%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT206^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:45:08 AM UTC 2026
% Result : Theorem 3.02s 1.62s
% Output : Refutation 3.02s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : DAT206^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n011.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 06:10:32 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.21/0.45 ---- Original DTF file ---
% 0.21/0.45 thf(spec,logic,$$dhol).
% 0.21/0.45 %------------------------------------------------------------------------------
% 0.21/0.45 % File : DAT206^1 : TPTP v9.2.1. Released v7.0.0.
% 0.21/0.45 % Domain : Data Structures
% 0.21/0.45 % Problem : Sorted list operations 216
% 0.21/0.45 % Version : [Bla16] axioms : Especial.
% 0.21/0.45 % English :
% 0.21/0.45
% 0.21/0.45 % Refs : [Lam09] Lammich (2009), Collections Framework
% 0.21/0.45 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.21/0.45 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.21/0.45 % Source : [Bla16]
% 0.21/0.45 % Names : sorted_list_operations__216.p [Bla16]
% 0.21/0.45
% 0.21/0.45 % Status : Theorem
% 0.21/0.45 % Rating : 0.33 v9.1.0, 0.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% 0.21/0.45 % Syntax : Number of formulae : 333 ( 60 unt; 53 typ; 0 def)
% 0.21/0.45 % Number of atoms : 934 ( 212 equ; 0 cnn)
% 0.21/0.45 % Maximal formula atoms : 13 ( 3 avg)
% 0.21/0.45 % Number of connectives : 4212 ( 132 ~; 19 |; 64 &;3410 @)
% 0.21/0.45 % ( 0 <=>; 587 =>; 0 <=; 0 <~>)
% 0.21/0.45 % Maximal formula depth : 24 ( 9 avg)
% 0.21/0.45 % Number of types : 2 ( 1 usr)
% 0.21/0.45 % Number of type conns : 249 ( 249 >; 0 *; 0 +; 0 <<)
% 0.21/0.45 % Number of symbols : 55 ( 52 usr; 8 con; 0-4 aty)
% 0.21/0.45 % Number of variables : 1153 ( 48 ^;1018 !; 43 ?;1153 :)
% 0.21/0.45 % ( 44 !>; 0 ?*; 0 @-; 0 @+)
% 0.21/0.45 % SPC : TH1_THM_EQU_NAR
% 0.21/0.45
% 0.21/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.21/0.45 % 2016-07-13 14:41:16.069
% 0.21/0.45 %------------------------------------------------------------------------------
% 0.21/0.45 %----Could-be-implicit typings (4)
% 0.21/0.45 thf(ty_t_List_Olist,type,
% 0.21/0.45 list: $tType > $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_t_Set_Oset,type,
% 0.21/0.45 set: $tType > $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_t_itself,type,
% 0.21/0.45 itself: $tType > $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_tf_a,type,
% 0.21/0.45 a: $tType ).
% 0.21/0.45
% 0.21/0.45 %----Explicit typings (49)
% 0.21/0.45 thf(sy_cl_HOL_Otype,type,
% 0.21/0.45 type:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.21/0.45 ord:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Otop,type,
% 0.21/0.45 top:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.21/0.45 order:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Ono__bot,type,
% 0.21/0.45 no_bot:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Ono__top,type,
% 0.21/0.45 no_top:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.21/0.45 linorder:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.21/0.45 preorder:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Oorder__top,type,
% 0.21/0.45 order_top:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.21/0.45 wellorder:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Odense__order,type,
% 0.21/0.45 dense_order:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.21/0.45 linordered_idom:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Fields_Olinordered__field,type,
% 0.21/0.45 linordered_field:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.21/0.45 dense_linorder:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.21/0.45 condit1656338222tinuum:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.21/0.45 condit1037483654norder:
% 0.21/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_If,type,
% 0.21/0.45 if:
% 0.21/0.45 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Oappend,type,
% 0.21/0.45 append:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Ocoset,type,
% 0.21/0.45 coset:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Odistinct,type,
% 0.21/0.45 distinct:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Oinsert,type,
% 0.21/0.45 insert:
% 0.21/0.45 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Olinorder__class_Osorted,type,
% 0.21/0.45 linorder_sorted:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Olist_OCons,type,
% 0.21/0.45 cons:
% 0.21/0.45 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Olist_ONil,type,
% 0.21/0.45 nil:
% 0.21/0.45 !>[A: $tType] : ( list @ A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Olist_Oset,type,
% 0.21/0.45 set2:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Oproduct__lists,type,
% 0.21/0.45 product_lists:
% 0.21/0.45 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Osublists,type,
% 0.21/0.45 sublists:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List_Ounion,type,
% 0.21/0.45 union:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_List__More_Ocombinatorial__product,type,
% 0.21/0.45 list_c659805718roduct:
% 0.21/0.45 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Misc_Omergesort__remdups,type,
% 0.21/0.45 mergesort_remdups:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord_Omax,type,
% 0.21/0.45 max:
% 0.21/0.45 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord_Omin,type,
% 0.21/0.45 min:
% 0.21/0.45 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.21/0.45 ord_less:
% 0.21/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.21/0.45 ord_less_eq:
% 0.21/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.21/0.45 order_strict_mono:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oordering__top,type,
% 0.21/0.45 ordering_top:
% 0.21/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.21/0.45 top_top:
% 0.21/0.45 !>[A: $tType] : A ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Pure_Otype,type,
% 0.21/0.45 type2:
% 0.21/0.45 !>[A: $tType] : ( itself @ A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set_OCollect,type,
% 0.21/0.45 collect:
% 0.21/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set_Oremove,type,
% 0.21/0.45 remove:
% 0.21/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set_Othe__elem,type,
% 0.21/0.45 the_elem:
% 0.21/0.45 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Osubset__sorted,type,
% 0.21/0.45 sorted1061247458sorted:
% 0.21/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_member,type,
% 0.21/0.45 member:
% 0.21/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_l1,type,
% 0.21/0.45 l1: list @ a ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_l1a____,type,
% 0.21/0.45 l1a: list @ a ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_l2,type,
% 0.21/0.45 l2: list @ a ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_l2b____,type,
% 0.21/0.45 l2b: list @ a ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_x1____,type,
% 0.21/0.45 x1: a ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_x2____,type,
% 0.21/0.45 x2: a ).
% 0.21/0.45
% 0.21/0.45 %----Relevant facts (255)
% 0.21/0.45 thf(fact_0_x1__neq__x2,axiom,
% 0.21/0.45 x1 != x2 ).
% 0.21/0.45
% 0.21/0.45 % x1_neq_x2
% 0.21/0.45 thf(fact_1_x2__le__x1,axiom,
% 0.21/0.45 ~ ( ord_less @ a @ x1 @ x2 ) ).
% 0.21/0.45
% 0.21/0.45 % x2_le_x1
% 0.21/0.45 thf(fact_2_minf_I11_J,axiom,
% 0.21/0.45 ! [C: $tType,D: $tType] :
% 0.21/0.45 ( ( ord @ C @ ( type2 @ C ) )
% 0.21/0.45 => ! [F: D] :
% 0.21/0.45 ? [Z: C] :
% 0.21/0.45 ! [X: C] :
% 0.21/0.45 ( ( ord_less @ C @ X @ Z )
% 0.21/0.45 => ( F = F ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(11)
% 0.21/0.45 thf(fact_3_minf_I7_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ X @ Z )
% 0.21/0.45 => ~ ( ord_less @ A @ T @ X ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(7)
% 0.21/0.45 thf(fact_4_minf_I5_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ X @ Z )
% 0.21/0.45 => ( ord_less @ A @ X @ T ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(5)
% 0.21/0.45 thf(fact_5_minf_I4_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ X @ Z )
% 0.21/0.45 => ( X != T ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(4)
% 0.21/0.45 thf(fact_6_minf_I3_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ X @ Z )
% 0.21/0.45 => ( X != T ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(3)
% 0.21/0.45 thf(fact_7_minf_I2_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.21/0.45 ( ? [Z2: A] :
% 0.21/0.45 ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Z2 )
% 0.21/0.45 => ( ( P @ X2 )
% 0.21/0.45 = ( P2 @ X2 ) ) )
% 0.21/0.45 => ( ? [Z2: A] :
% 0.21/0.45 ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Z2 )
% 0.21/0.45 => ( ( Q @ X2 )
% 0.21/0.45 = ( Q2 @ X2 ) ) )
% 0.21/0.45 => ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ X @ Z )
% 0.21/0.45 => ( ( ( P @ X )
% 0.21/0.45 | ( Q @ X ) )
% 0.21/0.45 = ( ( P2 @ X )
% 0.21/0.45 | ( Q2 @ X ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(2)
% 0.21/0.45 thf(fact_8_minf_I1_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.21/0.45 ( ? [Z2: A] :
% 0.21/0.45 ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Z2 )
