux__length,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,K: nat,N: nat] :
% 0.28/0.47 ( ( size_size @ ( list @ ( list @ A ) ) @ ( listSl1123830174ce_aux @ A @ Xs @ K @ N ) )
% 0.28/0.47 = N ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice_aux_length
% 0.28/0.47 thf(fact_40_nth__list__update,axiom,
% 0.28/0.47 ! [A: $tType,I3: nat,Xs: list @ A,J2: nat,X2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( ( I3 = J2 )
% 0.28/0.47 => ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ J2 )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 & ( ( I3 != J2 )
% 0.28/0.47 => ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ J2 )
% 0.28/0.47 = ( nth @ A @ Xs @ J2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nth_list_update
% 0.28/0.47 thf(fact_41_list__update__same__conv,axiom,
% 0.28/0.47 ! [A: $tType,I3: nat,Xs: list @ A,X2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( ( list_update @ A @ Xs @ I3 @ X2 )
% 0.28/0.47 = Xs )
% 0.28/0.47 = ( ( nth @ A @ Xs @ I3 )
% 0.28/0.47 = X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_update_same_conv
% 0.28/0.47 thf(fact_42_set__swap,axiom,
% 0.28/0.47 ! [A: $tType,I3: nat,Xs: list @ A,J2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ J2 ) ) @ J2 @ ( nth @ A @ Xs @ I3 ) ) )
% 0.28/0.47 = ( set2 @ A @ Xs ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % set_swap
% 0.28/0.47 thf(fact_43_list__desc__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ( ( list_desc @ A )
% 0.28/0.47 = ( ^ [Xs2: list @ A] :
% 0.28/0.47 ! [J: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_desc_trans
% 0.28/0.47 thf(fact_44_list__asc__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ( ( list_asc @ A )
% 0.28/0.47 = ( ^ [Xs2: list @ A] :
% 0.28/0.47 ! [J: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_asc_trans
% 0.28/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.28/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.28/0.47 = ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_Collect_eq
% 0.28/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( collect @ A
% 0.28/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.28/0.47 = A3 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mem_eq
% 0.28/0.47 thf(fact_47_Collect__cong,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( P @ X3 )
% 0.28/0.47 = ( Q @ X3 ) )
% 0.28/0.47 => ( ( collect @ A @ P )
% 0.28/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_cong
% 0.28/0.47 thf(fact_48_ext,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( F @ X3 )
% 0.28/0.47 = ( G @ X3 ) )
% 0.28/0.47 => ( F = G ) ) ).
% 0.28/0.47
% 0.28/0.47 % ext
% 0.28/0.47 thf(fact_49_distinct__swap,axiom,
% 0.28/0.47 ! [A: $tType,I3: nat,Xs: list @ A,J2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ J2 ) ) @ J2 @ ( nth @ A @ Xs @ I3 ) ) )
% 0.28/0.47 = ( distinct @ A @ Xs ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % distinct_swap
% 0.28/0.47 thf(fact_50_f__rangeE,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Xs: list @ A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( f_image @ A @ Xs @ ( top_top @ ( set @ nat ) ) ) )
% 0.28/0.47 => ~ ! [N2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( X2
% 0.28/0.47 != ( nth @ A @ Xs @ N2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_rangeE
% 0.28/0.47 thf(fact_51_f__rangeI,axiom,
% 0.28/0.47 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.28/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( member @ A @ ( nth @ A @ Xs @ N ) @ ( f_image @ A @ Xs @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_rangeI
% 0.28/0.47 thf(fact_52_f__range__eqI,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Xs: list @ A,N: nat] :
% 0.28/0.47 ( ( X2
% 0.28/0.47 = ( nth @ A @ Xs @ N ) )
% 0.28/0.47 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( member @ A @ X2 @ ( f_image @ A @ Xs @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_range_eqI
% 0.28/0.47 thf(fact_53_f__image__subsetI,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ nat,Xs: list @ A,B2: set @ A] :
% 0.28/0.47 ( ! [N2: nat] :
% 0.28/0.47 ( ( ( member @ nat @ N2 @ A3 )
% 0.28/0.47 & ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 0.28/0.47 => ( member @ A @ ( nth @ A @ Xs @ N2 ) @ B2 ) )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( f_image @ A @ Xs @ A3 ) @ B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_image_subsetI
% 0.28/0.47 thf(fact_54_f__image__subset__iff,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,A3: set @ nat,B2: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( f_image @ A @ Xs @ A3 ) @ B2 )
% 0.28/0.47 = ( ! [X4: nat] :
% 0.28/0.47 ( ( member @ nat @ X4 @ A3 )
% 0.28/0.47 => ( ( ord_less @ nat @ X4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( member @ A @ ( nth @ A @ Xs @ X4 ) @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_image_subset_iff
% 0.28/0.47 thf(fact_55_list__update__beyond,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,I3: nat,X2: A] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I3 )
% 0.28/0.47 => ( ( list_update @ A @ Xs @ I3 @ X2 )
% 0.28/0.47 = Xs ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_update_beyond
% 0.28/0.47 thf(fact_56_set__update__subsetI,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,A3: set @ A,X2: A,I3: nat] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
% 0.28/0.47 => ( ( member @ A @ X2 @ A3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) ) @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % set_update_subsetI
% 0.28/0.47 thf(fact_57_subset__code_I1_J,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,B2: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
% 0.28/0.47 = ( ! [X4: A] :
% 0.28/0.47 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.28/0.47 => ( member @ A @ X4 @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_code(1)
% 0.28/0.47 thf(fact_58_f__range__eq__set,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A] :
% 0.28/0.47 ( ( f_image @ A @ Xs @ ( top_top @ ( set @ nat ) ) )
% 0.28/0.47 = ( set2 @ A @ Xs ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_range_eq_set
% 0.28/0.47 thf(fact_59_subset__f__image__iff,axiom,
% 0.28/0.47 ! [A: $tType,B2: set @ A,Xs: list @ A,A3: set @ nat] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( f_image @ A @ Xs @ A3 ) )
% 0.28/0.47 = ( ? [A4: set @ nat] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ nat ) @ A4 @ A3 )
% 0.28/0.47 & ( B2
% 0.28/0.47 = ( f_image @ A @ Xs @ A4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_f_image_iff
% 0.28/0.47 thf(fact_60_f__image__mono,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ nat,B2: set @ nat,Xs: list @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ nat ) @ A3 @ B2 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( f_image @ A @ Xs @ A3 ) @ ( f_image @ A @ Xs @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_image_mono
% 0.28/0.47 thf(fact_61_less__mono__imp__le__mono,axiom,
% 0.28/0.47 ! [F: nat > nat,I3: nat,J2: nat] :
% 0.28/0.47 ( ! [I: nat,J3: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I @ J3 )
% 0.28/0.47 => ( ord_less @ nat @ ( F @ I ) @ ( F @ J3 ) ) )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ I3 @ J2 )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_mono_imp_le_mono
% 0.28/0.47 thf(fact_62_le__neq__implies__less,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ( M != N )
% 0.28/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_neq_implies_less
% 0.28/0.47 thf(fact_63_less__or__eq__imp__le,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 | ( M = N ) )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_or_eq_imp_le
% 0.28/0.47 thf(fact_64_le__eq__less__or__eq,axiom,
% 0.28/0.47 ( ( ord_less_eq @ nat )
% 0.28/0.47 = ( ^ [M3: nat,N3: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.28/0.47 | ( M3 = N3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_eq_less_or_eq
% 0.28/0.47 thf(fact_65_less__imp__le__nat,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_le_nat
% 0.28/0.47 thf(fact_66_nat__less__le,axiom,
% 0.28/0.47 ( ( ord_less @ nat )
% 0.28/0.47 = ( ^ [M3: nat,N3: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 0.28/0.47 & ( M3 != N3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_less_le
% 0.28/0.47 thf(fact_67_distinct__butlast,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A] :
% 0.28/0.47 ( ( distinct @ A @ Xs )
% 0.28/0.47 => ( distinct @ A @ ( butlast @ A @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % distinct_butlast
% 0.28/0.47 thf(fact_68_in__set__butlastD,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Xs: list @ A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
% 0.28/0.47 => ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % in_set_butlastD
% 0.28/0.47 thf(fact_69_distinct__Ex1,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,X2: A] :
% 0.28/0.47 ( ( distinct @ A @ Xs )
% 0.28/0.47 => ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.28/0.47 => ? [X3: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 & ( ( nth @ A @ Xs @ X3 )
% 0.28/0.47 = X2 )
% 0.28/0.47 & ! [Y3: nat] :
% 0.28/0.47 ( ( ( ord_less @ nat @ Y3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 & ( ( nth @ A @ Xs @ Y3 )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 => ( Y3 = X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % distinct_Ex1
% 0.28/0.47 thf(fact_70_list__ex__cong,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,F: A > $o,G: A > $o] :
% 0.28/0.47 ( ( Xs = Ys )
% 0.28/0.47 => ( ! [X3: A] :
% 0.28/0.47 ( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
% 0.28/0.47 => ( ( F @ X3 )
% 0.28/0.47 = ( G @ X3 ) ) )
% 0.28/0.47 => ( ( list_ex @ A @ F @ Xs )
% 0.28/0.47 = ( list_ex @ A @ G @ Ys ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_ex_cong
% 0.28/0.47 thf(fact_71_list__strict__asc__distinct,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [Xs: list @ A] :
% 0.28/0.47 ( ( list_strict_asc @ A @ Xs )
% 0.28/0.47 => ( distinct @ A @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_strict_asc_distinct
% 0.28/0.47 thf(fact_72_list__strict__desc__distinct,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [Xs: list @ A] :
% 0.28/0.47 ( ( list_strict_desc @ A @ Xs )
% 0.28/0.47 => ( distinct @ A @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_strict_desc_distinct
% 0.28/0.47 thf(fact_73_list__strict__asc__trans__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [Xs: list @ A] :
% 0.28/0.47 ( ( list_strict_asc @ A @ Xs )
% 0.28/0.47 => ! [J4: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ! [I5: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I5 @ J4 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_strict_asc_trans_le
% 0.28/0.47 thf(fact_74_list__asc__trans__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ( ( list_asc @ A )
% 0.28/0.47 = ( ^ [Xs2: list @ A] :
% 0.28/0.47 ! [J: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ! [I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_asc_trans_le
% 0.28/0.47 thf(fact_75_list__desc__trans__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ( ( list_desc @ A )
% 0.28/0.47 = ( ^ [Xs2: list @ A] :
% 0.28/0.47 ! [J: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ! [I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_desc_trans_le
% 0.28/0.47 thf(fact_76_nth__eq__iff__index__eq,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,I3: nat,J2: nat] :
% 0.28/0.47 ( ( distinct @ A @ Xs )
% 0.28/0.47 => ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ( ( nth @ A @ Xs @ I3 )
% 0.28/0.47 = ( nth @ A @ Xs @ J2 ) )
% 0.28/0.47 = ( I3 = J2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nth_eq_iff_index_eq
% 0.28/0.47 thf(fact_77_distinct__conv__nth,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( distinct @ A )
% 0.28/0.47 = ( ^ [Xs2: list @ A] :
% 0.28/0.47 ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ! [J: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.47 => ( ( I2 != J )
% 0.28/0.47 => ( ( nth @ A @ Xs2 @ I2 )
% 0.28/0.47 != ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % distinct_conv_nth
% 0.28/0.47 thf(fact_78_all__set__conv__all__nth,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.28/0.47 ( ( ! [X4: A] :
% 0.28/0.47 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.28/0.47 => ( P @ X4 ) ) )
% 0.28/0.47 = ( ! [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( P @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % all_set_conv_all_nth
% 0.28/0.47 thf(fact_79_all__nth__imp__all__set,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,P: A > $o,X2: A] :
% 0.28/0.47 ( ! [I: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( P @ ( nth @ A @ Xs @ I ) ) )
% 0.28/0.47 => ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.28/0.47 => ( P @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % all_nth_imp_all_set
% 0.28/0.47 thf(fact_80_in__set__conv__nth,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Xs: list @ A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.28/0.47 = ( ? [I2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 & ( ( nth @ A @ Xs @ I2 )
% 0.28/0.47 = X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % in_set_conv_nth
% 0.28/0.47 thf(fact_81_list__ball__nth,axiom,
% 0.28/0.47 ! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
% 0.28/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( ! [X3: A] :
% 0.28/0.47 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.28/0.47 => ( P @ X3 ) )
% 0.28/0.47 => ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_ball_nth
% 0.28/0.47 thf(fact_82_nth__mem,axiom,
% 0.28/0.47 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.28/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nth_mem
% 0.28/0.47 thf(fact_83_set__update__memI,axiom,
% 0.28/0.47 ! [A: $tType,N: nat,Xs: list @ A,X2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % set_update_memI
% 0.28/0.47 thf(fact_84_f__image__eq__set,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,A3: set @ nat] :
% 0.28/0.47 ( ! [N2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.47 => ( member @ nat @ N2 @ A3 ) )
% 0.28/0.47 => ( ( f_image @ A @ Xs @ A3 )
% 0.28/0.47 = ( set2 @ A @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f_image_eq_set
% 0.28/0.47 thf(fact_85_subsetI,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ( member @ A @ X3 @ B2 ) )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % subsetI
% 0.28/0.47 thf(fact_86_subset__antisym,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.28/0.47 => ( A3 = B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_antisym
% 0.28/0.47 thf(fact_87_psubsetI,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( A3 != B2 )
% 0.28/0.47 => ( ord_less @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubsetI
% 0.28/0.47 thf(fact_88_UNIV__I,axiom,
% 0.28/0.47 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % UNIV_I
% 0.28/0.47 thf(fact_89_iso__tuple__UNIV__I,axiom,
% 0.28/0.47 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % iso_tuple_UNIV_I
% 0.28/0.47 thf(fact_90_top__apply,axiom,
% 0.28/0.47 ! [C: $tType,D: $tType] :
% 0.28/0.47 ( ( top @ C )
% 0.28/0.47 => ( ( top_top @ ( D > C ) )
% 0.28/0.47 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % top_apply
% 0.28/0.47 thf(fact_91_distinct__union,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.28/0.47 ( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
% 0.28/0.47 = ( distinct @ A @ Ys ) ) ).
% 0.28/0.47
% 0.28/0.47 % distinct_union
% 0.28/0.47 thf(fact_92_order__refl,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_refl
% 0.28/0.47 thf(fact_93_top__set__def,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( top_top @ ( set @ A ) )
% 0.28/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % top_set_def
% 0.28/0.47 thf(fact_94_Nat_Oex__has__greatest__nat,axiom,
% 0.28/0.47 ! [P: nat > $o,K: nat,B3: nat] :
% 0.28/0.47 ( ( P @ K )
% 0.28/0.47 => ( ! [Y4: nat] :
% 0.28/0.47 ( ( P @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ nat @ Y4 @ B3 ) )
% 0.28/0.47 => ? [X3: nat] :
% 0.28/0.47 ( ( P @ X3 )
% 0.28/0.47 & ! [Y3: nat] :
% 0.28/0.47 ( ( P @ Y3 )
% 0.28/0.47 => ( ord_less_eq @ nat @ Y3 @ X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.ex_has_greatest_nat
% 0.28/0.47 thf(fact_95_nat__le__linear,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_le_linear
% 0.28/0.47 thf(fact_96_le__antisym,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.28/0.47 => ( M = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_antisym
% 0.28/0.47 thf(fact_97_eq__imp__le,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( M = N )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_imp_le
% 0.28/0.47 thf(fact_98_le__trans,axiom,
% 0.28/0.47 ! [I3: nat,J2: nat,K: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I3 @ J2 )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ J2 @ K )
% 0.28/0.47 => ( ord_less_eq @ nat @ I3 @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_trans
% 0.28/0.47 thf(fact_99_le__refl,axiom,
% 0.28/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.28/0.47
% 0.28/0.47 % le_refl
% 0.28/0.47 thf(fact_100_psubsetD,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
% 0.28/0.47 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( member @ A @ C2 @ A3 )
% 0.28/0.47 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubsetD
% 0.28/0.47 thf(fact_101_psubset__trans,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.47 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.28/0.47 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_trans
% 0.28/0.47 thf(fact_102_dual__order_Oantisym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B3: A,A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( A2 = B3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.antisym
% 0.28/0.47 thf(fact_103_dual__order_Oeq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.28/0.47 = ( ^ [A5: A,B4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.28/0.47 & ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.eq_iff
% 0.28/0.47 thf(fact_104_dual__order_Otrans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B3: A,A2: A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.28/0.47 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.trans
% 0.28/0.47 thf(fact_105_linorder__wlog,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [P: A > A > $o,A2: A,B3: A] :
% 0.28/0.47 ( ! [A6: A,B5: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.28/0.47 => ( P @ A6 @ B5 ) )
% 0.28/0.47 => ( ! [A6: A,B5: A] :
% 0.28/0.47 ( ( P @ B5 @ A6 )
% 0.28/0.47 => ( P @ A6 @ B5 ) )
% 0.28/0.47 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linorder_wlog
% 0.28/0.47 thf(fact_106_dual__order_Orefl,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.refl
% 0.28/0.47 thf(fact_107_order__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A,Z2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.28/0.47 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_trans
% 0.28/0.47 thf(fact_108_order__class_Oorder_Oantisym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.28/0.47 => ( A2 = B3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_class.order.antisym
% 0.28/0.47 thf(fact_109_ord__le__eq__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord @ A )
% 0.28/0.47 => ! [A2: A,B3: A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( B3 = C2 )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_le_eq_trans
% 0.28/0.47 thf(fact_110_ord__eq__le__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord @ A )
% 0.28/0.47 => ! [A2: A,B3: A,C2: A] :
% 0.28/0.47 ( ( A2 = B3 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_eq_le_trans
% 0.28/0.47 thf(fact_111_order__class_Oorder_Oeq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.28/0.47 = ( ^ [A5: A,B4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.28/0.47 & ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_class.order.eq_iff
% 0.28/0.47 thf(fact_112_antisym__conv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [Y2: A,X2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 = ( X2 = Y2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % antisym_conv
% 0.28/0.47 thf(fact_113_le__cases3,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A,Z2: A] :
% 0.28/0.47 ( ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.28/0.47 => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.28/0.47 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.28/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
% 0.28/0.47 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 0.28/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_cases3
% 0.28/0.47 thf(fact_114_order_Otrans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.trans
% 0.28/0.47 thf(fact_115_le__cases,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_cases
% 0.28/0.47 thf(fact_116_eq__refl,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( X2 = Y2 )
% 0.28/0.47 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_refl
% 0.28/0.47 thf(fact_117_linear,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linear
% 0.28/0.47 thf(fact_118_antisym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.28/0.47 => ( X2 = Y2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % antisym
% 0.28/0.47 thf(fact_119_eq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.28/0.47 = ( ^ [X4: A,Y5: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.28/0.47 & ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_iff
% 0.28/0.47 thf(fact_120_ord__le__eq__subst,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( ord @ B )
% 0.28/0.47 & ( ord @ A ) )
% 0.28/0.47 => ! [A2: A,B3: A,F: A > B,C2: B] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ( F @ B3 )
% 0.28/0.47 = C2 )
% 0.28/0.47 => ( ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_le_eq_subst
% 0.28/0.47 thf(fact_121_ord__eq__le__subst,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( ord @ B )
% 0.28/0.47 & ( ord @ A ) )
% 0.28/0.47 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 = ( F @ B3 ) )
% 0.28/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.28/0.47 => ( ! [X3: B,Y4: B] :
% 0.28/0.47 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_eq_le_subst
% 0.28/0.47 thf(fact_122_order__subst2,axiom,
% 0.28/0.47 ! [A: $tType,C: $tType] :
% 0.28/0.47 ( ( ( order @ C )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,B3: A,F: A > C,C2: C] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.28/0.47 => ( ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_subst2
% 0.28/0.47 thf(fact_123_order__subst1,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( order @ B )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 0.28/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.28/0.47 => ( ! [X3: B,Y4: B] :
% 0.28/0.47 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_subst1
% 0.28/0.47 thf(fact_124_le__fun__def,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.28/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.28/0.47 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_fun_def
% 0.28/0.47 thf(fact_125_le__funI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ! [F: A > B,G: A > B] :
% 0.28/0.47 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.28/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_funI
% 0.28/0.47 thf(fact_126_le__funE,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ! [F: A > B,G: A > B,X2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.28/0.47 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_funE
% 0.28/0.47 thf(fact_127_le__funD,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ! [F: A > B,G: A > B,X2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.28/0.47 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_funD
% 0.28/0.47 thf(fact_128_ord__eq__less__subst,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( ord @ B )
% 0.28/0.47 & ( ord @ A ) )
% 0.28/0.47 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 = ( F @ B3 ) )
% 0.28/0.47 => ( ( ord_less @ B @ B3 @ C2 )
% 0.28/0.47 => ( ! [X3: B,Y4: B] :
% 0.28/0.47 ( ( ord_less @ B @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_eq_less_subst
% 0.28/0.47 thf(fact_129_ord__less__eq__subst,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( ord @ B )
% 0.28/0.47 & ( ord @ A ) )
% 0.28/0.47 => ! [A2: A,B3: A,F: A > B,C2: B] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ( F @ B3 )
% 0.28/0.47 = C2 )
% 0.28/0.47 => ( ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_less_eq_subst
% 0.28/0.47 thf(fact_130_order__less__subst1,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( order @ B )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 0.28/0.47 => ( ( ord_less @ B @ B3 @ C2 )
% 0.28/0.47 => ( ! [X3: B,Y4: B] :
% 0.28/0.47 ( ( ord_less @ B @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_less_subst1
% 0.28/0.47 thf(fact_131_order__less__subst2,axiom,
% 0.28/0.47 ! [A: $tType,C: $tType] :
% 0.28/0.47 ( ( ( order @ C )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,B3: A,F: A > C,C2: C] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 0.28/0.47 => ( ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_less_subst2
% 0.28/0.47 thf(fact_132_lt__ex,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( no_bot @ A )
% 0.28/0.47 => ! [X2: A] :
% 0.28/0.47 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % lt_ex
% 0.28/0.47 thf(fact_133_gt__ex,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( no_top @ A )
% 0.28/0.47 => ! [X2: A] :
% 0.28/0.47 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 0.28/0.47
% 0.28/0.47 % gt_ex
% 0.28/0.47 thf(fact_134_neqE,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( X2 != Y2 )
% 0.28/0.47 => ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % neqE
% 0.28/0.47 thf(fact_135_neq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( X2 != Y2 )
% 0.28/0.47 = ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 | ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % neq_iff
% 0.28/0.47 thf(fact_136_order_Oasym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.asym
% 0.28/0.47 thf(fact_137_dense,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( dense_order @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ? [Z3: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Z3 )
% 0.28/0.47 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dense
% 0.28/0.47 thf(fact_138_less__imp__neq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( X2 != Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_neq
% 0.28/0.47 thf(fact_139_less__asym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_asym
% 0.28/0.47 thf(fact_140_less__asym_H,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [A2: A,B3: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_asym'
% 0.28/0.47 thf(fact_141_less__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A,Z2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.28/0.47 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_trans
% 0.28/0.47 thf(fact_142_less__linear,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 | ( X2 = Y2 )
% 0.28/0.47 | ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_linear
% 0.28/0.47 thf(fact_143_less__irrefl,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A] :
% 0.28/0.47 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_irrefl
% 0.28/0.47 thf(fact_144_ord__eq__less__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord @ A )
% 0.28/0.47 => ! [A2: A,B3: A,C2: A] :
% 0.28/0.47 ( ( A2 = B3 )
% 0.28/0.47 => ( ( ord_less @ A @ B3 @ C2 )
% 0.28/0.47 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_eq_less_trans
% 0.28/0.47 thf(fact_145_ord__less__eq__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord @ A )
% 0.28/0.47 => ! [A2: A,B3: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( B3 = C2 )
% 0.28/0.47 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ord_less_eq_trans
% 0.28/0.47 thf(fact_146_dual__order_Oasym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B3: A,A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.28/0.47 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.asym
% 0.28/0.47 thf(fact_147_less__imp__not__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( X2 != Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_not_eq
% 0.28/0.47 thf(fact_148_less__not__sym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_not_sym
% 0.28/0.47 thf(fact_149_less__induct,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( wellorder @ A )
% 0.28/0.47 => ! [P: A > $o,A2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ! [Y3: A] :
% 0.28/0.47 ( ( ord_less @ A @ Y3 @ X3 )
% 0.28/0.47 => ( P @ Y3 ) )
% 0.28/0.47 => ( P @ X3 ) )
% 0.28/0.47 => ( P @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_induct
% 0.28/0.47 thf(fact_150_antisym__conv3,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [Y2: A,X2: A] :
% 0.28/0.47 ( ~ ( ord_less @ A @ Y2 @ X2 )
% 0.28/0.47 => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 0.28/0.47 = ( X2 = Y2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % antisym_conv3
% 0.28/0.47 thf(fact_151_less__imp__not__eq2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( Y2 != X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_not_eq2
% 0.28/0.47 thf(fact_152_less__imp__triv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A,P: $o] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ord_less @ A @ Y2 @ X2 )
% 0.28/0.47 => P ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_triv
% 0.28/0.47 thf(fact_153_linorder__cases,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( X2 != Y2 )
% 0.28/0.47 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linorder_cases
% 0.28/0.47 thf(fact_154_dual__order_Oirrefl,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.irrefl
% 0.28/0.47 thf(fact_155_order_Ostrict__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less @ A @ B3 @ C2 )
% 0.28/0.47 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.strict_trans
% 0.28/0.47 thf(fact_156_less__imp__not__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_not_less
% 0.28/0.47 thf(fact_157_exists__least__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( wellorder @ A )
% 0.28/0.47 => ( ( ^ [P3: A > $o] :
% 0.28/0.47 ? [X: A] : ( P3 @ X ) )
% 0.28/0.47 = ( ^ [P2: A > $o] :
% 0.28/0.47 ? [N3: A] :
% 0.28/0.47 ( ( P2 @ N3 )
% 0.28/0.47 & ! [M3: A] :
% 0.28/0.47 ( ( ord_less @ A @ M3 @ N3 )
% 0.28/0.47 => ~ ( P2 @ M3 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % exists_least_iff
% 0.28/0.47 thf(fact_158_linorder__less__wlog,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [P: A > A > $o,A2: A,B3: A] :
% 0.28/0.47 ( ! [A6: A,B5: A] :
% 0.28/0.47 ( ( ord_less @ A @ A6 @ B5 )
% 0.28/0.47 => ( P @ A6 @ B5 ) )
% 0.28/0.47 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 0.28/0.47 => ( ! [A6: A,B5: A] :
% 0.28/0.47 ( ( P @ B5 @ A6 )
% 0.28/0.47 => ( P @ A6 @ B5 ) )
% 0.28/0.47 => ( P @ A2 @ B3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linorder_less_wlog
% 0.28/0.47 thf(fact_159_dual__order_Ostrict__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B3: A,A2: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ C2 @ B3 )
% 0.28/0.47 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.strict_trans
% 0.28/0.47 thf(fact_160_not__less__iff__gr__or__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 0.28/0.47 = ( ( ord_less @ A @ Y2 @ X2 )
% 0.28/0.47 | ( X2 = Y2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_less_iff_gr_or_eq
% 0.28/0.47 thf(fact_161_order_Ostrict__implies__not__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( A2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.strict_implies_not_eq
% 0.28/0.47 thf(fact_162_dual__order_Ostrict__implies__not__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B3: A,A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.28/0.47 => ( A2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.strict_implies_not_eq
% 0.28/0.47 thf(fact_163_UNIV__witness,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % UNIV_witness
% 0.28/0.47 thf(fact_164_UNIV__eq__I,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ! [X3: A] : ( member @ A @ X3 @ A3 )
% 0.28/0.47 => ( ( top_top @ ( set @ A ) )
% 0.28/0.47 = A3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % UNIV_eq_I
% 0.28/0.47 thf(fact_165_psubsetE,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.47 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubsetE
% 0.28/0.47 thf(fact_166_psubset__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord_less @ ( set @ A ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.28/0.47 & ( A7 != B6 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_eq
% 0.28/0.47 thf(fact_167_psubset__imp__subset,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.47 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_imp_subset
% 0.28/0.47 thf(fact_168_psubset__subset__trans,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.47 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.28/0.47 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_subset_trans
% 0.28/0.47 thf(fact_169_subset__not__subset__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord_less @ ( set @ A ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.28/0.47 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_not_subset_eq
% 0.28/0.47 thf(fact_170_subset__psubset__trans,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.28/0.47 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_psubset_trans
% 0.28/0.47 thf(fact_171_subset__iff__psubset__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 0.28/0.47 | ( A7 = B6 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_iff_psubset_eq
% 0.28/0.47 thf(fact_172_Collect__mono__iff,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.28/0.47 = ( ! [X4: A] :
% 0.28/0.47 ( ( P @ X4 )
% 0.28/0.47 => ( Q @ X4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mono_iff
% 0.28/0.47 thf(fact_173_set__eq__subset,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.28/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % set_eq_subset
% 0.28/0.47 thf(fact_174_subset__trans,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_trans
% 0.28/0.47 thf(fact_175_Collect__mono,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( P @ X3 )
% 0.28/0.47 => ( Q @ X3 ) )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mono
% 0.28/0.47 thf(fact_176_subset__refl,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.28/0.47
% 0.28/0.47 % subset_refl
% 0.28/0.47 thf(fact_177_subset__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ! [T2: A] :
% 0.28/0.47 ( ( member @ A @ T2 @ A7 )
% 0.28/0.47 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_iff
% 0.28/0.47 thf(fact_178_equalityD2,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.47 ( ( A3 = B2 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % equalityD2
% 0.28/0.47 thf(fact_179_equalityD1,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.47 ( ( A3 = B2 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % equalityD1
% 0.28/0.47 thf(fact_180_subset__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.28/0.47 ! [X4: A] :
% 0.28/0.47 ( ( member @ A @ X4 @ A7 )
% 0.28/0.47 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_eq
% 0.28/0.47 thf(fact_181_equalityE,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.47 ( ( A3 = B2 )
% 0.28/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % equalityE
% 0.28/0.47 thf(fact_182_subsetD,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( member @ A @ C2 @ A3 )
% 0.28/0.47 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subsetD
% 0.28/0.47 thf(fact_183_in__mono,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,X2: A] :
% 0.28/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.47 => ( ( member @ A @ X2 @ A3 )
% 0.28/0.47 => ( member @ A @ X2 @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % in_mono
% 0.28/0.47 thf(fact_184_order_Onot__eq__order__implies__strict,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A] :
% 0.28/0.47 ( ( A2 != B3 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.not_eq_order_implies_strict
% 0.28/0.47 thf(fact_185_dual__order_Ostrict__implies__order,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B3: A,A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.28/0.47 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.strict_implies_order
% 0.28/0.47 thf(fact_186_dual__order_Ostrict__iff__order,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ord_less @ A )
% 0.28/0.47 = ( ^ [B4: A,A5: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.28/0.47 & ( A5 != B4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.strict_iff_order
% 0.28/0.47 thf(fact_187_dual__order_Oorder__iff__strict,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [B4: A,A5: A] :
% 0.28/0.47 ( ( ord_less @ A @ B4 @ A5 )
% 0.28/0.47 | ( A5 = B4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.order_iff_strict
% 0.28/0.47 thf(fact_188_order_Ostrict__implies__order,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.strict_implies_order
% 0.28/0.47 thf(fact_189_dense__le__bounded,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( dense_linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A,Z2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ! [W: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ W )
% 0.28/0.47 => ( ( ord_less @ A @ W @ Y2 )
% 0.28/0.47 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.28/0.47 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dense_le_bounded
% 0.28/0.47 thf(fact_190_dense__ge__bounded,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( dense_linorder @ A )
% 0.28/0.47 => ! [Z2: A,X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z2 @ X2 )
% 0.28/0.47 => ( ! [W: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z2 @ W )
% 0.28/0.47 => ( ( ord_less @ A @ W @ X2 )
% 0.28/0.47 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 0.28/0.47 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dense_ge_bounded
% 0.28/0.47 thf(fact_191_dual__order_Ostrict__trans2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B3: A,A2: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.28/0.47 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.strict_trans2
% 0.28/0.47 thf(fact_192_dual__order_Ostrict__trans1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [B3: A,A2: A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ C2 @ B3 )
% 0.28/0.47 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dual_order.strict_trans1
% 0.28/0.47 thf(fact_193_order_Ostrict__iff__order,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ord_less @ A )
% 0.28/0.47 = ( ^ [A5: A,B4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.28/0.47 & ( A5 != B4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.strict_iff_order
% 0.28/0.47 thf(fact_194_order_Oorder__iff__strict,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [A5: A,B4: A] :
% 0.28/0.47 ( ( ord_less @ A @ A5 @ B4 )
% 0.28/0.47 | ( A5 = B4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.order_iff_strict
% 0.28/0.47 thf(fact_195_order_Ostrict__trans2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.28/0.47 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.strict_trans2
% 0.28/0.47 thf(fact_196_order_Ostrict__trans1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A,C2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less @ A @ B3 @ C2 )
% 0.28/0.47 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order.strict_trans1
% 0.28/0.47 thf(fact_197_not__le__imp__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [Y2: A,X2: A] :
% 0.28/0.47 ( ~ ( ord_less_eq @ A @ Y2 @ X2 )
% 0.28/0.47 => ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_le_imp_less
% 0.28/0.47 thf(fact_198_less__le__not__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ( ( ord_less @ A )
% 0.28/0.47 = ( ^ [X4: A,Y5: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.28/0.47 & ~ ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_le_not_le
% 0.28/0.47 thf(fact_199_le__imp__less__or__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 | ( X2 = Y2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_imp_less_or_eq
% 0.28/0.47 thf(fact_200_le__less__linear,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 | ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_less_linear
% 0.28/0.47 thf(fact_201_dense__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( dense_linorder @ A )
% 0.28/0.47 => ! [Y2: A,Z2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.47 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.28/0.47 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dense_le
% 0.28/0.47 thf(fact_202_dense__ge,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( dense_linorder @ A )
% 0.28/0.47 => ! [Z2: A,Y2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.28/0.47 => ( ord_less_eq @ A @ Y2 @ X3 ) )
% 0.28/0.47 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % dense_ge
% 0.28/0.47 thf(fact_203_less__le__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A,Z2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.28/0.47 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_le_trans
% 0.28/0.47 thf(fact_204_le__less__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A,Z2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.28/0.47 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_less_trans
% 0.28/0.47 thf(fact_205_less__imp__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( preorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_le
% 0.28/0.47 thf(fact_206_antisym__conv2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 0.28/0.47 = ( X2 = Y2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % antisym_conv2
% 0.28/0.47 thf(fact_207_antisym__conv1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.47 = ( X2 = Y2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % antisym_conv1
% 0.28/0.47 thf(fact_208_le__neq__trans,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A,B3: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( A2 != B3 )
% 0.28/0.47 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_neq_trans
% 0.28/0.47 thf(fact_209_not__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 0.28/0.47 = ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_less
% 0.28/0.47 thf(fact_210_not__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ( ~ ( ord_less_eq @ A @ X2 @ Y2 ) )
% 0.28/0.47 = ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_le
% 0.28/0.47 thf(fact_211_order__less__le__subst2,axiom,
% 0.28/0.47 ! [A: $tType,C: $tType] :
% 0.28/0.47 ( ( ( order @ C )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,B3: A,F: A > C,C2: C] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.28/0.47 => ( ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_less_le_subst2
% 0.28/0.47 thf(fact_212_order__less__le__subst1,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( order @ B )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 0.28/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.28/0.47 => ( ! [X3: B,Y4: B] :
% 0.28/0.47 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_less_le_subst1
% 0.28/0.47 thf(fact_213_order__le__less__subst2,axiom,
% 0.28/0.47 ! [A: $tType,C: $tType] :
% 0.28/0.47 ( ( ( order @ C )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,B3: A,F: A > C,C2: C] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 0.28/0.47 => ( ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_le_less_subst2
% 0.28/0.47 thf(fact_214_order__le__less__subst1,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType] :
% 0.28/0.47 ( ( ( order @ B )
% 0.28/0.47 & ( order @ A ) )
% 0.28/0.47 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 0.28/0.47 => ( ( ord_less @ B @ B3 @ C2 )
% 0.28/0.47 => ( ! [X3: B,Y4: B] :
% 0.28/0.47 ( ( ord_less @ B @ X3 @ Y4 )
% 0.28/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.28/0.47 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % order_le_less_subst1
% 0.28/0.47 thf(fact_215_less__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ord_less @ A )
% 0.28/0.47 = ( ^ [X4: A,Y5: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.28/0.47 & ( X4 != Y5 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_le
% 0.28/0.47 thf(fact_216_le__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [X4: A,Y5: A] :
% 0.28/0.47 ( ( ord_less @ A @ X4 @ Y5 )
% 0.28/0.47 | ( X4 = Y5 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_less
% 0.28/0.47 thf(fact_217_leI,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [X2: A,Y2: A] :
% 0.28/0.47 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.47 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % leI
% 0.28/0.47 thf(fact_218_leD,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [Y2: A,X2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.28/0.47 => ~ ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % leD
% 0.28/0.47 thf(fact_219_top_Oextremum__uniqueI,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order_top @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.28/0.47 => ( A2
% 0.28/0.47 = ( top_top @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % top.extremum_uniqueI
% 0.28/0.47 thf(fact_220_top_Oextremum__unique,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order_top @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( top_top @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % top.extremum_unique
% 0.28/0.47 thf(fact_221_top__greatest,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order_top @ A )
% 0.28/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % top_greatest
% 0.28/0.47 thf(fact_222_top_Onot__eq__extremum,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order_top @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 != ( top_top @ A ) )
% 0.28/0.47 = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % top.not_eq_extremum
% 0.28/0.47 thf(fact_223_top_Oextremum__strict,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order_top @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % top.extremum_strict
% 0.28/0.47 thf(fact_224_subset__UNIV,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_UNIV
% 0.28/0.47 thf(fact_225_div__gr__imp__gr__divisor,axiom,
% 0.28/0.47 ! [X2: nat,N: nat,M: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ X2 @ ( divide_divide @ nat @ N @ M ) )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_gr_imp_gr_divisor
% 0.28/0.47 thf(fact_226_div__le__mono,axiom,
% 0.28/0.47 ! [M: nat,N: nat,K: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_le_mono
% 0.28/0.47 thf(fact_227_div__le__dividend,axiom,
% 0.28/0.47 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 0.28/0.47
% 0.28/0.47 % div_le_dividend
% 0.28/0.47 thf(fact_228_Lattices__Big_Oex__has__greatest__nat,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
% 0.28/0.47 ( ( P @ K )
% 0.28/0.47 => ( ! [Y4: A] :
% 0.28/0.47 ( ( P @ Y4 )
% 0.28/0.47 => ( ord_less @ nat @ ( F @ Y4 ) @ B3 ) )
% 0.28/0.47 => ? [X3: A] :
% 0.28/0.47 ( ( P @ X3 )
% 0.28/0.47 & ! [Y3: A] :
% 0.28/0.47 ( ( P @ Y3 )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( F @ Y3 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Lattices_Big.ex_has_greatest_nat
% 0.28/0.47 thf(fact_229_nat__descend__induct,axiom,
% 0.28/0.47 ! [N: nat,P: nat > $o,M: nat] :
% 0.28/0.47 ( ! [K3: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ N @ K3 )
% 0.28/0.47 => ( P @ K3 ) )
% 0.28/0.47 => ( ! [K3: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K3 @ N )
% 0.28/0.47 => ( ! [I5: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ K3 @ I5 )
% 0.28/0.47 => ( P @ I5 ) )
% 0.28/0.47 => ( P @ K3 ) ) )
% 0.28/0.47 => ( P @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_descend_induct
% 0.28/0.47 thf(fact_230_top1I,axiom,
% 0.28/0.47 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 0.28/0.47
% 0.28/0.47 % top1I
% 0.28/0.47 thf(fact_231_less__fun__def,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( ord @ B )
% 0.28/0.47 => ( ( ord_less @ ( A > B ) )
% 0.28/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.28/0.47 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.28/0.47 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_fun_def
% 0.28/0.47 thf(fact_232_ex__has__least__nat,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.28/0.47 ( ( P @ K )
% 0.28/0.47 => ? [X3: A] :
% 0.28/0.47 ( ( P @ X3 )
% 0.28/0.47 & ! [Y3: A] :
% 0.28/0.47 ( ( P @ Y3 )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ex_has_least_nat
% 0.28/0.47 thf(fact_233_top__empty__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( top_top @ ( A > $o ) )
% 0.28/0.47 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % top_empty_eq
% 0.28/0.47 thf(fact_234_minf_I8_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ X6 @ Z3 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(8)
% 0.28/0.47 thf(fact_235_minf_I11_J,axiom,
% 0.28/0.47 ! [C: $tType,D: $tType] :
% 0.28/0.47 ( ( ord @ C )
% 0.28/0.47 => ! [F3: D] :
% 0.28/0.47 ? [Z3: C] :
% 0.28/0.47 ! [X6: C] :
% 0.28/0.47 ( ( ord_less @ C @ X6 @ Z3 )
% 0.28/0.47 => ( F3 = F3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(11)
% 0.28/0.47 thf(fact_236_minf_I7_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ X6 @ Z3 )
% 0.28/0.47 => ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(7)
% 0.28/0.47 thf(fact_237_minf_I5_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ X6 @ Z3 )
% 0.28/0.47 => ( ord_less @ A @ X6 @ T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(5)
% 0.28/0.47 thf(fact_238_minf_I4_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ X6 @ Z3 )
% 0.28/0.47 => ( X6 != T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(4)
% 0.28/0.47 thf(fact_239_minf_I3_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ X6 @ Z3 )
% 0.28/0.47 => ( X6 != T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(3)
% 0.28/0.47 thf(fact_240_minf_I2_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.28/0.47 ( ? [Z4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ Z4 )
% 0.28/0.47 => ( ( P @ X3 )
% 0.28/0.47 = ( P4 @ X3 ) ) )
% 0.28/0.47 => ( ? [Z4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ Z4 )
% 0.28/0.47 => ( ( Q @ X3 )
% 0.28/0.47 = ( Q2 @ X3 ) ) )
% 0.28/0.47 => ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ X6 @ Z3 )
% 0.28/0.47 => ( ( ( P @ X6 )
% 0.28/0.47 | ( Q @ X6 ) )
% 0.28/0.47 = ( ( P4 @ X6 )
% 0.28/0.47 | ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(2)
% 0.28/0.47 thf(fact_241_minf_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.28/0.47 ( ? [Z4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ Z4 )
% 0.28/0.47 => ( ( P @ X3 )
% 0.28/0.47 = ( P4 @ X3 ) ) )
% 0.28/0.47 => ( ? [Z4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ Z4 )
% 0.28/0.47 => ( ( Q @ X3 )
% 0.28/0.47 = ( Q2 @ X3 ) ) )
% 0.28/0.47 => ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ X6 @ Z3 )
% 0.28/0.47 => ( ( ( P @ X6 )
% 0.28/0.47 & ( Q @ X6 ) )
% 0.28/0.47 = ( ( P4 @ X6 )
% 0.28/0.47 & ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(1)
% 0.28/0.47 thf(fact_242_pinf_I11_J,axiom,
% 0.28/0.47 ! [C: $tType,D: $tType] :
% 0.28/0.47 ( ( ord @ C )
% 0.28/0.47 => ! [F3: D] :
% 0.28/0.47 ? [Z3: C] :
% 0.28/0.47 ! [X6: C] :
% 0.28/0.47 ( ( ord_less @ C @ Z3 @ X6 )
% 0.28/0.47 => ( F3 = F3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(11)
% 0.28/0.47 thf(fact_243_pinf_I7_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z3 @ X6 )
% 0.28/0.47 => ( ord_less @ A @ T @ X6 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(7)
% 0.28/0.47 thf(fact_244_pinf_I5_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z3 @ X6 )
% 0.28/0.47 => ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(5)
% 0.28/0.47 thf(fact_245_pinf_I4_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z3 @ X6 )
% 0.28/0.47 => ( X6 != T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(4)
% 0.28/0.47 thf(fact_246_pinf_I3_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z3 @ X6 )
% 0.28/0.47 => ( X6 != T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(3)
% 0.28/0.47 thf(fact_247_pinf_I2_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.28/0.47 ( ? [Z4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z4 @ X3 )
% 0.28/0.47 => ( ( P @ X3 )
% 0.28/0.47 = ( P4 @ X3 ) ) )
% 0.28/0.47 => ( ? [Z4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z4 @ X3 )
% 0.28/0.47 => ( ( Q @ X3 )
% 0.28/0.47 = ( Q2 @ X3 ) ) )
% 0.28/0.47 => ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z3 @ X6 )
% 0.28/0.47 => ( ( ( P @ X6 )
% 0.28/0.47 | ( Q @ X6 ) )
% 0.28/0.47 = ( ( P4 @ X6 )
% 0.28/0.47 | ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(2)
% 0.28/0.47 thf(fact_248_pinf_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.28/0.47 ( ? [Z4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z4 @ X3 )
% 0.28/0.47 => ( ( P @ X3 )
% 0.28/0.47 = ( P4 @ X3 ) ) )
% 0.28/0.47 => ( ? [Z4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z4 @ X3 )
% 0.28/0.47 => ( ( Q @ X3 )
% 0.28/0.47 = ( Q2 @ X3 ) ) )
% 0.28/0.47 => ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z3 @ X6 )
% 0.28/0.47 => ( ( ( P @ X6 )
% 0.28/0.47 & ( Q @ X6 ) )
% 0.28/0.47 = ( ( P4 @ X6 )
% 0.28/0.47 & ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(1)
% 0.28/0.47 thf(fact_249_pinf_I6_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z3 @ X6 )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(6)
% 0.28/0.47 thf(fact_250_pinf_I8_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ Z3 @ X6 )
% 0.28/0.47 => ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pinf(8)
% 0.28/0.47 thf(fact_251_minf_I6_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [T: A] :
% 0.28/0.47 ? [Z3: A] :
% 0.28/0.47 ! [X6: A] :
% 0.28/0.47 ( ( ord_less @ A @ X6 @ Z3 )
% 0.28/0.47 => ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minf(6)
% 0.28/0.47 thf(fact_252_complete__interval,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( condit1037483654norder @ A )
% 0.28/0.47 => ! [A2: A,B3: A,P: A > $o] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.28/0.47 => ( ( P @ A2 )
% 0.28/0.47 => ( ~ ( P @ B3 )
% 0.28/0.47 => ? [C4: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ C4 )
% 0.28/0.47 & ( ord_less_eq @ A @ C4 @ B3 )
% 0.28/0.47 & ! [X6: A] :
% 0.28/0.47 ( ( ( ord_less_eq @ A @ A2 @ X6 )
% 0.28/0.47 & ( ord_less @ A @ X6 @ C4 ) )
% 0.28/0.47 => ( P @ X6 ) )
% 0.28/0.47 & ! [D2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 0.28/0.47 & ( ord_less @ A @ X3 @ D2 ) )
% 0.28/0.47 => ( P @ X3 ) )
% 0.28/0.47 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % complete_interval
% 0.28/0.47 thf(fact_253_le__greater__neq__conv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [N3: A,A5: A] :
% 0.28/0.47 ! [X4: A] :
% 0.28/0.47 ( ( ord_less @ A @ A5 @ X4 )
% 0.28/0.47 => ( N3 != X4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_greater_neq_conv
% 0.28/0.47 thf(fact_254_ex__gt__or__lt,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( condit1656338222tinuum @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ? [B5: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B5 )
% 0.28/0.47 | ( ord_less @ A @ B5 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ex_gt_or_lt
% 0.28/0.47 thf(fact_255_ge__less__neq__conv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [A5: A,N3: A] :
% 0.28/0.47 ! [X4: A] :
% 0.28/0.47 ( ( ord_less @ A @ X4 @ A5 )
% 0.28/0.47 => ( N3 != X4 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ge_less_neq_conv
% 0.28/0.47
% 0.28/0.47 % Type constructors (25)
% 0.28/0.47 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.28/0.47 ! [A8: $tType,A9: $tType] :
% 0.28/0.47 ( ( order_top @ A9 )
% 0.28/0.47 => ( order_top @ ( A8 > A9 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.28/0.47 ! [A8: $tType,A9: $tType] :
% 0.28/0.47 ( ( preorder @ A9 )
% 0.28/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.28/0.47 ! [A8: $tType,A9: $tType] :
% 0.28/0.47 ( ( order @ A9 )
% 0.28/0.47 => ( order @ ( A8 > A9 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.28/0.47 ! [A8: $tType,A9: $tType] :
% 0.28/0.47 ( ( top @ A9 )
% 0.28/0.47 => ( top @ ( A8 > A9 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.28/0.47 ! [A8: $tType,A9: $tType] :
% 0.28/0.47 ( ( ord @ A9 )
% 0.28/0.47 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.28/0.47 condit1037483654norder @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.28/0.47 wellorder @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.28/0.47 preorder @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.28/0.47 linorder @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.28/0.47 no_top @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.28/0.47 order @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.28/0.47 ord @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.28/0.47 size @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
% 0.28/0.47 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.28/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.28/0.47 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 0.28/0.47 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.28/0.47 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
% 0.28/0.47 order_top @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.28/0.47 preorder @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
% 0.28/0.47 linorder @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 0.28/0.47 order @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
% 0.28/0.47 top @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
% 0.28/0.47 ord @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_List_Olist___Nat_Osize_15,axiom,
% 0.28/0.47 ! [A8: $tType] : ( size @ ( list @ A8 ) ) ).
% 0.28/0.47
% 0.28/0.47 % Conjectures (2)
% 0.28/0.47 thf(conj_0,hypothesis,
% 0.28/0.47 ord_less @ nat @ m @ ( divide_divide @ nat @ ( size_size @ ( list @ a ) @ xs ) @ k ) ).
% 0.28/0.47
% 0.28/0.47 thf(conj_1,conjecture,
% 0.28/0.47 ( ( size_size @ ( list @ a ) @ ( nth @ ( list @ a ) @ ( listSl776277612slice2 @ a @ xs @ k ) @ m ) )
% 0.28/0.47 = k ) ).
% 0.28/0.47
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 ------------------------
% 2.85/1.52 ---- Embedded in THF ---
% 2.85/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.85/1.52 %%% Generated on Tue Feb 24 08:26:22 EST 2026
% 2.85/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.85/1.52 %%% Logic specification used:
% 2.85/1.52 %%% thf(spec, logic, $$dhol).
% 2.85/1.52
% 2.85/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.9zNorDTVBi/DTF2DTF4125.p
% 2.85/1.52 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.85/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.85/1.52 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.85/1.52 thf(ty_tf_a, type, a: $tType).
% 2.85/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 2.85/1.52 thf(sy_c_List2_Of__image, type, f_image: (!> [A:$tType]: (((list @ A) > ((set @ nat) > (set @ A)))))).
% 2.85/1.52 thf(sy_c_List2_Olist__asc, type, list_asc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.85/1.52 thf(sy_c_List2_Olist__desc, type, list_desc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.85/1.52 thf(sy_c_List2_Olist__strict__asc, type, list_strict_asc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.85/1.52 thf(sy_c_List2_Olist__strict__desc, type, list_strict_desc: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.85/1.52 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice, type, listSl1903280966_slice: (!> [A:$tType]: (((list @ A) > (nat > (list @ (list @ A))))))).
% 2.85/1.52 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice2, type, listSl776277612slice2: (!> [A:$tType]: (((list @ A) > (nat > (list @ (list @ A))))))).
% 2.85/1.52 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice__aux, type, listSl1123830174ce_aux: (!> [A:$tType]: (((list @ A) > (nat > (nat > (list @ (list @ A)))))))).
% 2.85/1.52 thf(sy_c_List_Obutlast, type, butlast: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.85/1.52 thf(sy_c_List_Odistinct, type, distinct: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.85/1.52 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.85/1.52 thf(sy_c_List_Olist__ex, type, list_ex: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 2.85/1.52 thf(sy_c_List_Olist__update, type, list_update: (!> [A:$tType]: (((list @ A) > (nat > (A > (list @ A))))))).
% 2.85/1.52 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 2.85/1.52 thf(sy_c_List_Ounion, type, union: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.85/1.52 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.85/1.52 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.85/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.85/1.52 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.85/1.52 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 2.85/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.85/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.85/1.52 thf(sy_v_k, type, k: nat).
% 2.85/1.52 thf(sy_v_m, type, m: nat).
% 2.85/1.52 thf(sy_v_xs, type, xs: (list @ a)).
% 2.85/1.52 thf(fact_0_nth__equalityI, axiom, (! [A:$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)))))).
% 2.85/1.52 thf(fact_1_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => (? [X:A]: (((P @ I2) @ X)))))) = (? [Xs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = K) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => ((P @ I2) @ (((nth @ A) @ Xs2) @ I2)))))))))))).
% 2.85/1.52 thf(fact_2_list__eq__iff__nth__eq, axiom, (! [A:$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)))))))))))).
% 2.85/1.52 thf(fact_3_list__slice2__list__slice__nth, axiom, (! [A:$tType,M:nat,Xs:(list @ A),K:nat]: (((((ord_less @ nat) @ M) @ (((divide_divide @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ K)) => ((((nth @ (list @ A)) @ (((listSl776277612slice2 @ A) @ Xs) @ K)) @ M) = (((nth @ (list @ A)) @ (((listSl1903280966_slice @ A) @ Xs) @ K)) @ M)))))).
% 2.85/1.52 thf(fact_4_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs3:(list @ A)]: (((! [Ys3:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys3)) @ ((size_size @ (list @ A)) @ Xs3)) => (P @ Ys3)))) => (P @ Xs3)))) => (P @ Xs))))).
% 2.85/1.52 thf(fact_5_list__strict__desc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_strict_desc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ A) @ (((nth @ A) @ Xs2) @ J)) @ (((nth @ A) @ Xs2) @ I2))))))))))))))).
% 2.85/1.52 thf(fact_6_list__slice__nth__length, axiom, (! [A:$tType,M:nat,Xs:(list @ A),K:nat]: (((((ord_less @ nat) @ M) @ (((divide_divide @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ K)) => (((size_size @ (list @ A)) @ (((nth @ (list @ A)) @ (((listSl1903280966_slice @ A) @ Xs) @ K)) @ M)) = K))))).
% 2.85/1.52 thf(fact_7_list__strict__asc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_strict_asc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J))))))))))))))).
% 2.85/1.52 thf(fact_8_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs3:(list @ A)]: ((((size_size @ (list @ A)) @ Xs3) = N)))))).
% 2.85/1.52 thf(fact_9_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys)) => (Xs != Ys))))).
% 2.85/1.52 thf(fact_10_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X2:A,Y2:A]: (((((size_size @ A) @ X2) != ((size_size @ A) @ Y2)) => (X2 != Y2)))))))).
% 2.85/1.52 thf(fact_11_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.85/1.52 thf(fact_12_list__slice__length, axiom, (! [A:$tType,Xs:(list @ A),K:nat]: ((((size_size @ (list @ (list @ A))) @ (((listSl1903280966_slice @ A) @ Xs) @ K)) = (((divide_divide @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ K))))).
% 2.85/1.52 thf(fact_13_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y3:A]: (((((ord_less @ B) @ (F @ Y3)) @ (F @ X3)) => (P @ Y3)))) => (P @ X3)))) => (P @ A2)))))))).
% 2.85/1.52 thf(fact_14_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y3:A]: (((((ord_less @ B) @ (F @ Y3)) @ (F @ X3)) => (P @ Y3)))) => (P @ X3)))) => (P @ A2)))))))).
% 2.85/1.52 thf(fact_15_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X2:A]: (((! [X3:A]: (((~ ((P @ X3))) => (? [Y3:A]: (((((ord_less @ nat) @ (V @ Y3)) @ (V @ X3)) & (~ ((P @ Y3))))))))) => (P @ X2))))).
% 2.85/1.52 thf(fact_16_linorder__neqE__nat, axiom, (! [X2:nat,Y2:nat]: (((X2 != Y2) => ((~ ((((ord_less @ nat) @ X2) @ Y2))) => (((ord_less @ nat) @ Y2) @ X2)))))).
% 2.85/1.52 thf(fact_17_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2))))))))) => (P @ N))))).
% 2.85/1.52 thf(fact_18_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 2.85/1.52 thf(fact_19_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.85/1.52 thf(fact_20_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.85/1.52 thf(fact_21_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.85/1.52 thf(fact_22_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.85/1.52 thf(fact_23_list__slice__def, axiom, (! [A:$tType]: (((listSl1903280966_slice @ A) = (^ [Xs2:(list @ A),K2:nat]: (((((listSl1123830174ce_aux @ A) @ Xs2) @ K2) @ (((divide_divide @ nat) @ ((size_size @ (list @ A)) @ Xs2)) @ K2)))))))).
% 2.85/1.52 thf(fact_24_list__strict__desc__imp__list__desc, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_desc @ A) @ Xs) => ((list_desc @ A) @ Xs)))))))).
% 2.85/1.52 thf(fact_25_list__ex__length, axiom, (! [A:$tType]: (((list_ex @ A) = (^ [P2:(A > $o),Xs2:(list @ A)]: ((? [N3:nat]: (((((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs2)) & (P2 @ (((nth @ A) @ Xs2) @ N3))))))))))).
% 2.85/1.52 thf(fact_26_f__image__eqI, axiom, (! [A:$tType,X2:A,Xs:(list @ A),N:nat,A3:(set @ nat)]: (((X2 = (((nth @ A) @ Xs) @ N)) => ((((member @ nat) @ N) @ A3) => ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ X2) @ (((f_image @ A) @ Xs) @ A3)))))))).
% 2.85/1.52 thf(fact_27_list__strict__asc__imp__list__asc, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_asc @ A) @ Xs) => ((list_asc @ A) @ Xs)))))))).
% 2.85/1.52 thf(fact_28_nth__list__update__eq, axiom, (! [A:$tType,I3:nat,Xs:(list @ A),X2:A]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I3) @ X2)) @ I3) = X2))))).
% 2.85/1.52 thf(fact_29_rev__f__imageI, axiom, (! [A:$tType,N:nat,A3:(set @ nat),Xs:(list @ A),X2:A]: (((((member @ nat) @ N) @ A3) => ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((X2 = (((nth @ A) @ Xs) @ N)) => (((member @ A) @ X2) @ (((f_image @ A) @ Xs) @ A3)))))))).
% 2.85/1.52 thf(fact_30_f__image__iff, axiom, (! [A:$tType,X2:A,Xs:(list @ A),A3:(set @ nat)]: (((((member @ A) @ X2) @ (((f_image @ A) @ Xs) @ A3)) = (? [X4:nat]: (((((member @ nat) @ X4) @ A3) & ((((ord_less @ nat) @ X4) @ ((size_size @ (list @ A)) @ Xs)) & (X2 = (((nth @ A) @ Xs) @ X4)))))))))).
% 2.85/1.52 thf(fact_31_nth__butlast, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ ((butlast @ A) @ Xs))) => ((((nth @ A) @ ((butlast @ A) @ Xs)) @ N) = (((nth @ A) @ Xs) @ N)))))).
% 2.85/1.52 thf(fact_32_f__imageI, axiom, (! [A:$tType,N:nat,A3:(set @ nat),Xs:(list @ A)]: (((((member @ nat) @ N) @ A3) => ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ (((nth @ A) @ Xs) @ N)) @ (((f_image @ A) @ Xs) @ A3))))))).
% 2.85/1.52 thf(fact_33_f__imageE, axiom, (! [A:$tType,X2:A,Xs:(list @ A),A3:(set @ nat)]: (((((member @ A) @ X2) @ (((f_image @ A) @ Xs) @ A3)) => (~ ((! [N2:nat]: (((X2 = (((nth @ A) @ Xs) @ N2)) => ((((member @ nat) @ N2) @ A3) => (~ ((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)))))))))))))).
% 2.85/1.52 thf(fact_34_list__update__overwrite, axiom, (! [A:$tType,Xs:(list @ A),I3:nat,X2:A,Y2:A]: ((((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I3) @ X2)) @ I3) @ Y2) = ((((list_update @ A) @ Xs) @ I3) @ Y2))))).
% 2.85/1.52 thf(fact_35_length__list__update, axiom, (! [A:$tType,Xs:(list @ A),I3:nat,X2:A]: ((((size_size @ (list @ A)) @ ((((list_update @ A) @ Xs) @ I3) @ X2)) = ((size_size @ (list @ A)) @ Xs))))).
% 2.85/1.52 thf(fact_36_list__update__id, axiom, (! [A:$tType,Xs:(list @ A),I3:nat]: ((((((list_update @ A) @ Xs) @ I3) @ (((nth @ A) @ Xs) @ I3)) = Xs)))).
% 2.85/1.52 thf(fact_37_nth__list__update__neq, axiom, (! [A:$tType,I3:nat,J2:nat,Xs:(list @ A),X2:A]: (((I3 != J2) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I3) @ X2)) @ J2) = (((nth @ A) @ Xs) @ J2)))))).
% 2.85/1.52 thf(fact_38_list__update__swap, axiom, (! [A:$tType,I3:nat,I4:nat,Xs:(list @ A),X2:A,X5:A]: (((I3 != I4) => (((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I3) @ X2)) @ I4) @ X5) = ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I4) @ X5)) @ I3) @ X2)))))).
% 2.85/1.52 thf(fact_39_list__slice__aux__length, axiom, (! [A:$tType,Xs:(list @ A),K:nat,N:nat]: ((((size_size @ (list @ (list @ A))) @ ((((listSl1123830174ce_aux @ A) @ Xs) @ K) @ N)) = N)))).
% 2.85/1.52 thf(fact_40_nth__list__update, axiom, (! [A:$tType,I3:nat,Xs:(list @ A),J2:nat,X2:A]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => (((I3 = J2) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I3) @ X2)) @ J2) = X2)) & ((I3 != J2) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I3) @ X2)) @ J2) = (((nth @ A) @ Xs) @ J2)))))))).
% 2.85/1.52 thf(fact_41_list__update__same__conv, axiom, (! [A:$tType,I3:nat,Xs:(list @ A),X2:A]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((((list_update @ A) @ Xs) @ I3) @ X2) = Xs) = ((((nth @ A) @ Xs) @ I3) = X2)))))).
% 2.85/1.52 thf(fact_42_set__swap, axiom, (! [A:$tType,I3:nat,Xs:(list @ A),J2:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J2) @ ((size_size @ (list @ A)) @ Xs)) => (((set2 @ A) @ ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I3) @ (((nth @ A) @ Xs) @ J2))) @ J2) @ (((nth @ A) @ Xs) @ I3))) = ((set2 @ A) @ Xs))))))).
% 2.85/1.53 thf(fact_43_list__desc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_desc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ J)) @ (((nth @ A) @ Xs2) @ I2))))))))))))))).
% 2.85/1.53 thf(fact_44_list__asc__trans, axiom, (! [A:$tType]: (((preorder @ A) => ((list_asc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J))))))))))))))).
% 2.85/1.53 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.85/1.53 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3)))) = A3)))).
% 2.85/1.53 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.85/1.53 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.85/1.53 thf(fact_49_distinct__swap, axiom, (! [A:$tType,I3:nat,Xs:(list @ A),J2:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J2) @ ((size_size @ (list @ A)) @ Xs)) => (((distinct @ A) @ ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I3) @ (((nth @ A) @ Xs) @ J2))) @ J2) @ (((nth @ A) @ Xs) @ I3))) = ((distinct @ A) @ Xs))))))).
% 2.85/1.53 thf(fact_50_f__rangeE, axiom, (! [A:$tType,X2:A,Xs:(list @ A)]: (((((member @ A) @ X2) @ (((f_image @ A) @ Xs) @ (top_top @ (set @ nat)))) => (~ ((! [N2:nat]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (X2 != (((nth @ A) @ Xs) @ N2))))))))))).
% 2.85/1.53 thf(fact_51_f__rangeI, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ (((nth @ A) @ Xs) @ N)) @ (((f_image @ A) @ Xs) @ (top_top @ (set @ nat)))))))).
% 2.85/1.53 thf(fact_52_f__range__eqI, axiom, (! [A:$tType,X2:A,Xs:(list @ A),N:nat]: (((X2 = (((nth @ A) @ Xs) @ N)) => ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ X2) @ (((f_image @ A) @ Xs) @ (top_top @ (set @ nat))))))))).
% 2.85/1.53 thf(fact_53_f__image__subsetI, axiom, (! [A:$tType,A3:(set @ nat),Xs:(list @ A),B2:(set @ A)]: (((! [N2:nat]: ((((((member @ nat) @ N2) @ A3) & (((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs))) => (((member @ A) @ (((nth @ A) @ Xs) @ N2)) @ B2)))) => (((ord_less_eq @ (set @ A)) @ (((f_image @ A) @ Xs) @ A3)) @ B2))))).
% 2.85/1.53 thf(fact_54_f__image__subset__iff, axiom, (! [A:$tType,Xs:(list @ A),A3:(set @ nat),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((f_image @ A) @ Xs) @ A3)) @ B2) = (! [X4:nat]: (((((member @ nat) @ X4) @ A3) => ((((ord_less @ nat) @ X4) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ (((nth @ A) @ Xs) @ X4)) @ B2))))))))).
% 2.85/1.53 thf(fact_55_list__update__beyond, axiom, (! [A:$tType,Xs:(list @ A),I3:nat,X2:A]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ I3) => (((((list_update @ A) @ Xs) @ I3) @ X2) = Xs))))).
% 2.85/1.53 thf(fact_56_set__update__subsetI, axiom, (! [A:$tType,Xs:(list @ A),A3:(set @ A),X2:A,I3:nat]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ A3) => ((((member @ A) @ X2) @ A3) => (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ ((((list_update @ A) @ Xs) @ I3) @ X2))) @ A3)))))).
% 2.85/1.53 thf(fact_57_subset__code_I1_J, axiom, (! [A:$tType,Xs:(list @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ B2) = (! [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) => (((member @ A) @ X4) @ B2)))))))).
% 2.85/1.53 thf(fact_58_f__range__eq__set, axiom, (! [A:$tType,Xs:(list @ A)]: (((((f_image @ A) @ Xs) @ (top_top @ (set @ nat))) = ((set2 @ A) @ Xs))))).
% 2.85/1.53 thf(fact_59_subset__f__image__iff, axiom, (! [A:$tType,B2:(set @ A),Xs:(list @ A),A3:(set @ nat)]: (((((ord_less_eq @ (set @ A)) @ B2) @ (((f_image @ A) @ Xs) @ A3)) = (? [A4:(set @ nat)]: (((((ord_less_eq @ (set @ nat)) @ A4) @ A3) & (B2 = (((f_image @ A) @ Xs) @ A4))))))))).
% 2.85/1.53 thf(fact_60_f__image__mono, axiom, (! [A:$tType,A3:(set @ nat),B2:(set @ nat),Xs:(list @ A)]: (((((ord_less_eq @ (set @ nat)) @ A3) @ B2) => (((ord_less_eq @ (set @ A)) @ (((f_image @ A) @ Xs) @ A3)) @ (((f_image @ A) @ Xs) @ B2)))))).
% 2.85/1.53 thf(fact_61_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I3:nat,J2:nat]: (((! [I:nat,J3:nat]: (((((ord_less @ nat) @ I) @ J3) => (((ord_less @ nat) @ (F @ I)) @ (F @ J3))))) => ((((ord_less_eq @ nat) @ I3) @ J2) => (((ord_less_eq @ nat) @ (F @ I3)) @ (F @ J2))))))).
% 2.85/1.53 thf(fact_62_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.85/1.53 thf(fact_63_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.85/1.53 thf(fact_64_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M3:nat,N3:nat]: (((((ord_less @ nat) @ M3) @ N3) | (M3 = N3)))))).
% 2.85/1.53 thf(fact_65_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.85/1.53 thf(fact_66_nat__less__le, axiom, ((ord_less @ nat) = (^ [M3:nat,N3:nat]: (((((ord_less_eq @ nat) @ M3) @ N3) & (M3 != N3)))))).
% 2.85/1.53 thf(fact_67_distinct__butlast, axiom, (! [A:$tType,Xs:(list @ A)]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ ((butlast @ A) @ Xs)))))).
% 2.85/1.53 thf(fact_68_in__set__butlastD, axiom, (! [A:$tType,X2:A,Xs:(list @ A)]: (((((member @ A) @ X2) @ ((set2 @ A) @ ((butlast @ A) @ Xs))) => (((member @ A) @ X2) @ ((set2 @ A) @ Xs)))))).
% 2.85/1.53 thf(fact_69_distinct__Ex1, axiom, (! [A:$tType,Xs:(list @ A),X2:A]: ((((distinct @ A) @ Xs) => ((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (? [X3:nat]: (((((ord_less @ nat) @ X3) @ ((size_size @ (list @ A)) @ Xs)) & (((((nth @ A) @ Xs) @ X3) = X2) & (! [Y3:nat]: ((((((ord_less @ nat) @ Y3) @ ((size_size @ (list @ A)) @ Xs)) & ((((nth @ A) @ Xs) @ Y3) = X2)) => (Y3 = X3))))))))))))).
% 2.85/1.53 thf(fact_70_list__ex__cong, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),F:(A > $o),G:(A > $o)]: (((Xs = Ys) => ((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Ys)) => ((F @ X3) = (G @ X3))))) => ((((list_ex @ A) @ F) @ Xs) = (((list_ex @ A) @ G) @ Ys))))))).
% 2.85/1.53 thf(fact_71_list__strict__asc__distinct, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_asc @ A) @ Xs) => ((distinct @ A) @ Xs)))))))).
% 2.85/1.53 thf(fact_72_list__strict__desc__distinct, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_desc @ A) @ Xs) => ((distinct @ A) @ Xs)))))))).
% 2.85/1.53 thf(fact_73_list__strict__asc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_asc @ A) @ Xs) => (! [J4:nat]: (((((ord_less @ nat) @ J4) @ ((size_size @ (list @ A)) @ Xs)) => (! [I5:nat]: (((((ord_less_eq @ nat) @ I5) @ J4) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I5)) @ (((nth @ A) @ Xs) @ J4))))))))))))))).
% 2.85/1.53 thf(fact_74_list__asc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => ((list_asc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ I2)) @ (((nth @ A) @ Xs2) @ J))))))))))))))).
% 2.85/1.53 thf(fact_75_list__desc__trans__le, axiom, (! [A:$tType]: (((preorder @ A) => ((list_desc @ A) = (^ [Xs2:(list @ A)]: ((! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => (! [I2:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs2) @ J)) @ (((nth @ A) @ Xs2) @ I2))))))))))))))).
% 2.85/1.53 thf(fact_76_nth__eq__iff__index__eq, axiom, (! [A:$tType,Xs:(list @ A),I3:nat,J2:nat]: ((((distinct @ A) @ Xs) => ((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J2) @ ((size_size @ (list @ A)) @ Xs)) => (((((nth @ A) @ Xs) @ I3) = (((nth @ A) @ Xs) @ J2)) = (I3 = J2)))))))).
% 2.85/1.53 thf(fact_77_distinct__conv__nth, axiom, (! [A:$tType]: (((distinct @ A) = (^ [Xs2:(list @ A)]: ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs2)) => (! [J:nat]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs2)) => ((I2 != J) => ((((nth @ A) @ Xs2) @ I2) != (((nth @ A) @ Xs2) @ J))))))))))))))).
% 2.85/1.53 thf(fact_78_all__set__conv__all__nth, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) => (P @ X4)))) = (! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ I2))))))))).
% 2.85/1.53 thf(fact_79_all__nth__imp__all__set, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),X2:A]: (((! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ I))))) => ((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (P @ X2)))))).
% 2.85/1.53 thf(fact_80_in__set__conv__nth, axiom, (! [A:$tType,X2:A,Xs:(list @ A)]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) = (? [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs)) & ((((nth @ A) @ Xs) @ I2) = X2)))))))).
% 2.85/1.53 thf(fact_81_list__ball__nth, axiom, (! [A:$tType,N:nat,Xs:(list @ A),P:(A > $o)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (P @ X3)))) => (P @ (((nth @ A) @ Xs) @ N))))))).
% 2.85/1.53 thf(fact_82_nth__mem, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ (((nth @ A) @ Xs) @ N)) @ ((set2 @ A) @ Xs)))))).
% 2.85/1.53 thf(fact_83_set__update__memI, axiom, (! [A:$tType,N:nat,Xs:(list @ A),X2:A]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ X2) @ ((set2 @ A) @ ((((list_update @ A) @ Xs) @ N) @ X2))))))).
% 2.85/1.53 thf(fact_84_f__image__eq__set, axiom, (! [A:$tType,Xs:(list @ A),A3:(set @ nat)]: (((! [N2:nat]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ nat) @ N2) @ A3)))) => ((((f_image @ A) @ Xs) @ A3) = ((set2 @ A) @ Xs)))))).
% 2.85/1.53 thf(fact_85_subsetI, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 2.85/1.53 thf(fact_86_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A3) => (A3 = B2)))))).
% 2.85/1.53 thf(fact_87_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((A3 != B2) => (((ord_less @ (set @ A)) @ A3) @ B2)))))).
% 2.85/1.53 thf(fact_88_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.85/1.53 thf(fact_89_iso__tuple__UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.85/1.53 thf(fact_90_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X4:D]: ((top_top @ C)))))))).
% 2.85/1.53 thf(fact_91_distinct__union, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((distinct @ A) @ (((union @ A) @ Xs) @ Ys)) = ((distinct @ A) @ Ys))))).
% 2.85/1.53 thf(fact_92_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.85/1.53 thf(fact_93_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.85/1.53 thf(fact_94_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B3:nat]: (((P @ K) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B3)))) => (? [X3:nat]: (((P @ X3) & (! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ X3)))))))))))).
% 2.85/1.53 thf(fact_95_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.85/1.53 thf(fact_96_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.85/1.53 thf(fact_97_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.85/1.53 thf(fact_98_le__trans, axiom, (! [I3:nat,J2:nat,K:nat]: (((((ord_less_eq @ nat) @ I3) @ J2) => ((((ord_less_eq @ nat) @ J2) @ K) => (((ord_less_eq @ nat) @ I3) @ K)))))).
% 2.85/1.53 thf(fact_99_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.85/1.53 thf(fact_100_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B2)))))).
% 2.85/1.53 thf(fact_101_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 2.85/1.53 thf(fact_102_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ A2) @ B3) => (A2 = B3))))))))).
% 2.85/1.53 thf(fact_103_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A5) & (((ord_less_eq @ A) @ A5) @ B4))))))))).
% 2.85/1.53 thf(fact_104_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.85/1.53 thf(fact_105_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B3:A]: (((! [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) => ((P @ A6) @ B5)))) => ((! [A6:A,B5:A]: ((((P @ B5) @ A6) => ((P @ A6) @ B5)))) => ((P @ A2) @ B3))))))))).
% 2.85/1.53 thf(fact_106_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.85/1.53 thf(fact_107_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X2) @ Z2))))))))).
% 2.85/1.53 thf(fact_108_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ A2) => (A2 = B3))))))))).
% 2.85/1.53 thf(fact_109_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.85/1.53 thf(fact_110_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B3:A,C2:A]: (((A2 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.85/1.53 thf(fact_111_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) & (((ord_less_eq @ A) @ B4) @ A5))))))))).
% 2.85/1.53 thf(fact_112_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X2:A]: (((((ord_less_eq @ A) @ Y2) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y2) = (X2 = Y2))))))))).
% 2.85/1.53 thf(fact_113_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z2)))) => (((((ord_less_eq @ A) @ X2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X2)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y2))))))))))))))))).
% 2.85/1.53 thf(fact_114_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.85/1.53 thf(fact_115_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.85/1.53 thf(fact_116_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((X2 = Y2) => (((ord_less_eq @ A) @ X2) @ Y2)))))))).
% 2.85/1.53 thf(fact_117_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.85/1.53 thf(fact_118_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X2) => (X2 = Y2))))))))).
% 2.85/1.53 thf(fact_119_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X4))))))))).
% 2.85/1.53 thf(fact_120_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B3:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B3) => (((F @ B3) = C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 2.85/1.53 thf(fact_121_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B3:B,C2:B]: (((A2 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.85/1.53 thf(fact_122_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ C) @ (F @ A2)) @ C2)))))))))).
% 2.85/1.53 thf(fact_123_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.85/1.53 thf(fact_124_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F2 @ X4)) @ (G2 @ X4))))))))))).
% 2.85/1.53 thf(fact_125_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.85/1.53 thf(fact_126_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.85/1.53 thf(fact_127_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.85/1.53 thf(fact_128_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B3:B,C2:B]: (((A2 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less @ B) @ X3) @ Y4) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.85/1.53 thf(fact_129_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B3:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A2) @ B3) => (((F @ B3) = C2) => ((! [X3:A,Y4:A]: (((((ord_less @ A) @ X3) @ Y4) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less @ B) @ (F @ A2)) @ C2)))))))))).
% 2.85/1.53 thf(fact_130_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B3:B,C2:B]: (((((ord_less @ A) @ A2) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less @ B) @ X3) @ Y4) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.85/1.53 thf(fact_131_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B3:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y4:A]: (((((ord_less @ A) @ X3) @ Y4) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A2)) @ C2)))))))))).
% 2.85/1.53 thf(fact_132_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X2:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X2))))))))).
% 2.85/1.53 thf(fact_133_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X2:A]: ((? [X_1:A]: ((((ord_less @ A) @ X2) @ X_1))))))))).
% 2.85/1.53 thf(fact_134_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((X2 != Y2) => ((~ ((((ord_less @ A) @ X2) @ Y2))) => (((ord_less @ A) @ Y2) @ X2))))))))).
% 2.85/1.53 thf(fact_135_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((X2 != Y2) = ((((ord_less @ A) @ X2) @ Y2) | (((ord_less @ A) @ Y2) @ X2))))))))).
% 2.85/1.53 thf(fact_136_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.85/1.53 thf(fact_137_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (? [Z3:A]: (((((ord_less @ A) @ X2) @ Z3) & (((ord_less @ A) @ Z3) @ Y2))))))))))).
% 2.85/1.53 thf(fact_138_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (X2 != Y2)))))))).
% 2.85/1.53 thf(fact_139_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X2)))))))))).
% 2.85/1.53 thf(fact_140_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.85/1.53 thf(fact_141_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X2) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X2) @ Z2))))))))).
% 2.85/1.53 thf(fact_142_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) | ((X2 = Y2) | (((ord_less @ A) @ Y2) @ X2))))))))).
% 2.85/1.53 thf(fact_143_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((~ ((((ord_less @ A) @ X2) @ X2))))))))).
% 2.85/1.53 thf(fact_144_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B3:A,C2:A]: (((A2 = B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.85/1.53 thf(fact_145_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B3:A,C2:A]: (((((ord_less @ A) @ A2) @ B3) => ((B3 = C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.85/1.53 thf(fact_146_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (~ ((((ord_less @ A) @ A2) @ B3)))))))))).
% 2.85/1.53 thf(fact_147_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (X2 != Y2)))))))).
% 2.85/1.53 thf(fact_148_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X2)))))))))).
% 2.85/1.53 thf(fact_149_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y3:A]: (((((ord_less @ A) @ Y3) @ X3) => (P @ Y3)))) => (P @ X3)))) => (P @ A2)))))))).
% 2.85/1.53 thf(fact_150_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y2:A,X2:A]: (((~ ((((ord_less @ A) @ Y2) @ X2))) => ((~ ((((ord_less @ A) @ X2) @ Y2))) = (X2 = Y2))))))))).
% 2.85/1.53 thf(fact_151_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (Y2 != X2)))))))).
% 2.85/1.53 thf(fact_152_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,P:$o]: (((((ord_less @ A) @ X2) @ Y2) => ((((ord_less @ A) @ Y2) @ X2) => P)))))))).
% 2.85/1.53 thf(fact_153_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) => ((X2 != Y2) => (((ord_less @ A) @ Y2) @ X2))))))))).
% 2.85/1.53 thf(fact_154_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.85/1.53 thf(fact_155_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A,C2:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.85/1.53 thf(fact_156_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X2)))))))))).
% 2.85/1.53 thf(fact_157_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P3:(A > $o)]: ((? [X:A]: ((P3 @ X))))) = (^ [P2:(A > $o)]: ((? [N3:A]: (((P2 @ N3) & (! [M3:A]: (((((ord_less @ A) @ M3) @ N3) => (~ ((P2 @ M3)))))))))))))))).
% 2.85/1.53 thf(fact_158_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B3:A]: (((! [A6:A,B5:A]: (((((ord_less @ A) @ A6) @ B5) => ((P @ A6) @ B5)))) => ((! [A6:A]: (((P @ A6) @ A6))) => ((! [A6:A,B5:A]: ((((P @ B5) @ A6) => ((P @ A6) @ B5)))) => ((P @ A2) @ B3)))))))))).
% 2.85/1.53 thf(fact_159_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A,C2:A]: (((((ord_less @ A) @ B3) @ A2) => ((((ord_less @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.85/1.53 thf(fact_160_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) = ((((ord_less @ A) @ Y2) @ X2) | (X2 = Y2))))))))).
% 2.85/1.53 thf(fact_161_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (A2 != B3)))))))).
% 2.85/1.53 thf(fact_162_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (A2 != B3)))))))).
% 2.85/1.53 thf(fact_163_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.85/1.53 thf(fact_164_UNIV__eq__I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A3))) => ((top_top @ (set @ A)) = A3))))).
% 2.85/1.53 thf(fact_165_psubsetE, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B2) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A3)))))))).
% 2.85/1.53 thf(fact_166_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) & (A7 != B6)))))))).
% 2.85/1.53 thf(fact_167_psubset__imp__subset, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B2) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 2.85/1.53 thf(fact_168_psubset__subset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 2.85/1.53 thf(fact_169_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) & (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A7)))))))))).
% 2.85/1.53 thf(fact_170_subset__psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 2.85/1.53 thf(fact_171_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B6) | (A7 = B6)))))))).
% 2.85/1.53 thf(fact_172_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 2.85/1.53 thf(fact_173_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y:(set @ A),Z:(set @ A)]: ((Y = Z))) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A7)))))))).
% 2.85/1.53 thf(fact_174_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less_eq @ (set @ A)) @ A3) @ C3)))))).
% 2.85/1.53 thf(fact_175_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.85/1.53 thf(fact_176_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.85/1.53 thf(fact_177_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A7) => (((member @ A) @ T2) @ B6)))))))))).
% 2.85/1.53 thf(fact_178_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A3))))).
% 2.85/1.53 thf(fact_179_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 2.85/1.53 thf(fact_180_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A7) => (((member @ A) @ X4) @ B6)))))))))).
% 2.85/1.53 thf(fact_181_equalityE, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A3)))))))))).
% 2.85/1.53 thf(fact_182_subsetD, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B2)))))).
% 2.85/1.53 thf(fact_183_in__mono, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B2)))))).
% 2.85/1.53 thf(fact_184_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((A2 != B3) => ((((ord_less_eq @ A) @ A2) @ B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.85/1.53 thf(fact_185_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (((ord_less_eq @ A) @ B3) @ A2)))))))).
% 2.85/1.53 thf(fact_186_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B4:A,A5:A]: (((((ord_less_eq @ A) @ B4) @ A5) & (A5 != B4))))))))).
% 2.85/1.53 thf(fact_187_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B4:A,A5:A]: (((((ord_less @ A) @ B4) @ A5) | (A5 = B4))))))))).
% 2.85/1.53 thf(fact_188_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (((ord_less_eq @ A) @ A2) @ B3)))))))).
% 2.85/1.53 thf(fact_189_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X2) @ Y2) => ((! [W:A]: (((((ord_less @ A) @ X2) @ W) => ((((ord_less @ A) @ W) @ Y2) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.85/1.53 thf(fact_190_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X2:A,Y2:A]: (((((ord_less @ A) @ Z2) @ X2) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X2) => (((ord_less_eq @ A) @ Y2) @ W))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.85/1.53 thf(fact_191_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A,C2:A]: (((((ord_less @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.85/1.53 thf(fact_192_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.85/1.53 thf(fact_193_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) & (A5 != B4))))))))).
% 2.85/1.53 thf(fact_194_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ B4) | (A5 = B4))))))))).
% 2.85/1.53 thf(fact_195_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A,C2:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.85/1.53 thf(fact_196_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.85/1.53 thf(fact_197_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y2:A,X2:A]: (((~ ((((ord_less_eq @ A) @ Y2) @ X2))) => (((ord_less @ A) @ X2) @ Y2)))))))).
% 2.85/1.53 thf(fact_198_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) & (~ ((((ord_less_eq @ A) @ Y5) @ X4))))))))))).
% 2.85/1.53 thf(fact_199_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less @ A) @ X2) @ Y2) | (X2 = Y2))))))))).
% 2.85/1.53 thf(fact_200_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) | (((ord_less @ A) @ Y2) @ X2)))))))).
% 2.85/1.53 thf(fact_201_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y2:A,Z2:A]: (((! [X3:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less_eq @ A) @ X3) @ Z2)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.85/1.53 thf(fact_202_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,Y2:A]: (((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => (((ord_less_eq @ A) @ Y2) @ X3)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.85/1.53 thf(fact_203_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X2) @ Z2))))))))).
% 2.85/1.53 thf(fact_204_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X2) @ Z2))))))))).
% 2.85/1.53 thf(fact_205_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less_eq @ A) @ X2) @ Y2)))))))).
% 2.85/1.53 thf(fact_206_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((~ ((((ord_less @ A) @ X2) @ Y2))) = (X2 = Y2))))))))).
% 2.85/1.53 thf(fact_207_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) => ((((ord_less_eq @ A) @ X2) @ Y2) = (X2 = Y2))))))))).
% 2.85/1.53 thf(fact_208_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((A2 != B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.85/1.53 thf(fact_209_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) = (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.85/1.53 thf(fact_210_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y2))) = (((ord_less @ A) @ Y2) @ X2)))))))).
% 2.85/1.53 thf(fact_211_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B3:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y4:A]: (((((ord_less @ A) @ X3) @ Y4) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A2)) @ C2)))))))))).
% 2.85/1.53 thf(fact_212_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B3:B,C2:B]: (((((ord_less @ A) @ A2) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.85/1.53 thf(fact_213_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A2)) @ C2)))))))))).
% 2.85/1.53 thf(fact_214_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less @ B) @ X3) @ Y4) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.85/1.53 thf(fact_215_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) & (X4 != Y5))))))))).
% 2.85/1.53 thf(fact_216_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X4:A,Y5:A]: (((((ord_less @ A) @ X4) @ Y5) | (X4 = Y5))))))))).
% 2.85/1.53 thf(fact_217_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less @ A) @ X2) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.85/1.53 thf(fact_218_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X2:A]: (((((ord_less_eq @ A) @ Y2) @ X2) => (~ ((((ord_less @ A) @ X2) @ Y2)))))))))).
% 2.85/1.53 thf(fact_219_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))).
% 2.85/1.53 thf(fact_220_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))).
% 2.85/1.53 thf(fact_221_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))).
% 2.85/1.53 thf(fact_222_top_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((A2 != (top_top @ A)) = (((ord_less @ A) @ A2) @ (top_top @ A))))))))).
% 2.85/1.53 thf(fact_223_top_Oextremum__strict, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ (top_top @ A)) @ A2))))))))).
% 2.85/1.53 thf(fact_224_subset__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ (top_top @ (set @ A)))))).
% 2.85/1.53 thf(fact_225_div__gr__imp__gr__divisor, axiom, (! [X2:nat,N:nat,M:nat]: (((((ord_less @ nat) @ X2) @ (((divide_divide @ nat) @ N) @ M)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.85/1.53 thf(fact_226_div__le__mono, axiom, (! [M:nat,N:nat,K:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ K)) @ (((divide_divide @ nat) @ N) @ K)))))).
% 2.85/1.53 thf(fact_227_div__le__dividend, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))).
% 2.85/1.53 thf(fact_228_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),B3:nat]: (((P @ K) => ((! [Y4:A]: (((P @ Y4) => (((ord_less @ nat) @ (F @ Y4)) @ B3)))) => (? [X3:A]: (((P @ X3) & (! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ nat) @ (F @ Y3)) @ (F @ X3))))))))))))).
% 2.85/1.53 thf(fact_229_nat__descend__induct, axiom, (! [N:nat,P:(nat > $o),M:nat]: (((! [K3:nat]: (((((ord_less @ nat) @ N) @ K3) => (P @ K3)))) => ((! [K3:nat]: (((((ord_less_eq @ nat) @ K3) @ N) => ((! [I5:nat]: (((((ord_less @ nat) @ K3) @ I5) => (P @ I5)))) => (P @ K3))))) => (P @ M)))))).
% 2.85/1.53 thf(fact_230_top1I, axiom, (! [A:$tType,X2:A]: (((top_top @ (A > $o)) @ X2)))).
% 2.85/1.53 thf(fact_231_less__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F2))))))))))).
% 2.85/1.53 thf(fact_232_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (? [X3:A]: (((P @ X3) & (! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ nat) @ (M @ X3)) @ (M @ Y3)))))))))))).
% 2.85/1.53 thf(fact_233_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A))))))))).
% 2.85/1.53 thf(fact_234_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X6)))))))))))))).
% 2.85/1.53 thf(fact_235_minf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F3:D]: ((? [Z3:C]: ((! [X6:C]: (((((ord_less @ C) @ X6) @ Z3) => (F3 = F3)))))))))))).
% 2.85/1.53 thf(fact_236_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (~ ((((ord_less @ A) @ T) @ X6)))))))))))))).
% 2.85/1.53 thf(fact_237_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (((ord_less @ A) @ X6) @ T)))))))))))).
% 2.85/1.53 thf(fact_238_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (X6 != T)))))))))))).
% 2.85/1.53 thf(fact_239_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (X6 != T)))))))))))).
% 2.85/1.53 thf(fact_240_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((P @ X3) = (P4 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (((P @ X6) | (Q @ X6)) = ((P4 @ X6) | (Q2 @ X6)))))))))))))))).
% 2.85/1.53 thf(fact_241_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((P @ X3) = (P4 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (((P @ X6) & (Q @ X6)) = ((P4 @ X6) & (Q2 @ X6)))))))))))))))).
% 2.85/1.53 thf(fact_242_pinf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F3:D]: ((? [Z3:C]: ((! [X6:C]: (((((ord_less @ C) @ Z3) @ X6) => (F3 = F3)))))))))))).
% 2.85/1.53 thf(fact_243_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (((ord_less @ A) @ T) @ X6)))))))))))).
% 2.85/1.53 thf(fact_244_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (~ ((((ord_less @ A) @ X6) @ T)))))))))))))).
% 2.85/1.53 thf(fact_245_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (X6 != T)))))))))))).
% 2.85/1.53 thf(fact_246_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (X6 != T)))))))))))).
% 2.85/1.53 thf(fact_247_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((P @ X3) = (P4 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (((P @ X6) | (Q @ X6)) = ((P4 @ X6) | (Q2 @ X6)))))))))))))))).
% 2.85/1.53 thf(fact_248_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((P @ X3) = (P4 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (((P @ X6) & (Q @ X6)) = ((P4 @ X6) & (Q2 @ X6)))))))))))))))).
% 2.85/1.53 thf(fact_249_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (~ ((((ord_less_eq @ A) @ X6) @ T)))))))))))))).
% 2.85/1.53 thf(fact_250_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (((ord_less_eq @ A) @ T) @ X6)))))))))))).
% 2.85/1.53 thf(fact_251_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (((ord_less_eq @ A) @ X6) @ T)))))))))))).
% 2.85/1.53 thf(fact_252_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A2:A,B3:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B3) => ((P @ A2) => ((~ ((P @ B3))) => (? [C4:A]: (((((ord_less_eq @ A) @ A2) @ C4) & ((((ord_less_eq @ A) @ C4) @ B3) & ((! [X6:A]: ((((((ord_less_eq @ A) @ A2) @ X6) & (((ord_less @ A) @ X6) @ C4)) => (P @ X6)))) & (! [D2:A]: (((! [X3:A]: ((((((ord_less_eq @ A) @ A2) @ X3) & (((ord_less @ A) @ X3) @ D2)) => (P @ X3)))) => (((ord_less_eq @ A) @ D2) @ C4)))))))))))))))))).
% 2.85/1.53 thf(fact_253_le__greater__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [N3:A,A5:A]: ((! [X4:A]: (((((ord_less @ A) @ A5) @ X4) => (N3 != X4))))))))))).
% 2.85/1.53 thf(fact_254_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A2:A]: ((? [B5:A]: (((((ord_less @ A) @ A2) @ B5) | (((ord_less @ A) @ B5) @ A2)))))))))).
% 2.85/1.53 thf(fact_255_ge__less__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A5:A,N3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ A5) => (N3 != X4))))))))))).
% 2.85/1.53 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A8:$tType,A9:$tType]: (((order_top @ A9) => (order_top @ (A8 > A9)))))).
% 2.85/1.53 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.85/1.53 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.85/1.53 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: (((top @ A9) => (top @ (A8 > A9)))))).
% 2.85/1.53 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.85/1.53 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ nat)).
% 2.85/1.53 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.85/1.53 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.85/1.53 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.85/1.53 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 2.85/1.53 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.85/1.53 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.85/1.53 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 2.85/1.53 thf(tcon_Set_Oset___Orderings_Oorder__top_4, axiom, (! [A8:$tType]: ((order_top @ (set @ A8))))).
% 2.85/1.53 thf(tcon_Set_Oset___Orderings_Opreorder_5, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.85/1.53 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.85/1.53 thf(tcon_Set_Oset___Orderings_Otop_7, axiom, (! [A8:$tType]: ((top @ (set @ A8))))).
% 2.85/1.53 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.85/1.53 thf(tcon_HOL_Obool___Orderings_Oorder__top_9, axiom, (order_top @ $o)).
% 2.85/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_10, axiom, (preorder @ $o)).
% 2.85/1.53 thf(tcon_HOL_Obool___Orderings_Olinorder_11, axiom, (linorder @ $o)).
% 2.85/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_12, axiom, (order @ $o)).
% 2.85/1.53 thf(tcon_HOL_Obool___Orderings_Otop_13, axiom, (top @ $o)).
% 2.85/1.53 thf(tcon_HOL_Obool___Orderings_Oord_14, axiom, (ord @ $o)).
% 2.85/1.53 thf(tcon_List_Olist___Nat_Osize_15, axiom, (! [A8:$tType]: ((size @ (list @ A8))))).
% 2.85/1.53 thf(conj_0, hypothesis, (((ord_less @ nat) @ m) @ (((divide_divide @ nat) @ ((size_size @ (list @ a)) @ xs)) @ k))).
% 2.85/1.53 thf(conj_1, conjecture, (((size_size @ (list @ a)) @ (((nth @ (list @ a)) @ (((listSl776277612slice2 @ a) @ xs) @ k)) @ m)) = k)).
% 2.85/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.9zNorDTVBi/DTF2DTF4125.p
% 2.85/1.53 ------------------------
% 2.85/1.56 ---- Cleaned THF ---
% 2.85/1.57 thf(ty_t_List_Olist,type,
% 2.85/1.57 list: $tType > $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_t_Set_Oset,type,
% 2.85/1.57 set: $tType > $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_t_Nat_Onat,type,
% 2.85/1.57 nat: $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_tf_a,type,
% 2.85/1.57 a: $tType ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_HOL_Otype,type,
% 2.85/1.57 type:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Nat_Osize,type,
% 2.85/1.57 size:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Oord,type,
% 2.85/1.57 ord:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Otop,type,
% 2.85/1.57 top:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Oorder,type,
% 2.85/1.57 order:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Ono__bot,type,
% 2.85/1.57 no_bot:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Ono__top,type,
% 2.85/1.57 no_top:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Olinorder,type,
% 2.85/1.57 linorder:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Opreorder,type,
% 2.85/1.57 preorder:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Oorder__top,type,
% 2.85/1.57 order_top:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Owellorder,type,
% 2.85/1.57 wellorder:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Odense__order,type,
% 2.85/1.57 dense_order:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.85/1.57 dense_linorder:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.85/1.57 condit1656338222tinuum:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.85/1.57 condit1037483654norder:
% 2.85/1.57 !>[A: $tType] : $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List2_Of__image,type,
% 2.85/1.57 f_image:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( set @ A ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List2_Olist__asc,type,
% 2.85/1.57 list_asc:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List2_Olist__desc,type,
% 2.85/1.57 list_desc:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List2_Olist__strict__asc,type,
% 2.85/1.57 list_strict_asc:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List2_Olist__strict__desc,type,
% 2.85/1.57 list_strict_desc:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice,type,
% 2.85/1.57 listSl1903280966_slice:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > nat > ( list @ ( list @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice2,type,
% 2.85/1.57 listSl776277612slice2:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > nat > ( list @ ( list @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice__aux,type,
% 2.85/1.57 listSl1123830174ce_aux:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > nat > nat > ( list @ ( list @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List_Obutlast,type,
% 2.85/1.57 butlast:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List_Odistinct,type,
% 2.85/1.57 distinct:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List_Olist_Oset,type,
% 2.85/1.57 set2:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List_Olist__ex,type,
% 2.85/1.57 list_ex:
% 2.85/1.57 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List_Olist__update,type,
% 2.85/1.57 list_update:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List_Onth,type,
% 2.85/1.57 nth:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_List_Ounion,type,
% 2.85/1.57 union:
% 2.85/1.57 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.85/1.57 size_size:
% 2.85/1.57 !>[A: $tType] : ( A > nat ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.85/1.57 ord_less:
% 2.85/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.85/1.57 ord_less_eq:
% 2.85/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.85/1.57 top_top:
% 2.85/1.57 !>[A: $tType] : A ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 2.85/1.57 divide_divide:
% 2.85/1.57 !>[A: $tType] : ( A > A > A ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Set_OCollect,type,
% 2.85/1.57 collect:
% 2.85/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_member,type,
% 2.85/1.57 member:
% 2.85/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_v_k,type,
% 2.85/1.57 k: nat ).
% 2.85/1.57
% 2.85/1.57 thf(sy_v_m,type,
% 2.85/1.57 m: nat ).
% 2.85/1.57
% 2.85/1.57 thf(sy_v_xs,type,
% 2.85/1.57 xs: list @ a ).
% 2.85/1.57
% 2.85/1.57 thf(fact_0_nth__equalityI,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.85/1.57 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.85/1.57 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.85/1.57 => ( ! [I: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( nth @ A @ Xs @ I )
% 2.85/1.57 = ( nth @ A @ Ys @ I ) ) )
% 2.85/1.57 => ( Xs = Ys ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_1_Skolem__list__nth,axiom,
% 2.85/1.57 ! [A: $tType,K: nat,P: nat > A > $o] :
% 2.85/1.57 ( ( ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ K )
% 2.85/1.57 => ? [X: A] : ( P @ I2 @ X ) ) )
% 2.85/1.57 = ( ? [Xs2: list @ A] :
% 2.85/1.57 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.85/1.57 = K )
% 2.85/1.57 & ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ K )
% 2.85/1.57 => ( P @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_2_list__eq__iff__nth__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ^ [Y: list @ A,Z: list @ A] : ( Y = Z ) )
% 2.85/1.57 = ( ^ [Xs2: list @ A,Ys2: list @ A] :
% 2.85/1.57 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.85/1.57 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 2.85/1.57 & ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ( ( nth @ A @ Xs2 @ I2 )
% 2.85/1.57 = ( nth @ A @ Ys2 @ I2 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_3_list__slice2__list__slice__nth,axiom,
% 2.85/1.57 ! [A: $tType,M: nat,Xs: list @ A,K: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) )
% 2.85/1.57 => ( ( nth @ ( list @ A ) @ ( listSl776277612slice2 @ A @ Xs @ K ) @ M )
% 2.85/1.57 = ( nth @ ( list @ A ) @ ( listSl1903280966_slice @ A @ Xs @ K ) @ M ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_4_length__induct,axiom,
% 2.85/1.57 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.85/1.57 ( ! [Xs3: list @ A] :
% 2.85/1.57 ( ! [Ys3: list @ A] :
% 2.85/1.57 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.85/1.57 => ( P @ Ys3 ) )
% 2.85/1.57 => ( P @ Xs3 ) )
% 2.85/1.57 => ( P @ Xs ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_5_list__strict__desc__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ( ( list_strict_desc @ A )
% 2.85/1.57 = ( ^ [Xs2: list @ A] :
% 2.85/1.57 ! [J: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ J )
% 2.85/1.57 => ( ord_less @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_6_list__slice__nth__length,axiom,
% 2.85/1.57 ! [A: $tType,M: nat,Xs: list @ A,K: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) )
% 2.85/1.57 => ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ ( listSl1903280966_slice @ A @ Xs @ K ) @ M ) )
% 2.85/1.57 = K ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_7_list__strict__asc__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ( ( list_strict_asc @ A )
% 2.85/1.57 = ( ^ [Xs2: list @ A] :
% 2.85/1.57 ! [J: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ J )
% 2.85/1.57 => ( ord_less @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_8_Ex__list__of__length,axiom,
% 2.85/1.57 ! [A: $tType,N: nat] :
% 2.85/1.57 ? [Xs3: list @ A] :
% 2.85/1.57 ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.85/1.57 = N ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_9_neq__if__length__neq,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.85/1.57 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.85/1.57 != ( size_size @ ( list @ A ) @ Ys ) )
% 2.85/1.57 => ( Xs != Ys ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_10_size__neq__size__imp__neq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( size @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ( size_size @ A @ X2 )
% 2.85/1.57 != ( size_size @ A @ Y2 ) )
% 2.85/1.57 => ( X2 != Y2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_11_nat__neq__iff,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( M != N )
% 2.85/1.57 = ( ( ord_less @ nat @ M @ N )
% 2.85/1.57 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_12_list__slice__length,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,K: nat] :
% 2.85/1.57 ( ( size_size @ ( list @ ( list @ A ) ) @ ( listSl1903280966_slice @ A @ Xs @ K ) )
% 2.85/1.57 = ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_13_measure__induct__rule,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType] :
% 2.85/1.57 ( ( wellorder @ B )
% 2.85/1.57 => ! [F: A > B,P: A > $o,A2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ! [Y3: A] :
% 2.85/1.57 ( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
% 2.85/1.57 => ( P @ Y3 ) )
% 2.85/1.57 => ( P @ X3 ) )
% 2.85/1.57 => ( P @ A2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_14_measure__induct,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType] :
% 2.85/1.57 ( ( wellorder @ B )
% 2.85/1.57 => ! [F: A > B,P: A > $o,A2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ! [Y3: A] :
% 2.85/1.57 ( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
% 2.85/1.57 => ( P @ Y3 ) )
% 2.85/1.57 => ( P @ X3 ) )
% 2.85/1.57 => ( P @ A2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_15_infinite__descent__measure,axiom,
% 2.85/1.57 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ~ ( P @ X3 )
% 2.85/1.57 => ? [Y3: A] :
% 2.85/1.57 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X3 ) )
% 2.85/1.57 & ~ ( P @ Y3 ) ) )
% 2.85/1.57 => ( P @ X2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_16_linorder__neqE__nat,axiom,
% 2.85/1.57 ! [X2: nat,Y2: nat] :
% 2.85/1.57 ( ( X2 != Y2 )
% 2.85/1.57 => ( ~ ( ord_less @ nat @ X2 @ Y2 )
% 2.85/1.57 => ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_17_infinite__descent,axiom,
% 2.85/1.57 ! [P: nat > $o,N: nat] :
% 2.85/1.57 ( ! [N2: nat] :
% 2.85/1.57 ( ~ ( P @ N2 )
% 2.85/1.57 => ? [M2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M2 @ N2 )
% 2.85/1.57 & ~ ( P @ M2 ) ) )
% 2.85/1.57 => ( P @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_18_nat__less__induct,axiom,
% 2.85/1.57 ! [P: nat > $o,N: nat] :
% 2.85/1.57 ( ! [N2: nat] :
% 2.85/1.57 ( ! [M2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M2 @ N2 )
% 2.85/1.57 => ( P @ M2 ) )
% 2.85/1.57 => ( P @ N2 ) )
% 2.85/1.57 => ( P @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_19_less__irrefl__nat,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ~ ( ord_less @ nat @ N @ N ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_20_less__not__refl3,axiom,
% 2.85/1.57 ! [S: nat,T: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ S @ T )
% 2.85/1.57 => ( S != T ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_21_less__not__refl2,axiom,
% 2.85/1.57 ! [N: nat,M: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ N @ M )
% 2.85/1.57 => ( M != N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_22_less__not__refl,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ~ ( ord_less @ nat @ N @ N ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_23_list__slice__def,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( listSl1903280966_slice @ A )
% 2.85/1.57 = ( ^ [Xs2: list @ A,K2: nat] : ( listSl1123830174ce_aux @ A @ Xs2 @ K2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ K2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_24_list__strict__desc__imp__list__desc,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [Xs: list @ A] :
% 2.85/1.57 ( ( list_strict_desc @ A @ Xs )
% 2.85/1.57 => ( list_desc @ A @ Xs ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_25_list__ex__length,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( list_ex @ A )
% 2.85/1.57 = ( ^ [P2: A > $o,Xs2: list @ A] :
% 2.85/1.57 ? [N3: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 & ( P2 @ ( nth @ A @ Xs2 @ N3 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_26_f__image__eqI,axiom,
% 2.85/1.57 ! [A: $tType,X2: A,Xs: list @ A,N: nat,A3: set @ nat] :
% 2.85/1.57 ( ( X2
% 2.85/1.57 = ( nth @ A @ Xs @ N ) )
% 2.85/1.57 => ( ( member @ nat @ N @ A3 )
% 2.85/1.57 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( member @ A @ X2 @ ( f_image @ A @ Xs @ A3 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_27_list__strict__asc__imp__list__asc,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [Xs: list @ A] :
% 2.85/1.57 ( ( list_strict_asc @ A @ Xs )
% 2.85/1.57 => ( list_asc @ A @ Xs ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_28_nth__list__update__eq,axiom,
% 2.85/1.57 ! [A: $tType,I3: nat,Xs: list @ A,X2: A] :
% 2.85/1.57 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ I3 )
% 2.85/1.57 = X2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_29_rev__f__imageI,axiom,
% 2.85/1.57 ! [A: $tType,N: nat,A3: set @ nat,Xs: list @ A,X2: A] :
% 2.85/1.57 ( ( member @ nat @ N @ A3 )
% 2.85/1.57 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( X2
% 2.85/1.57 = ( nth @ A @ Xs @ N ) )
% 2.85/1.57 => ( member @ A @ X2 @ ( f_image @ A @ Xs @ A3 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_30_f__image__iff,axiom,
% 2.85/1.57 ! [A: $tType,X2: A,Xs: list @ A,A3: set @ nat] :
% 2.85/1.57 ( ( member @ A @ X2 @ ( f_image @ A @ Xs @ A3 ) )
% 2.85/1.57 = ( ? [X4: nat] :
% 2.85/1.57 ( ( member @ nat @ X4 @ A3 )
% 2.85/1.57 & ( ord_less @ nat @ X4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 & ( X2
% 2.85/1.57 = ( nth @ A @ Xs @ X4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_31_nth__butlast,axiom,
% 2.85/1.57 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.85/1.57 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
% 2.85/1.57 => ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
% 2.85/1.57 = ( nth @ A @ Xs @ N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_32_f__imageI,axiom,
% 2.85/1.57 ! [A: $tType,N: nat,A3: set @ nat,Xs: list @ A] :
% 2.85/1.57 ( ( member @ nat @ N @ A3 )
% 2.85/1.57 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( member @ A @ ( nth @ A @ Xs @ N ) @ ( f_image @ A @ Xs @ A3 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_33_f__imageE,axiom,
% 2.85/1.57 ! [A: $tType,X2: A,Xs: list @ A,A3: set @ nat] :
% 2.85/1.57 ( ( member @ A @ X2 @ ( f_image @ A @ Xs @ A3 ) )
% 2.85/1.57 => ~ ! [N2: nat] :
% 2.85/1.57 ( ( X2
% 2.85/1.57 = ( nth @ A @ Xs @ N2 ) )
% 2.85/1.57 => ( ( member @ nat @ N2 @ A3 )
% 2.85/1.57 => ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_34_list__update__overwrite,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,I3: nat,X2: A,Y2: A] :
% 2.85/1.57 ( ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ I3 @ Y2 )
% 2.85/1.57 = ( list_update @ A @ Xs @ I3 @ Y2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_35_length__list__update,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,I3: nat,X2: A] :
% 2.85/1.57 ( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I3 @ X2 ) )
% 2.85/1.57 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_36_list__update__id,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,I3: nat] :
% 2.85/1.57 ( ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ I3 ) )
% 2.85/1.57 = Xs ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_37_nth__list__update__neq,axiom,
% 2.85/1.57 ! [A: $tType,I3: nat,J2: nat,Xs: list @ A,X2: A] :
% 2.85/1.57 ( ( I3 != J2 )
% 2.85/1.57 => ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ J2 )
% 2.85/1.57 = ( nth @ A @ Xs @ J2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_38_list__update__swap,axiom,
% 2.85/1.57 ! [A: $tType,I3: nat,I4: nat,Xs: list @ A,X2: A,X5: A] :
% 2.85/1.57 ( ( I3 != I4 )
% 2.85/1.57 => ( ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ I4 @ X5 )
% 2.85/1.57 = ( list_update @ A @ ( list_update @ A @ Xs @ I4 @ X5 ) @ I3 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_39_list__slice__aux__length,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,K: nat,N: nat] :
% 2.85/1.57 ( ( size_size @ ( list @ ( list @ A ) ) @ ( listSl1123830174ce_aux @ A @ Xs @ K @ N ) )
% 2.85/1.57 = N ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_40_nth__list__update,axiom,
% 2.85/1.57 ! [A: $tType,I3: nat,Xs: list @ A,J2: nat,X2: A] :
% 2.85/1.57 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( ( I3 = J2 )
% 2.85/1.57 => ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ J2 )
% 2.85/1.57 = X2 ) )
% 2.85/1.57 & ( ( I3 != J2 )
% 2.85/1.57 => ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ J2 )
% 2.85/1.57 = ( nth @ A @ Xs @ J2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_41_list__update__same__conv,axiom,
% 2.85/1.57 ! [A: $tType,I3: nat,Xs: list @ A,X2: A] :
% 2.85/1.57 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( ( list_update @ A @ Xs @ I3 @ X2 )
% 2.85/1.57 = Xs )
% 2.85/1.57 = ( ( nth @ A @ Xs @ I3 )
% 2.85/1.57 = X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_42_set__swap,axiom,
% 2.85/1.57 ! [A: $tType,I3: nat,Xs: list @ A,J2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ J2 ) ) @ J2 @ ( nth @ A @ Xs @ I3 ) ) )
% 2.85/1.57 = ( set2 @ A @ Xs ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_43_list__desc__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ( ( list_desc @ A )
% 2.85/1.57 = ( ^ [Xs2: list @ A] :
% 2.85/1.57 ! [J: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ J )
% 2.85/1.57 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_44_list__asc__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ( ( list_asc @ A )
% 2.85/1.57 = ( ^ [Xs2: list @ A] :
% 2.85/1.57 ! [J: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ J )
% 2.85/1.57 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_45_mem__Collect__eq,axiom,
% 2.85/1.57 ! [A: $tType,A2: A,P: A > $o] :
% 2.85/1.57 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.85/1.57 = ( P @ A2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_46_Collect__mem__eq,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A] :
% 2.85/1.57 ( ( collect @ A
% 2.85/1.57 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.85/1.57 = A3 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_47_Collect__cong,axiom,
% 2.85/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( P @ X3 )
% 2.85/1.57 = ( Q @ X3 ) )
% 2.85/1.57 => ( ( collect @ A @ P )
% 2.85/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_48_ext,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( F @ X3 )
% 2.85/1.57 = ( G @ X3 ) )
% 2.85/1.57 => ( F = G ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_49_distinct__swap,axiom,
% 2.85/1.57 ! [A: $tType,I3: nat,Xs: list @ A,J2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ J2 ) ) @ J2 @ ( nth @ A @ Xs @ I3 ) ) )
% 2.85/1.57 = ( distinct @ A @ Xs ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_50_f__rangeE,axiom,
% 2.85/1.57 ! [A: $tType,X2: A,Xs: list @ A] :
% 2.85/1.57 ( ( member @ A @ X2 @ ( f_image @ A @ Xs @ ( top_top @ ( set @ nat ) ) ) )
% 2.85/1.57 => ~ ! [N2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( X2
% 2.85/1.57 != ( nth @ A @ Xs @ N2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_51_f__rangeI,axiom,
% 2.85/1.57 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.85/1.57 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( member @ A @ ( nth @ A @ Xs @ N ) @ ( f_image @ A @ Xs @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_52_f__range__eqI,axiom,
% 2.85/1.57 ! [A: $tType,X2: A,Xs: list @ A,N: nat] :
% 2.85/1.57 ( ( X2
% 2.85/1.57 = ( nth @ A @ Xs @ N ) )
% 2.85/1.57 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( member @ A @ X2 @ ( f_image @ A @ Xs @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_53_f__image__subsetI,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ nat,Xs: list @ A,B2: set @ A] :
% 2.85/1.57 ( ! [N2: nat] :
% 2.85/1.57 ( ( ( member @ nat @ N2 @ A3 )
% 2.85/1.57 & ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 2.85/1.57 => ( member @ A @ ( nth @ A @ Xs @ N2 ) @ B2 ) )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ ( f_image @ A @ Xs @ A3 ) @ B2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_54_f__image__subset__iff,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,A3: set @ nat,B2: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( f_image @ A @ Xs @ A3 ) @ B2 )
% 2.85/1.57 = ( ! [X4: nat] :
% 2.85/1.57 ( ( member @ nat @ X4 @ A3 )
% 2.85/1.57 => ( ( ord_less @ nat @ X4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( member @ A @ ( nth @ A @ Xs @ X4 ) @ B2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_55_list__update__beyond,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,I3: nat,X2: A] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I3 )
% 2.85/1.57 => ( ( list_update @ A @ Xs @ I3 @ X2 )
% 2.85/1.57 = Xs ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_56_set__update__subsetI,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,A3: set @ A,X2: A,I3: nat] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
% 2.85/1.57 => ( ( member @ A @ X2 @ A3 )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) ) @ A3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_57_subset__code_I1_J,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,B2: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
% 2.85/1.57 = ( ! [X4: A] :
% 2.85/1.57 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.85/1.57 => ( member @ A @ X4 @ B2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_58_f__range__eq__set,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A] :
% 2.85/1.57 ( ( f_image @ A @ Xs @ ( top_top @ ( set @ nat ) ) )
% 2.85/1.57 = ( set2 @ A @ Xs ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_59_subset__f__image__iff,axiom,
% 2.85/1.57 ! [A: $tType,B2: set @ A,Xs: list @ A,A3: set @ nat] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( f_image @ A @ Xs @ A3 ) )
% 2.85/1.57 = ( ? [A4: set @ nat] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ nat ) @ A4 @ A3 )
% 2.85/1.57 & ( B2
% 2.85/1.57 = ( f_image @ A @ Xs @ A4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_60_f__image__mono,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ nat,B2: set @ nat,Xs: list @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ nat ) @ A3 @ B2 )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ ( f_image @ A @ Xs @ A3 ) @ ( f_image @ A @ Xs @ B2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_61_less__mono__imp__le__mono,axiom,
% 2.85/1.57 ! [F: nat > nat,I3: nat,J2: nat] :
% 2.85/1.57 ( ! [I: nat,J3: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I @ J3 )
% 2.85/1.57 => ( ord_less @ nat @ ( F @ I ) @ ( F @ J3 ) ) )
% 2.85/1.57 => ( ( ord_less_eq @ nat @ I3 @ J2 )
% 2.85/1.57 => ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_62_le__neq__implies__less,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ M @ N )
% 2.85/1.57 => ( ( M != N )
% 2.85/1.57 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_63_less__or__eq__imp__le,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ( ord_less @ nat @ M @ N )
% 2.85/1.57 | ( M = N ) )
% 2.85/1.57 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_64_le__eq__less__or__eq,axiom,
% 2.85/1.57 ( ( ord_less_eq @ nat )
% 2.85/1.57 = ( ^ [M3: nat,N3: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M3 @ N3 )
% 2.85/1.57 | ( M3 = N3 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_65_less__imp__le__nat,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M @ N )
% 2.85/1.57 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_66_nat__less__le,axiom,
% 2.85/1.57 ( ( ord_less @ nat )
% 2.85/1.57 = ( ^ [M3: nat,N3: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 2.85/1.57 & ( M3 != N3 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_67_distinct__butlast,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A] :
% 2.85/1.57 ( ( distinct @ A @ Xs )
% 2.85/1.57 => ( distinct @ A @ ( butlast @ A @ Xs ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_68_in__set__butlastD,axiom,
% 2.85/1.57 ! [A: $tType,X2: A,Xs: list @ A] :
% 2.85/1.57 ( ( member @ A @ X2 @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
% 2.85/1.57 => ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_69_distinct__Ex1,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,X2: A] :
% 2.85/1.57 ( ( distinct @ A @ Xs )
% 2.85/1.57 => ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 2.85/1.57 => ? [X3: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 & ( ( nth @ A @ Xs @ X3 )
% 2.85/1.57 = X2 )
% 2.85/1.57 & ! [Y3: nat] :
% 2.85/1.57 ( ( ( ord_less @ nat @ Y3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 & ( ( nth @ A @ Xs @ Y3 )
% 2.85/1.57 = X2 ) )
% 2.85/1.57 => ( Y3 = X3 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_70_list__ex__cong,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,Ys: list @ A,F: A > $o,G: A > $o] :
% 2.85/1.57 ( ( Xs = Ys )
% 2.85/1.57 => ( ! [X3: A] :
% 2.85/1.57 ( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
% 2.85/1.57 => ( ( F @ X3 )
% 2.85/1.57 = ( G @ X3 ) ) )
% 2.85/1.57 => ( ( list_ex @ A @ F @ Xs )
% 2.85/1.57 = ( list_ex @ A @ G @ Ys ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_71_list__strict__asc__distinct,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [Xs: list @ A] :
% 2.85/1.57 ( ( list_strict_asc @ A @ Xs )
% 2.85/1.57 => ( distinct @ A @ Xs ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_72_list__strict__desc__distinct,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [Xs: list @ A] :
% 2.85/1.57 ( ( list_strict_desc @ A @ Xs )
% 2.85/1.57 => ( distinct @ A @ Xs ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_73_list__strict__asc__trans__le,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [Xs: list @ A] :
% 2.85/1.57 ( ( list_strict_asc @ A @ Xs )
% 2.85/1.57 => ! [J4: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ! [I5: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ I5 @ J4 )
% 2.85/1.57 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_74_list__asc__trans__le,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ( ( list_asc @ A )
% 2.85/1.57 = ( ^ [Xs2: list @ A] :
% 2.85/1.57 ! [J: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ! [I2: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.85/1.57 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_75_list__desc__trans__le,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ( ( list_desc @ A )
% 2.85/1.57 = ( ^ [Xs2: list @ A] :
% 2.85/1.57 ! [J: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ! [I2: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.85/1.57 => ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_76_nth__eq__iff__index__eq,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,I3: nat,J2: nat] :
% 2.85/1.57 ( ( distinct @ A @ Xs )
% 2.85/1.57 => ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ( ( nth @ A @ Xs @ I3 )
% 2.85/1.57 = ( nth @ A @ Xs @ J2 ) )
% 2.85/1.57 = ( I3 = J2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_77_distinct__conv__nth,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( distinct @ A )
% 2.85/1.57 = ( ^ [Xs2: list @ A] :
% 2.85/1.57 ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ! [J: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.85/1.57 => ( ( I2 != J )
% 2.85/1.57 => ( ( nth @ A @ Xs2 @ I2 )
% 2.85/1.57 != ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_78_all__set__conv__all__nth,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.85/1.57 ( ( ! [X4: A] :
% 2.85/1.57 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.85/1.57 => ( P @ X4 ) ) )
% 2.85/1.57 = ( ! [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( P @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_79_all__nth__imp__all__set,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,P: A > $o,X2: A] :
% 2.85/1.57 ( ! [I: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( P @ ( nth @ A @ Xs @ I ) ) )
% 2.85/1.57 => ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 2.85/1.57 => ( P @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_80_in__set__conv__nth,axiom,
% 2.85/1.57 ! [A: $tType,X2: A,Xs: list @ A] :
% 2.85/1.57 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 2.85/1.57 = ( ? [I2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 & ( ( nth @ A @ Xs @ I2 )
% 2.85/1.57 = X2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_81_list__ball__nth,axiom,
% 2.85/1.57 ! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
% 2.85/1.57 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( ! [X3: A] :
% 2.85/1.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.85/1.57 => ( P @ X3 ) )
% 2.85/1.57 => ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_82_nth__mem,axiom,
% 2.85/1.57 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.85/1.57 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_83_set__update__memI,axiom,
% 2.85/1.57 ! [A: $tType,N: nat,Xs: list @ A,X2: A] :
% 2.85/1.57 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_84_f__image__eq__set,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,A3: set @ nat] :
% 2.85/1.57 ( ! [N2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.85/1.57 => ( member @ nat @ N2 @ A3 ) )
% 2.85/1.57 => ( ( f_image @ A @ Xs @ A3 )
% 2.85/1.57 = ( set2 @ A @ Xs ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_85_subsetI,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( member @ A @ X3 @ A3 )
% 2.85/1.57 => ( member @ A @ X3 @ B2 ) )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_86_subset__antisym,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 2.85/1.57 => ( A3 = B2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_87_psubsetI,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( A3 != B2 )
% 2.85/1.57 => ( ord_less @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_88_UNIV__I,axiom,
% 2.85/1.57 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_89_iso__tuple__UNIV__I,axiom,
% 2.85/1.57 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_90_top__apply,axiom,
% 2.85/1.57 ! [C: $tType,D: $tType] :
% 2.85/1.57 ( ( top @ C )
% 2.85/1.57 => ( ( top_top @ ( D > C ) )
% 2.85/1.57 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_91_distinct__union,axiom,
% 2.85/1.57 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.85/1.57 ( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
% 2.85/1.57 = ( distinct @ A @ Ys ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_92_order__refl,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_93_top__set__def,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( top_top @ ( set @ A ) )
% 2.85/1.57 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_94_Nat_Oex__has__greatest__nat,axiom,
% 2.85/1.57 ! [P: nat > $o,K: nat,B3: nat] :
% 2.85/1.57 ( ( P @ K )
% 2.85/1.57 => ( ! [Y4: nat] :
% 2.85/1.57 ( ( P @ Y4 )
% 2.85/1.57 => ( ord_less_eq @ nat @ Y4 @ B3 ) )
% 2.85/1.57 => ? [X3: nat] :
% 2.85/1.57 ( ( P @ X3 )
% 2.85/1.57 & ! [Y3: nat] :
% 2.85/1.57 ( ( P @ Y3 )
% 2.85/1.57 => ( ord_less_eq @ nat @ Y3 @ X3 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_95_nat__le__linear,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ M @ N )
% 2.85/1.57 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_96_le__antisym,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ M @ N )
% 2.85/1.57 => ( ( ord_less_eq @ nat @ N @ M )
% 2.85/1.57 => ( M = N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_97_eq__imp__le,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( M = N )
% 2.85/1.57 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_98_le__trans,axiom,
% 2.85/1.57 ! [I3: nat,J2: nat,K: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ I3 @ J2 )
% 2.85/1.57 => ( ( ord_less_eq @ nat @ J2 @ K )
% 2.85/1.57 => ( ord_less_eq @ nat @ I3 @ K ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_99_le__refl,axiom,
% 2.85/1.57 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_100_psubsetD,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
% 2.85/1.57 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( member @ A @ C2 @ A3 )
% 2.85/1.57 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_101_psubset__trans,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 2.85/1.57 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.85/1.57 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_102_dual__order_Oantisym,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [B3: A,A2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( A2 = B3 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_103_dual__order_Oeq__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.85/1.57 = ( ^ [A5: A,B4: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.85/1.57 & ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_104_dual__order_Otrans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [B3: A,A2: A,C2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.85/1.57 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_105_linorder__wlog,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [P: A > A > $o,A2: A,B3: A] :
% 2.85/1.57 ( ! [A6: A,B5: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A6 @ B5 )
% 2.85/1.57 => ( P @ A6 @ B5 ) )
% 2.85/1.57 => ( ! [A6: A,B5: A] :
% 2.85/1.57 ( ( P @ B5 @ A6 )
% 2.85/1.57 => ( P @ A6 @ B5 ) )
% 2.85/1.57 => ( P @ A2 @ B3 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_106_dual__order_Orefl,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_107_order__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A,Z2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.85/1.57 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_108_order__class_Oorder_Oantisym,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.85/1.57 => ( A2 = B3 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_109_ord__le__eq__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord @ A )
% 2.85/1.57 => ! [A2: A,B3: A,C2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( B3 = C2 )
% 2.85/1.57 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_110_ord__eq__le__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord @ A )
% 2.85/1.57 => ! [A2: A,B3: A,C2: A] :
% 2.85/1.57 ( ( A2 = B3 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.85/1.57 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_111_order__class_Oorder_Oeq__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.85/1.57 = ( ^ [A5: A,B4: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.85/1.57 & ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_112_antisym__conv,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [Y2: A,X2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 = ( X2 = Y2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_113_le__cases3,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A,Z2: A] :
% 2.85/1.57 ( ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.85/1.57 => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.85/1.57 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.85/1.57 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.85/1.57 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.85/1.57 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.85/1.57 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
% 2.85/1.57 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.85/1.57 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 2.85/1.57 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 2.85/1.57 => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_114_order_Otrans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A,C2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.85/1.57 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_115_le__cases,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_116_eq__refl,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( X2 = Y2 )
% 2.85/1.57 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_117_linear,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_118_antisym,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.85/1.57 => ( X2 = Y2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_119_eq__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.85/1.57 = ( ^ [X4: A,Y5: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.85/1.57 & ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_120_ord__le__eq__subst,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType] :
% 2.85/1.57 ( ( ( ord @ B )
% 2.85/1.57 & ( ord @ A ) )
% 2.85/1.57 => ! [A2: A,B3: A,F: A > B,C2: B] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ( F @ B3 )
% 2.85/1.57 = C2 )
% 2.85/1.57 => ( ! [X3: A,Y4: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_121_ord__eq__le__subst,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType] :
% 2.85/1.57 ( ( ( ord @ B )
% 2.85/1.57 & ( ord @ A ) )
% 2.85/1.57 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 2.85/1.57 ( ( A2
% 2.85/1.57 = ( F @ B3 ) )
% 2.85/1.57 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.85/1.57 => ( ! [X3: B,Y4: B] :
% 2.85/1.57 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_122_order__subst2,axiom,
% 2.85/1.57 ! [A: $tType,C: $tType] :
% 2.85/1.57 ( ( ( order @ C )
% 2.85/1.57 & ( order @ A ) )
% 2.85/1.57 => ! [A2: A,B3: A,F: A > C,C2: C] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 2.85/1.57 => ( ! [X3: A,Y4: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_123_order__subst1,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType] :
% 2.85/1.57 ( ( ( order @ B )
% 2.85/1.57 & ( order @ A ) )
% 2.85/1.57 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 2.85/1.57 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.85/1.57 => ( ! [X3: B,Y4: B] :
% 2.85/1.57 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_124_le__fun__def,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType] :
% 2.85/1.57 ( ( ord @ B )
% 2.85/1.57 => ( ( ord_less_eq @ ( A > B ) )
% 2.85/1.57 = ( ^ [F2: A > B,G2: A > B] :
% 2.85/1.57 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_125_le__funI,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType] :
% 2.85/1.57 ( ( ord @ B )
% 2.85/1.57 => ! [F: A > B,G: A > B] :
% 2.85/1.57 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.85/1.57 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_126_le__funE,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType] :
% 2.85/1.57 ( ( ord @ B )
% 2.85/1.57 => ! [F: A > B,G: A > B,X2: A] :
% 2.85/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.85/1.57 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_127_le__funD,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType] :
% 2.85/1.57 ( ( ord @ B )
% 2.85/1.57 => ! [F: A > B,G: A > B,X2: A] :
% 2.85/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.85/1.57 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_128_ord__eq__less__subst,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType] :
% 2.85/1.57 ( ( ( ord @ B )
% 2.85/1.57 & ( ord @ A ) )
% 2.85/1.57 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 2.85/1.57 ( ( A2
% 2.85/1.57 = ( F @ B3 ) )
% 2.85/1.57 => ( ( ord_less @ B @ B3 @ C2 )
% 2.85/1.57 => ( ! [X3: B,Y4: B] :
% 2.85/1.57 ( ( ord_less @ B @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_129_ord__less__eq__subst,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType] :
% 2.85/1.57 ( ( ( ord @ B )
% 2.85/1.57 & ( ord @ A ) )
% 2.85/1.57 => ! [A2: A,B3: A,F: A > B,C2: B] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ( F @ B3 )
% 2.85/1.57 = C2 )
% 2.85/1.57 => ( ! [X3: A,Y4: A] :
% 2.85/1.57 ( ( ord_less @ A @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_130_order__less__subst1,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType] :
% 2.85/1.57 ( ( ( order @ B )
% 2.85/1.57 & ( order @ A ) )
% 2.85/1.57 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 2.85/1.57 => ( ( ord_less @ B @ B3 @ C2 )
% 2.85/1.57 => ( ! [X3: B,Y4: B] :
% 2.85/1.57 ( ( ord_less @ B @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_131_order__less__subst2,axiom,
% 2.85/1.57 ! [A: $tType,C: $tType] :
% 2.85/1.57 ( ( ( order @ C )
% 2.85/1.57 & ( order @ A ) )
% 2.85/1.57 => ! [A2: A,B3: A,F: A > C,C2: C] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 2.85/1.57 => ( ! [X3: A,Y4: A] :
% 2.85/1.57 ( ( ord_less @ A @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_132_lt__ex,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( no_bot @ A )
% 2.85/1.57 => ! [X2: A] :
% 2.85/1.57 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_133_gt__ex,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( no_top @ A )
% 2.85/1.57 => ! [X2: A] :
% 2.85/1.57 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_134_neqE,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( X2 != Y2 )
% 2.85/1.57 => ( ~ ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_135_neq__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( X2 != Y2 )
% 2.85/1.57 = ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 | ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_136_order_Oasym,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_137_dense,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( dense_order @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ? [Z3: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Z3 )
% 2.85/1.57 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_138_less__imp__neq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( X2 != Y2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_139_less__asym,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_140_less__asym_H,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [A2: A,B3: A] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_141_less__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A,Z2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.85/1.57 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_142_less__linear,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 | ( X2 = Y2 )
% 2.85/1.57 | ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_143_less__irrefl,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A] :
% 2.85/1.57 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_144_ord__eq__less__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord @ A )
% 2.85/1.57 => ! [A2: A,B3: A,C2: A] :
% 2.85/1.57 ( ( A2 = B3 )
% 2.85/1.57 => ( ( ord_less @ A @ B3 @ C2 )
% 2.85/1.57 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_145_ord__less__eq__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord @ A )
% 2.85/1.57 => ! [A2: A,B3: A,C2: A] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( B3 = C2 )
% 2.85/1.57 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_146_dual__order_Oasym,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [B3: A,A2: A] :
% 2.85/1.57 ( ( ord_less @ A @ B3 @ A2 )
% 2.85/1.57 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_147_less__imp__not__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( X2 != Y2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_148_less__not__sym,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_149_less__induct,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( wellorder @ A )
% 2.85/1.57 => ! [P: A > $o,A2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ! [Y3: A] :
% 2.85/1.57 ( ( ord_less @ A @ Y3 @ X3 )
% 2.85/1.57 => ( P @ Y3 ) )
% 2.85/1.57 => ( P @ X3 ) )
% 2.85/1.57 => ( P @ A2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_150_antisym__conv3,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [Y2: A,X2: A] :
% 2.85/1.57 ( ~ ( ord_less @ A @ Y2 @ X2 )
% 2.85/1.57 => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 2.85/1.57 = ( X2 = Y2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_151_less__imp__not__eq2,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( Y2 != X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_152_less__imp__triv,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A,P: $o] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ord_less @ A @ Y2 @ X2 )
% 2.85/1.57 => P ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_153_linorder__cases,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( X2 != Y2 )
% 2.85/1.57 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_154_dual__order_Oirrefl,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A] :
% 2.85/1.57 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_155_order_Ostrict__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A,C2: A] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less @ A @ B3 @ C2 )
% 2.85/1.57 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_156_less__imp__not__less,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_157_exists__least__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( wellorder @ A )
% 2.85/1.57 => ( ( ^ [P3: A > $o] :
% 2.85/1.57 ? [X: A] : ( P3 @ X ) )
% 2.85/1.57 = ( ^ [P2: A > $o] :
% 2.85/1.57 ? [N3: A] :
% 2.85/1.57 ( ( P2 @ N3 )
% 2.85/1.57 & ! [M3: A] :
% 2.85/1.57 ( ( ord_less @ A @ M3 @ N3 )
% 2.85/1.57 => ~ ( P2 @ M3 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_158_linorder__less__wlog,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [P: A > A > $o,A2: A,B3: A] :
% 2.85/1.57 ( ! [A6: A,B5: A] :
% 2.85/1.57 ( ( ord_less @ A @ A6 @ B5 )
% 2.85/1.57 => ( P @ A6 @ B5 ) )
% 2.85/1.57 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 2.85/1.57 => ( ! [A6: A,B5: A] :
% 2.85/1.57 ( ( P @ B5 @ A6 )
% 2.85/1.57 => ( P @ A6 @ B5 ) )
% 2.85/1.57 => ( P @ A2 @ B3 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_159_dual__order_Ostrict__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [B3: A,A2: A,C2: A] :
% 2.85/1.57 ( ( ord_less @ A @ B3 @ A2 )
% 2.85/1.57 => ( ( ord_less @ A @ C2 @ B3 )
% 2.85/1.57 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_160_not__less__iff__gr__or__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 2.85/1.57 = ( ( ord_less @ A @ Y2 @ X2 )
% 2.85/1.57 | ( X2 = Y2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_161_order_Ostrict__implies__not__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( A2 != B3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_162_dual__order_Ostrict__implies__not__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [B3: A,A2: A] :
% 2.85/1.57 ( ( ord_less @ A @ B3 @ A2 )
% 2.85/1.57 => ( A2 != B3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_163_UNIV__witness,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_164_UNIV__eq__I,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A] :
% 2.85/1.57 ( ! [X3: A] : ( member @ A @ X3 @ A3 )
% 2.85/1.57 => ( ( top_top @ ( set @ A ) )
% 2.85/1.57 = A3 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_165_psubsetE,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.85/1.57 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_166_psubset__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord_less @ ( set @ A ) )
% 2.85/1.57 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.85/1.57 & ( A7 != B6 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_167_psubset__imp__subset,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.85/1.57 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_168_psubset__subset__trans,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 2.85/1.57 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.85/1.57 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_169_subset__not__subset__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord_less @ ( set @ A ) )
% 2.85/1.57 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.85/1.57 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_170_subset__psubset__trans,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.85/1.57 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_171_subset__iff__psubset__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.85/1.57 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.85/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 2.85/1.57 | ( A7 = B6 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_172_Collect__mono__iff,axiom,
% 2.85/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.85/1.57 = ( ! [X4: A] :
% 2.85/1.57 ( ( P @ X4 )
% 2.85/1.57 => ( Q @ X4 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_173_set__eq__subset,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 2.85/1.57 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.85/1.57 & ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_174_subset__trans,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_175_Collect__mono,axiom,
% 2.85/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( P @ X3 )
% 2.85/1.57 => ( Q @ X3 ) )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_176_subset__refl,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_177_subset__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.85/1.57 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.85/1.57 ! [T2: A] :
% 2.85/1.57 ( ( member @ A @ T2 @ A7 )
% 2.85/1.57 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_178_equalityD2,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.85/1.57 ( ( A3 = B2 )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_179_equalityD1,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.85/1.57 ( ( A3 = B2 )
% 2.85/1.57 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_180_subset__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.85/1.57 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.85/1.57 ! [X4: A] :
% 2.85/1.57 ( ( member @ A @ X4 @ A7 )
% 2.85/1.57 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_181_equalityE,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.85/1.57 ( ( A3 = B2 )
% 2.85/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_182_subsetD,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( member @ A @ C2 @ A3 )
% 2.85/1.57 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_183_in__mono,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,X2: A] :
% 2.85/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.85/1.57 => ( ( member @ A @ X2 @ A3 )
% 2.85/1.57 => ( member @ A @ X2 @ B2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_184_order_Onot__eq__order__implies__strict,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A] :
% 2.85/1.57 ( ( A2 != B3 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_185_dual__order_Ostrict__implies__order,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [B3: A,A2: A] :
% 2.85/1.57 ( ( ord_less @ A @ B3 @ A2 )
% 2.85/1.57 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_186_dual__order_Ostrict__iff__order,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ord_less @ A )
% 2.85/1.57 = ( ^ [B4: A,A5: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.85/1.57 & ( A5 != B4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_187_dual__order_Oorder__iff__strict,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ord_less_eq @ A )
% 2.85/1.57 = ( ^ [B4: A,A5: A] :
% 2.85/1.57 ( ( ord_less @ A @ B4 @ A5 )
% 2.85/1.57 | ( A5 = B4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_188_order_Ostrict__implies__order,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_189_dense__le__bounded,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( dense_linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A,Z2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ! [W: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ W )
% 2.85/1.57 => ( ( ord_less @ A @ W @ Y2 )
% 2.85/1.57 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.85/1.57 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_190_dense__ge__bounded,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( dense_linorder @ A )
% 2.85/1.57 => ! [Z2: A,X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z2 @ X2 )
% 2.85/1.57 => ( ! [W: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z2 @ W )
% 2.85/1.57 => ( ( ord_less @ A @ W @ X2 )
% 2.85/1.57 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 2.85/1.57 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_191_dual__order_Ostrict__trans2,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [B3: A,A2: A,C2: A] :
% 2.85/1.57 ( ( ord_less @ A @ B3 @ A2 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.85/1.57 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_192_dual__order_Ostrict__trans1,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [B3: A,A2: A,C2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.85/1.57 => ( ( ord_less @ A @ C2 @ B3 )
% 2.85/1.57 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_193_order_Ostrict__iff__order,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ord_less @ A )
% 2.85/1.57 = ( ^ [A5: A,B4: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.85/1.57 & ( A5 != B4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_194_order_Oorder__iff__strict,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ord_less_eq @ A )
% 2.85/1.57 = ( ^ [A5: A,B4: A] :
% 2.85/1.57 ( ( ord_less @ A @ A5 @ B4 )
% 2.85/1.57 | ( A5 = B4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_195_order_Ostrict__trans2,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A,C2: A] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.85/1.57 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_196_order_Ostrict__trans1,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A,C2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less @ A @ B3 @ C2 )
% 2.85/1.57 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_197_not__le__imp__less,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [Y2: A,X2: A] :
% 2.85/1.57 ( ~ ( ord_less_eq @ A @ Y2 @ X2 )
% 2.85/1.57 => ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_198_less__le__not__le,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ( ( ord_less @ A )
% 2.85/1.57 = ( ^ [X4: A,Y5: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.85/1.57 & ~ ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_199_le__imp__less__or__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 | ( X2 = Y2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_200_le__less__linear,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 | ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_201_dense__le,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( dense_linorder @ A )
% 2.85/1.57 => ! [Y2: A,Z2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.85/1.57 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.85/1.57 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_202_dense__ge,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( dense_linorder @ A )
% 2.85/1.57 => ! [Z2: A,Y2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z2 @ X3 )
% 2.85/1.57 => ( ord_less_eq @ A @ Y2 @ X3 ) )
% 2.85/1.57 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_203_less__le__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A,Z2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.85/1.57 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_204_le__less__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A,Z2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.85/1.57 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_205_less__imp__le,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( preorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_206_antisym__conv2,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 2.85/1.57 = ( X2 = Y2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_207_antisym__conv1,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.85/1.57 = ( X2 = Y2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_208_le__neq__trans,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [A2: A,B3: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( A2 != B3 )
% 2.85/1.57 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_209_not__less,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
% 2.85/1.57 = ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_210_not__le,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ( ~ ( ord_less_eq @ A @ X2 @ Y2 ) )
% 2.85/1.57 = ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_211_order__less__le__subst2,axiom,
% 2.85/1.57 ! [A: $tType,C: $tType] :
% 2.85/1.57 ( ( ( order @ C )
% 2.85/1.57 & ( order @ A ) )
% 2.85/1.57 => ! [A2: A,B3: A,F: A > C,C2: C] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 2.85/1.57 => ( ! [X3: A,Y4: A] :
% 2.85/1.57 ( ( ord_less @ A @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_212_order__less__le__subst1,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType] :
% 2.85/1.57 ( ( ( order @ B )
% 2.85/1.57 & ( order @ A ) )
% 2.85/1.57 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 2.85/1.57 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.85/1.57 => ( ! [X3: B,Y4: B] :
% 2.85/1.57 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_213_order__le__less__subst2,axiom,
% 2.85/1.57 ! [A: $tType,C: $tType] :
% 2.85/1.57 ( ( ( order @ C )
% 2.85/1.57 & ( order @ A ) )
% 2.85/1.57 => ! [A2: A,B3: A,F: A > C,C2: C] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 2.85/1.57 => ( ! [X3: A,Y4: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_214_order__le__less__subst1,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType] :
% 2.85/1.57 ( ( ( order @ B )
% 2.85/1.57 & ( order @ A ) )
% 2.85/1.57 => ! [A2: A,F: B > A,B3: B,C2: B] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 2.85/1.57 => ( ( ord_less @ B @ B3 @ C2 )
% 2.85/1.57 => ( ! [X3: B,Y4: B] :
% 2.85/1.57 ( ( ord_less @ B @ X3 @ Y4 )
% 2.85/1.57 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.85/1.57 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_215_less__le,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ord_less @ A )
% 2.85/1.57 = ( ^ [X4: A,Y5: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.85/1.57 & ( X4 != Y5 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_216_le__less,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ( ( ord_less_eq @ A )
% 2.85/1.57 = ( ^ [X4: A,Y5: A] :
% 2.85/1.57 ( ( ord_less @ A @ X4 @ Y5 )
% 2.85/1.57 | ( X4 = Y5 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_217_leI,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [X2: A,Y2: A] :
% 2.85/1.57 ( ~ ( ord_less @ A @ X2 @ Y2 )
% 2.85/1.57 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_218_leD,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order @ A )
% 2.85/1.57 => ! [Y2: A,X2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.85/1.57 => ~ ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_219_top_Oextremum__uniqueI,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order_top @ A )
% 2.85/1.57 => ! [A2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.85/1.57 => ( A2
% 2.85/1.57 = ( top_top @ A ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_220_top_Oextremum__unique,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order_top @ A )
% 2.85/1.57 => ! [A2: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.85/1.57 = ( A2
% 2.85/1.57 = ( top_top @ A ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_221_top__greatest,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order_top @ A )
% 2.85/1.57 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_222_top_Onot__eq__extremum,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order_top @ A )
% 2.85/1.57 => ! [A2: A] :
% 2.85/1.57 ( ( A2
% 2.85/1.57 != ( top_top @ A ) )
% 2.85/1.57 = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_223_top_Oextremum__strict,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( order_top @ A )
% 2.85/1.57 => ! [A2: A] :
% 2.85/1.57 ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_224_subset__UNIV,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_225_div__gr__imp__gr__divisor,axiom,
% 2.85/1.57 ! [X2: nat,N: nat,M: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ X2 @ ( divide_divide @ nat @ N @ M ) )
% 2.85/1.57 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_226_div__le__mono,axiom,
% 2.85/1.57 ! [M: nat,N: nat,K: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ M @ N )
% 2.85/1.57 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_227_div__le__dividend,axiom,
% 2.85/1.57 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_228_Lattices__Big_Oex__has__greatest__nat,axiom,
% 2.85/1.57 ! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
% 2.85/1.57 ( ( P @ K )
% 2.85/1.57 => ( ! [Y4: A] :
% 2.85/1.57 ( ( P @ Y4 )
% 2.85/1.57 => ( ord_less @ nat @ ( F @ Y4 ) @ B3 ) )
% 2.85/1.57 => ? [X3: A] :
% 2.85/1.57 ( ( P @ X3 )
% 2.85/1.57 & ! [Y3: A] :
% 2.85/1.57 ( ( P @ Y3 )
% 2.85/1.57 => ( ord_less_eq @ nat @ ( F @ Y3 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_229_nat__descend__induct,axiom,
% 2.85/1.57 ! [N: nat,P: nat > $o,M: nat] :
% 2.85/1.57 ( ! [K3: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ N @ K3 )
% 2.85/1.57 => ( P @ K3 ) )
% 2.85/1.57 => ( ! [K3: nat] :
% 2.85/1.57 ( ( ord_less_eq @ nat @ K3 @ N )
% 2.85/1.57 => ( ! [I5: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ K3 @ I5 )
% 2.85/1.57 => ( P @ I5 ) )
% 2.85/1.57 => ( P @ K3 ) ) )
% 2.85/1.57 => ( P @ M ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_230_top1I,axiom,
% 2.85/1.57 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_231_less__fun__def,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType] :
% 2.85/1.57 ( ( ord @ B )
% 2.85/1.57 => ( ( ord_less @ ( A > B ) )
% 2.85/1.57 = ( ^ [F2: A > B,G2: A > B] :
% 2.85/1.57 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.85/1.57 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_232_ex__has__least__nat,axiom,
% 2.85/1.57 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 2.85/1.57 ( ( P @ K )
% 2.85/1.57 => ? [X3: A] :
% 2.85/1.57 ( ( P @ X3 )
% 2.85/1.57 & ! [Y3: A] :
% 2.85/1.57 ( ( P @ Y3 )
% 2.85/1.57 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y3 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_233_top__empty__eq,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( top_top @ ( A > $o ) )
% 2.85/1.57 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_234_minf_I8_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ X6 @ Z3 )
% 2.85/1.57 => ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_235_minf_I11_J,axiom,
% 2.85/1.57 ! [C: $tType,D: $tType] :
% 2.85/1.57 ( ( ord @ C )
% 2.85/1.57 => ! [F3: D] :
% 2.85/1.57 ? [Z3: C] :
% 2.85/1.57 ! [X6: C] :
% 2.85/1.57 ( ( ord_less @ C @ X6 @ Z3 )
% 2.85/1.57 => ( F3 = F3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_236_minf_I7_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ X6 @ Z3 )
% 2.85/1.57 => ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_237_minf_I5_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ X6 @ Z3 )
% 2.85/1.57 => ( ord_less @ A @ X6 @ T ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_238_minf_I4_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ X6 @ Z3 )
% 2.85/1.57 => ( X6 != T ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_239_minf_I3_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ X6 @ Z3 )
% 2.85/1.57 => ( X6 != T ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_240_minf_I2_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.85/1.57 ( ? [Z4: A] :
% 2.85/1.57 ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ X3 @ Z4 )
% 2.85/1.57 => ( ( P @ X3 )
% 2.85/1.57 = ( P4 @ X3 ) ) )
% 2.85/1.57 => ( ? [Z4: A] :
% 2.85/1.57 ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ X3 @ Z4 )
% 2.85/1.57 => ( ( Q @ X3 )
% 2.85/1.57 = ( Q2 @ X3 ) ) )
% 2.85/1.57 => ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ X6 @ Z3 )
% 2.85/1.57 => ( ( ( P @ X6 )
% 2.85/1.57 | ( Q @ X6 ) )
% 2.85/1.57 = ( ( P4 @ X6 )
% 2.85/1.57 | ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_241_minf_I1_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.85/1.57 ( ? [Z4: A] :
% 2.85/1.57 ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ X3 @ Z4 )
% 2.85/1.57 => ( ( P @ X3 )
% 2.85/1.57 = ( P4 @ X3 ) ) )
% 2.85/1.57 => ( ? [Z4: A] :
% 2.85/1.57 ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ X3 @ Z4 )
% 2.85/1.57 => ( ( Q @ X3 )
% 2.85/1.57 = ( Q2 @ X3 ) ) )
% 2.85/1.57 => ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ X6 @ Z3 )
% 2.85/1.57 => ( ( ( P @ X6 )
% 2.85/1.57 & ( Q @ X6 ) )
% 2.85/1.57 = ( ( P4 @ X6 )
% 2.85/1.57 & ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_242_pinf_I11_J,axiom,
% 2.85/1.57 ! [C: $tType,D: $tType] :
% 2.85/1.57 ( ( ord @ C )
% 2.85/1.57 => ! [F3: D] :
% 2.85/1.57 ? [Z3: C] :
% 2.85/1.57 ! [X6: C] :
% 2.85/1.57 ( ( ord_less @ C @ Z3 @ X6 )
% 2.85/1.57 => ( F3 = F3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_243_pinf_I7_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z3 @ X6 )
% 2.85/1.57 => ( ord_less @ A @ T @ X6 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_244_pinf_I5_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z3 @ X6 )
% 2.85/1.57 => ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_245_pinf_I4_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z3 @ X6 )
% 2.85/1.57 => ( X6 != T ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_246_pinf_I3_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z3 @ X6 )
% 2.85/1.57 => ( X6 != T ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_247_pinf_I2_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.85/1.57 ( ? [Z4: A] :
% 2.85/1.57 ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z4 @ X3 )
% 2.85/1.57 => ( ( P @ X3 )
% 2.85/1.57 = ( P4 @ X3 ) ) )
% 2.85/1.57 => ( ? [Z4: A] :
% 2.85/1.57 ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z4 @ X3 )
% 2.85/1.57 => ( ( Q @ X3 )
% 2.85/1.57 = ( Q2 @ X3 ) ) )
% 2.85/1.57 => ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z3 @ X6 )
% 2.85/1.57 => ( ( ( P @ X6 )
% 2.85/1.57 | ( Q @ X6 ) )
% 2.85/1.57 = ( ( P4 @ X6 )
% 2.85/1.57 | ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_248_pinf_I1_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.85/1.57 ( ? [Z4: A] :
% 2.85/1.57 ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z4 @ X3 )
% 2.85/1.57 => ( ( P @ X3 )
% 2.85/1.57 = ( P4 @ X3 ) ) )
% 2.85/1.57 => ( ? [Z4: A] :
% 2.85/1.57 ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z4 @ X3 )
% 2.85/1.57 => ( ( Q @ X3 )
% 2.85/1.57 = ( Q2 @ X3 ) ) )
% 2.85/1.57 => ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z3 @ X6 )
% 2.85/1.57 => ( ( ( P @ X6 )
% 2.85/1.57 & ( Q @ X6 ) )
% 2.85/1.57 = ( ( P4 @ X6 )
% 2.85/1.57 & ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_249_pinf_I6_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z3 @ X6 )
% 2.85/1.57 => ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_250_pinf_I8_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ Z3 @ X6 )
% 2.85/1.57 => ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_251_minf_I6_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ! [T: A] :
% 2.85/1.57 ? [Z3: A] :
% 2.85/1.57 ! [X6: A] :
% 2.85/1.57 ( ( ord_less @ A @ X6 @ Z3 )
% 2.85/1.57 => ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_252_complete__interval,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( condit1037483654norder @ A )
% 2.85/1.57 => ! [A2: A,B3: A,P: A > $o] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B3 )
% 2.85/1.57 => ( ( P @ A2 )
% 2.85/1.57 => ( ~ ( P @ B3 )
% 2.85/1.57 => ? [C4: A] :
% 2.85/1.57 ( ( ord_less_eq @ A @ A2 @ C4 )
% 2.85/1.57 & ( ord_less_eq @ A @ C4 @ B3 )
% 2.85/1.57 & ! [X6: A] :
% 2.85/1.57 ( ( ( ord_less_eq @ A @ A2 @ X6 )
% 2.85/1.57 & ( ord_less @ A @ X6 @ C4 ) )
% 2.85/1.57 => ( P @ X6 ) )
% 2.85/1.57 & ! [D2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 2.85/1.57 & ( ord_less @ A @ X3 @ D2 ) )
% 2.85/1.57 => ( P @ X3 ) )
% 2.85/1.57 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_253_le__greater__neq__conv,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ( ( ord_less_eq @ A )
% 2.85/1.57 = ( ^ [N3: A,A5: A] :
% 2.85/1.57 ! [X4: A] :
% 2.85/1.57 ( ( ord_less @ A @ A5 @ X4 )
% 2.85/1.57 => ( N3 != X4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_254_ex__gt__or__lt,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( condit1656338222tinuum @ A )
% 2.85/1.57 => ! [A2: A] :
% 2.85/1.57 ? [B5: A] :
% 2.85/1.57 ( ( ord_less @ A @ A2 @ B5 )
% 2.85/1.57 | ( ord_less @ A @ B5 @ A2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_255_ge__less__neq__conv,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linorder @ A )
% 2.85/1.57 => ( ( ord_less_eq @ A )
% 2.85/1.57 = ( ^ [A5: A,N3: A] :
% 2.85/1.57 ! [X4: A] :
% 2.85/1.57 ( ( ord_less @ A @ X4 @ A5 )
% 2.85/1.57 => ( N3 != X4 ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.85/1.57 ! [A8: $tType,A9: $tType] :
% 2.85/1.57 ( ( order_top @ A9 )
% 2.85/1.57 => ( order_top @ ( A8 > A9 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.85/1.57 ! [A8: $tType,A9: $tType] :
% 2.85/1.57 ( ( preorder @ A9 )
% 2.85/1.57 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.85/1.57 ! [A8: $tType,A9: $tType] :
% 2.85/1.57 ( ( order @ A9 )
% 2.85/1.57 => ( order @ ( A8 > A9 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_fun___Orderings_Otop,axiom,
% 2.85/1.57 ! [A8: $tType,A9: $tType] :
% 2.85/1.57 ( ( top @ A9 )
% 2.85/1.57 => ( top @ ( A8 > A9 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 2.85/1.57 ! [A8: $tType,A9: $tType] :
% 2.85/1.57 ( ( ord @ A9 )
% 2.85/1.57 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.85/1.57 condit1037483654norder @ nat ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.85/1.57 wellorder @ nat ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.85/1.57 preorder @ nat ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.85/1.57 linorder @ nat ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.85/1.57 no_top @ nat ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.85/1.57 order @ nat ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.85/1.57 ord @ nat ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 2.85/1.57 size @ nat ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
% 2.85/1.57 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 2.85/1.57 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 2.85/1.57 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 2.85/1.57 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 2.85/1.57 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
% 2.85/1.57 order_top @ $o ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 2.85/1.57 preorder @ $o ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
% 2.85/1.57 linorder @ $o ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 2.85/1.57 order @ $o ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
% 2.85/1.57 top @ $o ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
% 2.85/1.57 ord @ $o ).
% 2.85/1.57
% 2.85/1.57 thf(tcon_List_Olist___Nat_Osize_15,axiom,
% 2.85/1.57 ! [A8: $tType] : ( size @ ( list @ A8 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(conj_0,hypothesis,
% 2.85/1.57 ord_less @ nat @ m @ ( divide_divide @ nat @ ( size_size @ ( list @ a ) @ xs ) @ k ) ).
% 2.85/1.57
% 2.85/1.57 thf(conj_1,conjecture,
% 2.85/1.57 ( ( size_size @ ( list @ a ) @ ( nth @ ( list @ a ) @ ( listSl776277612slice2 @ a @ xs @ k ) @ m ) )
% 2.85/1.57 = k ) ).
% 2.85/1.57 ------------------------
% 2.94/1.60 % (4398)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_4125 for (2999ds/4Mi)
% 2.94/1.60 % (4399)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_4125 for (2999ds/27Mi)
% 2.94/1.60 % (4397)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/183Mi)
% 2.94/1.60 % (4401)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_4125 for (2999ds/2Mi)
% 2.94/1.60 % (4401)Instruction limit reached!
% 2.94/1.60 % (4401)------------------------------
% 2.94/1.60 % (4401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.60 % (4401)Termination reason: Unknown
% 2.94/1.60 % (4401)Termination phase: shuffling
% 2.94/1.60
% 2.94/1.60 % (4401)Memory used [KB]: 1407
% 2.94/1.60 % (4401)Time elapsed: 0.002 s
% 2.94/1.60 % (4401)Instructions burned: 3 (million)
% 2.94/1.60 % (4401)------------------------------
% 2.94/1.60 % (4401)------------------------------
% 2.94/1.60 % (4398)Instruction limit reached!
% 2.94/1.60 % (4398)------------------------------
% 2.94/1.60 % (4398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.60 % (4398)Termination reason: Unknown
% 2.94/1.60 % (4398)Termination phase: shuffling
% 2.94/1.60
% 2.94/1.60 % (4398)Memory used [KB]: 1407
% 2.94/1.60 % (4398)Time elapsed: 0.004 s
% 2.94/1.60 % (4398)Instructions burned: 6 (million)
% 2.94/1.60 % (4398)------------------------------
% 2.94/1.60 % (4398)------------------------------
% 2.94/1.60 % (4400)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/2Mi)
% 2.94/1.60 % (4402)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_4125 for (2999ds/275Mi)
% 2.94/1.60 % (4403)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/18Mi)
% 2.94/1.61 % (4400)Instruction limit reached!
% 2.94/1.61 % (4400)------------------------------
% 2.94/1.61 % (4400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (4400)Termination reason: Unknown
% 2.94/1.61 % (4400)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (4400)Memory used [KB]: 1407
% 2.94/1.61 % (4400)Time elapsed: 0.004 s
% 2.94/1.61 % (4400)Instructions burned: 3 (million)
% 2.94/1.61 % (4400)------------------------------
% 2.94/1.61 % (4400)------------------------------
% 2.94/1.61 % (4399)Instruction limit reached!
% 2.94/1.61 % (4399)------------------------------
% 2.94/1.61 % (4399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (4399)Termination reason: Unknown
% 2.94/1.61 % (4399)Termination phase: Property scanning
% 2.94/1.61
% 2.94/1.61 % (4399)Memory used [KB]: 1791
% 2.94/1.61 % (4399)Time elapsed: 0.011 s
% 2.94/1.61 % (4399)Instructions burned: 30 (million)
% 2.94/1.61 % (4399)------------------------------
% 2.94/1.61 % (4399)------------------------------
% 2.94/1.61 % (4404)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/3Mi)
% 2.94/1.61 % (4405)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_4125 for (2999ds/37Mi)
% 2.94/1.61 % (4404)Instruction limit reached!
% 2.94/1.61 % (4404)------------------------------
% 2.94/1.61 % (4404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (4404)Termination reason: Unknown
% 2.94/1.61 % (4404)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (4404)Memory used [KB]: 1407
% 2.94/1.61 % (4404)Time elapsed: 0.003 s
% 2.94/1.61 % (4404)Instructions burned: 3 (million)
% 2.94/1.61 % (4404)------------------------------
% 2.94/1.61 % (4404)------------------------------
% 2.94/1.62 % (4403)Instruction limit reached!
% 2.94/1.62 % (4403)------------------------------
% 2.94/1.62 % (4403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (4403)Termination reason: Unknown
% 2.94/1.62 % (4403)Termination phase: shuffling
% 2.94/1.62
% 2.94/1.62 % (4403)Memory used [KB]: 1791
% 2.94/1.62 % (4403)Time elapsed: 0.013 s
% 2.94/1.62 % (4403)Instructions burned: 18 (million)
% 2.94/1.62 % (4403)------------------------------
% 2.94/1.62 % (4403)------------------------------
% 2.94/1.62 % (4407)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/3Mi)
% 2.94/1.62 % (4407)Instruction limit reached!
% 2.94/1.62 % (4407)------------------------------
% 2.94/1.62 % (4407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (4407)Termination reason: Unknown
% 2.94/1.62 % (4407)Termination phase: shuffling
% 2.94/1.62
% 2.94/1.62 % (4407)Memory used [KB]: 1407
% 2.94/1.62 % (4407)Time elapsed: 0.003 s
% 2.94/1.62 % (4407)Instructions burned: 3 (million)
% 2.94/1.62 % (4407)------------------------------
% 2.94/1.62 % (4407)------------------------------
% 2.94/1.62 % (4408)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/1041Mi)
% 2.94/1.62 % (4405)Instruction limit reached!
% 2.94/1.62 % (4405)------------------------------
% 2.94/1.62 % (4405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (4405)Termination reason: Unknown
% 2.94/1.62 % (4405)Termination phase: shuffling
% 2.94/1.62
% 2.94/1.62 % (4405)Memory used [KB]: 2174
% 2.94/1.62 % (4405)Time elapsed: 0.012 s
% 2.94/1.62 % (4405)Instructions burned: 38 (million)
% 2.94/1.62 % (4405)------------------------------
% 2.94/1.62 % (4405)------------------------------
% 2.94/1.62 % (4406)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_4125 for (2999ds/15Mi)
% 2.94/1.63 % (4410)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/16Mi)
% 2.94/1.63 % (4411)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/3Mi)
% 2.94/1.63 % (4411)Instruction limit reached!
% 2.94/1.63 % (4411)------------------------------
% 2.94/1.63 % (4411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (4411)Termination reason: Unknown
% 2.94/1.63 % (4411)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (4411)Memory used [KB]: 1407
% 2.94/1.63 % (4411)Time elapsed: 0.003 s
% 2.94/1.63 % (4411)Instructions burned: 5 (million)
% 2.94/1.63 % (4411)------------------------------
% 2.94/1.63 % (4411)------------------------------
% 2.94/1.63 % (4406)Instruction limit reached!
% 2.94/1.63 % (4406)------------------------------
% 2.94/1.63 % (4406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (4406)Termination reason: Unknown
% 2.94/1.63 % (4406)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (4406)Memory used [KB]: 1663
% 2.94/1.63 % (4406)Time elapsed: 0.012 s
% 2.94/1.63 % (4406)Instructions burned: 16 (million)
% 2.94/1.63 % (4406)------------------------------
% 2.94/1.63 % (4406)------------------------------
% 2.94/1.63 % (4409)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4125 for (2999ds/7Mi)
% 2.94/1.63 % (4410)Instruction limit reached!
% 2.94/1.63 % (4410)------------------------------
% 2.94/1.63 % (4410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (4410)Termination reason: Unknown
% 2.94/1.63 % (4410)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (4410)Memory used [KB]: 1791
% 2.94/1.63 % (4410)Time elapsed: 0.008 s
% 2.94/1.63 % (4410)Instructions burned: 17 (million)
% 2.94/1.63 % (4410)------------------------------
% 2.94/1.63 % (4410)------------------------------
% 2.94/1.63 % (4402)First to succeed.
% 2.94/1.64 % (4402)Refutation found. Thanks to Tanya!
% 2.94/1.64 % SZS status Theorem for DTF2THF_4125
% 2.94/1.64 % SZS output start Proof for DTF2THF_4125
% 2.94/1.64 thf(type_def_5, type, list: ($tType) > $tType).
% 2.94/1.64 thf(type_def_6, type, nat: $tType).
% 2.94/1.64 thf(type_def_7, type, set: ($tType) > $tType).
% 2.94/1.64 thf(type_def_9, type, a: $tType).
% 2.94/1.64 thf(func_def_0, type, list: ($tType) > $tType).
% 2.94/1.64 thf(func_def_1, type, set: ($tType) > $tType).
% 2.94/1.64 thf(func_def_2, type, nat: $tType).
% 2.94/1.64 thf(func_def_3, type, a: $tType).
% 2.94/1.64 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_5, type, size: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_6, type, ord: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_7, type, top: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_8, type, order: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_9, type, no_bot: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_10, type, no_top: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_13, type, order_top: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_14, type, wellorder: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_15, type, dense_order: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_16, type, dense_linorder: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_17, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_18, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.94/1.64 thf(func_def_19, type, f_image: !>[X0: $tType]:(list @ X0 > set @ nat > set @ X0)).
% 2.94/1.64 thf(func_def_20, type, list_asc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.94/1.64 thf(func_def_21, type, list_desc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.94/1.64 thf(func_def_22, type, list_strict_asc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.94/1.64 thf(func_def_23, type, list_strict_desc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.94/1.64 thf(func_def_24, type, listSl1903280966_slice: !>[X0: $tType]:(list @ X0 > nat > list @ list @ X0)).
% 2.94/1.64 thf(func_def_25, type, listSl776277612slice2: !>[X0: $tType]:(list @ X0 > nat > list @ list @ X0)).
% 2.94/1.64 thf(func_def_26, type, listSl1123830174ce_aux: !>[X0: $tType]:(list @ X0 > nat > nat > list @ list @ X0)).
% 2.94/1.64 thf(func_def_27, type, butlast: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.94/1.64 thf(func_def_28, type, distinct: !>[X0: $tType]:(list @ X0 > $o)).
% 2.94/1.64 thf(func_def_29, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.94/1.64 thf(func_def_30, type, list_ex: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 2.94/1.64 thf(func_def_31, type, list_update: !>[X0: $tType]:(list @ X0 > nat > X0 > list @ X0)).
% 2.94/1.64 thf(func_def_32, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 2.94/1.64 thf(func_def_33, type, union: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.94/1.64 thf(func_def_34, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.94/1.64 thf(func_def_35, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.94/1.64 thf(func_def_36, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.94/1.64 thf(func_def_37, type, top_top: !>[X0: $tType]:(X0)).
% 2.94/1.64 thf(func_def_38, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.94/1.64 thf(func_def_39, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.94/1.64 thf(func_def_40, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.94/1.64 thf(func_def_41, type, k: nat).
% 2.94/1.64 thf(func_def_42, type, m: nat).
% 2.94/1.64 thf(func_def_43, type, xs: list @ a).
% 2.94/1.64 thf(func_def_58, type, sK0: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.94/1.64 thf(func_def_59, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.94/1.64 thf(func_def_60, type, sK2: (nat > $o) > nat).
% 2.94/1.64 thf(func_def_61, type, sK3: (nat > $o) > nat).
% 2.94/1.64 thf(func_def_62, type, sK4: nat > (nat > $o) > nat).
% 2.94/1.64 thf(func_def_63, type, sK5: !>[X0: $tType]:((nat > X0 > $o) > nat > list @ X0)).
% 2.94/1.64 thf(func_def_64, type, sK6: !>[X0: $tType]:((nat > X0 > $o) > nat > nat)).
% 2.94/1.64 thf(func_def_65, type, sK7: !>[X0: $tType]:(nat > (nat > X0 > $o) > X0)).
% 2.94/1.64 thf(func_def_66, type, sK8: !>[X0: $tType]:(list @ X0 > (nat > X0 > $o) > nat > nat)).
% 2.94/1.64 thf(func_def_67, type, sK9: !>[X0: $tType]:(nat > list @ X0)).
% 2.94/1.64 thf(func_def_68, type, sK10: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.94/1.64 thf(func_def_69, type, sK11: (nat > nat) > nat).
% 2.94/1.64 thf(func_def_70, type, sK12: (nat > nat) > nat).
% 2.94/1.64 thf(func_def_71, type, sK13: (nat > $o) > nat > nat).
% 2.94/1.64 thf(func_def_72, type, sK14: (nat > $o) > nat > nat).
% 2.94/1.64 thf(func_def_73, type, sK15: !>[X0: $tType]:(list @ X0 > list @ X0 > nat)).
% 2.94/1.64 thf(func_def_74, type, sK16: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.94/1.64 thf(func_def_75, type, sK17: !>[X0: $tType]:(nat > (X0 > nat) > (X0 > $o) > X0)).
% 2.94/1.64 thf(func_def_76, type, sK18: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.94/1.64 thf(func_def_77, type, sK19: (nat > $o) > nat).
% 2.94/1.64 thf(func_def_79, type, ph21: !>[X0: $tType]:(X0)).
% 2.94/1.64 thf(f990,plain,(
% 2.94/1.64 $false),
% 2.94/1.64 inference(subsumption_resolution,[status(thm)],[f989,f903])).
% 2.94/1.64 thf(f903,plain,(
% 2.94/1.64 ((ord_less @ nat @ m @ (divide_divide @ nat @ (size_size @ list @ a @ xs) @ k)) = $true)),
% 2.94/1.64 inference(cnf_transformation,[status(thm)],[f435])).
% 2.94/1.64 thf(f435,plain,(
% 2.94/1.64 ((ord_less @ nat @ m @ (divide_divide @ nat @ (size_size @ list @ a @ xs) @ k)) = $true)),
% 2.94/1.64 inference(fool_elimination,[status(thm)],[f434])).
% 2.94/1.64 thf(f434,plain,(
% 2.94/1.64 (ord_less @ nat @ m @ (divide_divide @ nat @ (size_size @ list @ a @ xs) @ k))),
% 2.94/1.64 inference(rectify,[status(thm)],[f282])).
% 2.94/1.64 thf(f282,axiom,(
% 2.94/1.64 (ord_less @ nat @ m @ (divide_divide @ nat @ (size_size @ list @ a @ xs) @ k))),
% 2.94/1.64 file('/export/starexec/sandbox2/tmp/tmp.9zNorDTVBi/DTF2THF_4125.p',conj_0)).
% 2.94/1.64 thf(f989,plain,(
% 2.94/1.64 ((ord_less @ nat @ m @ (divide_divide @ nat @ (size_size @ list @ a @ xs) @ k)) != $true)),
% 2.94/1.64 inference(trivial_inequality_removal,[status(thm)],[f988])).
% 2.94/1.64 thf(f988,plain,(
% 2.94/1.64 ((ord_less @ nat @ m @ (divide_divide @ nat @ (size_size @ list @ a @ xs) @ k)) != $true) | (k != k)),
% 2.94/1.64 inference(superposition,[status(thm)],[f976,f933])).
% 2.94/1.64 thf(f933,plain,(
% 2.94/1.64 ( ! [X0 : $tType,X2 : list @ X0,X3 : nat,X1 : nat] : (((size_size @ list @ X0 @ (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X2 @ X3) @ X1)) = X3) | ($true != (ord_less @ nat @ X1 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X2) @ X3)))) )),
% 2.94/1.64 inference(cnf_transformation,[status(thm)],[f852])).
% 2.94/1.64 thf(f852,plain,(
% 2.94/1.64 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat] : (($true != (ord_less @ nat @ X1 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X2) @ X3))) | ((size_size @ list @ X0 @ (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X2 @ X3) @ X1)) = X3))),
% 2.94/1.64 inference(ennf_transformation,[status(thm)],[f793])).
% 2.94/1.64 thf(f793,plain,(
% 2.94/1.64 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat] : (($true = (ord_less @ nat @ X1 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X2) @ X3))) => ((size_size @ list @ X0 @ (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X2 @ X3) @ X1)) = X3))),
% 2.94/1.64 inference(fool_elimination,[status(thm)],[f792])).
% 2.94/1.64 thf(f792,plain,(
% 2.94/1.64 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat] : ((ord_less @ nat @ X1 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X2) @ X3)) => ((size_size @ list @ X0 @ (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X2 @ X3) @ X1)) = X3))),
% 2.94/1.64 inference(rectify,[status(thm)],[f7])).
% 2.94/1.64 thf(f7,axiom,(
% 2.94/1.64 ! [X0 : $tType,X12 : nat,X1 : list @ X0,X4 : nat] : ((ord_less @ nat @ X12 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X1) @ X4)) => ((size_size @ list @ X0 @ (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X1 @ X4) @ X12)) = X4))),
% 2.94/1.64 file('/export/starexec/sandbox2/tmp/tmp.9zNorDTVBi/DTF2THF_4125.p',fact_6_list__slice__nth__length)).
% 2.94/1.64 thf(f976,plain,(
% 2.94/1.64 (k != (size_size @ list @ a @ (nth @ list @ a @ (listSl1903280966_slice @ a @ xs @ k) @ m)))),
% 2.94/1.64 inference(subsumption_resolution,[status(thm)],[f975,f903])).
% 2.94/1.64 thf(f975,plain,(
% 2.94/1.64 (k != (size_size @ list @ a @ (nth @ list @ a @ (listSl1903280966_slice @ a @ xs @ k) @ m))) | ((ord_less @ nat @ m @ (divide_divide @ nat @ (size_size @ list @ a @ xs) @ k)) != $true)),
% 2.94/1.64 inference(superposition,[status(thm)],[f959,f912])).
% 2.94/1.64 thf(f912,plain,(
% 2.94/1.64 ( ! [X0 : $tType,X2 : list @ X0,X3 : nat,X1 : nat] : (((nth @ list @ X0 @ (listSl776277612slice2 @ X0 @ X2 @ X3) @ X1) = (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X2 @ X3) @ X1)) | ($true != (ord_less @ nat @ X1 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X2) @ X3)))) )),
% 2.94/1.64 inference(cnf_transformation,[status(thm)],[f839])).
% 2.94/1.64 thf(f839,plain,(
% 2.94/1.64 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat] : (((nth @ list @ X0 @ (listSl776277612slice2 @ X0 @ X2 @ X3) @ X1) = (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X2 @ X3) @ X1)) | ($true != (ord_less @ nat @ X1 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X2) @ X3))))),
% 2.94/1.64 inference(ennf_transformation,[status(thm)],[f765])).
% 2.94/1.64 thf(f765,plain,(
% 2.94/1.64 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat] : (($true = (ord_less @ nat @ X1 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X2) @ X3))) => ((nth @ list @ X0 @ (listSl776277612slice2 @ X0 @ X2 @ X3) @ X1) = (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X2 @ X3) @ X1)))),
% 2.94/1.64 inference(fool_elimination,[status(thm)],[f764])).
% 2.94/1.64 thf(f764,plain,(
% 2.94/1.64 ! [X0 : $tType,X1 : nat,X2 : list @ X0,X3 : nat] : ((ord_less @ nat @ X1 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X2) @ X3)) => ((nth @ list @ X0 @ (listSl776277612slice2 @ X0 @ X2 @ X3) @ X1) = (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X2 @ X3) @ X1)))),
% 2.94/1.64 inference(rectify,[status(thm)],[f4])).
% 2.94/1.64 thf(f4,axiom,(
% 2.94/1.64 ! [X0 : $tType,X12 : nat,X1 : list @ X0,X4 : nat] : ((ord_less @ nat @ X12 @ (divide_divide @ nat @ (size_size @ list @ X0 @ X1) @ X4)) => ((nth @ list @ X0 @ (listSl776277612slice2 @ X0 @ X1 @ X4) @ X12) = (nth @ list @ X0 @ (listSl1903280966_slice @ X0 @ X1 @ X4) @ X12)))),
% 2.94/1.64 file('/export/starexec/sandbox2/tmp/tmp.9zNorDTVBi/DTF2THF_4125.p',fact_3_list__slice2__list__slice__nth)).
% 2.94/1.64 thf(f959,plain,(
% 2.94/1.64 (k != (size_size @ list @ a @ (nth @ list @ a @ (listSl776277612slice2 @ a @ xs @ k) @ m)))),
% 2.94/1.64 inference(cnf_transformation,[status(thm)],[f820])).
% 2.94/1.64 thf(f820,plain,(
% 2.94/1.64 (k != (size_size @ list @ a @ (nth @ list @ a @ (listSl776277612slice2 @ a @ xs @ k) @ m)))),
% 2.94/1.64 inference(flattening,[status(thm)],[f284])).
% 2.94/1.64 thf(f284,negated_conjecture,(
% 2.94/1.64 ~(k = (size_size @ list @ a @ (nth @ list @ a @ (listSl776277612slice2 @ a @ xs @ k) @ m)))),
% 2.94/1.64 inference(negated_conjecture,[status(cth)],[f283])).
% 2.94/1.64 thf(f283,conjecture,(
% 2.94/1.64 (k = (size_size @ list @ a @ (nth @ list @ a @ (listSl776277612slice2 @ a @ xs @ k) @ m)))),
% 2.94/1.64 file('/export/starexec/sandbox2/tmp/tmp.9zNorDTVBi/DTF2THF_4125.p',conj_1)).
% 2.94/1.64 % SZS output end Proof for DTF2THF_4125
% 2.94/1.64 % (4402)------------------------------
% 2.94/1.64 % (4402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (4402)Termination reason: Refutation
% 2.94/1.64
% 2.94/1.64 % (4402)Memory used [KB]: 6652
% 2.94/1.64 % (4402)Time elapsed: 0.034 s
% 2.94/1.64 % (4402)Instructions burned: 50 (million)
% 2.94/1.64 % (4402)------------------------------
% 2.94/1.64 % (4402)------------------------------
% 2.94/1.64 % (4396)Success in time 0.06 s
%------------------------------------------------------------------------------