% 0.21/0.45 => ( ( P @ X2 )
% 0.21/0.45 = ( P2 @ X2 ) ) )
% 0.21/0.45 => ( ? [Z2: A] :
% 0.21/0.45 ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Z2 )
% 0.21/0.45 => ( ( Q @ X2 )
% 0.21/0.45 = ( Q2 @ X2 ) ) )
% 0.21/0.45 => ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ X @ Z )
% 0.21/0.45 => ( ( ( P @ X )
% 0.21/0.45 & ( Q @ X ) )
% 0.21/0.45 = ( ( P2 @ X )
% 0.21/0.45 & ( Q2 @ X ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(1)
% 0.21/0.45 thf(fact_9_pinf_I11_J,axiom,
% 0.21/0.45 ! [C: $tType,D: $tType] :
% 0.21/0.45 ( ( ord @ C @ ( type2 @ C ) )
% 0.21/0.45 => ! [F: D] :
% 0.21/0.45 ? [Z: C] :
% 0.21/0.45 ! [X: C] :
% 0.21/0.45 ( ( ord_less @ C @ Z @ X )
% 0.21/0.45 => ( F = F ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(11)
% 0.21/0.45 thf(fact_10_pinf_I7_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z @ X )
% 0.21/0.45 => ( ord_less @ A @ T @ X ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(7)
% 0.21/0.45 thf(fact_11_pinf_I5_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z @ X )
% 0.21/0.45 => ~ ( ord_less @ A @ X @ T ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(5)
% 0.21/0.45 thf(fact_12_pinf_I4_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z @ X )
% 0.21/0.45 => ( X != T ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(4)
% 0.21/0.45 thf(fact_13_pinf_I3_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z @ X )
% 0.21/0.45 => ( X != T ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(3)
% 0.21/0.45 thf(fact_14_pinf_I1_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.21/0.45 ( ? [Z2: A] :
% 0.21/0.45 ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z2 @ X2 )
% 0.21/0.45 => ( ( P @ X2 )
% 0.21/0.45 = ( P2 @ X2 ) ) )
% 0.21/0.45 => ( ? [Z2: A] :
% 0.21/0.45 ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z2 @ X2 )
% 0.21/0.45 => ( ( Q @ X2 )
% 0.21/0.45 = ( Q2 @ X2 ) ) )
% 0.21/0.45 => ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z @ X )
% 0.21/0.45 => ( ( ( P @ X )
% 0.21/0.45 & ( Q @ X ) )
% 0.21/0.45 = ( ( P2 @ X )
% 0.21/0.45 & ( Q2 @ X ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(1)
% 0.21/0.45 thf(fact_15_pinf_I2_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.21/0.45 ( ? [Z2: A] :
% 0.21/0.45 ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z2 @ X2 )
% 0.21/0.45 => ( ( P @ X2 )
% 0.21/0.45 = ( P2 @ X2 ) ) )
% 0.21/0.45 => ( ? [Z2: A] :
% 0.21/0.45 ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z2 @ X2 )
% 0.21/0.45 => ( ( Q @ X2 )
% 0.21/0.45 = ( Q2 @ X2 ) ) )
% 0.21/0.45 => ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z @ X )
% 0.21/0.45 => ( ( ( P @ X )
% 0.21/0.45 | ( Q @ X ) )
% 0.21/0.45 = ( ( P2 @ X )
% 0.21/0.45 | ( Q2 @ X ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(2)
% 0.21/0.45 thf(fact_16_ex__gt__or__lt,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A] :
% 0.21/0.45 ? [B2: A] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.21/0.45 | ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ex_gt_or_lt
% 0.21/0.45 thf(fact_17_linorder__neqE__linordered__idom,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( X3 != Y )
% 0.21/0.45 => ( ~ ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % linorder_neqE_linordered_idom
% 0.21/0.45 thf(fact_18_dual__order_Ostrict__implies__not__eq,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [B3: A,A2: A] :
% 0.21/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.21/0.45 => ( A2 != B3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.strict_implies_not_eq
% 0.21/0.45 thf(fact_19_linordered__field__no__ub,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X: A] :
% 0.21/0.45 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.21/0.45
% 0.21/0.45 % linordered_field_no_ub
% 0.21/0.45 thf(fact_20_linordered__field__no__lb,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X: A] :
% 0.21/0.45 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 0.21/0.45
% 0.21/0.45 % linordered_field_no_lb
% 0.21/0.45 thf(fact_21_order_Ostrict__implies__not__eq,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( A2 != B3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.strict_implies_not_eq
% 0.21/0.45 thf(fact_22_dependent__wellorder__choice,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [P: ( A > B ) > A > B > $o] :
% 0.21/0.45 ( ! [R: B,F2: A > B,G: A > B,X2: A] :
% 0.21/0.45 ( ! [Y3: A] :
% 0.21/0.45 ( ( ord_less @ A @ Y3 @ X2 )
% 0.21/0.45 => ( ( F2 @ Y3 )
% 0.21/0.45 = ( G @ Y3 ) ) )
% 0.21/0.45 => ( ( P @ F2 @ X2 @ R )
% 0.21/0.45 = ( P @ G @ X2 @ R ) ) )
% 0.21/0.45 => ( ! [X2: A,F2: A > B] :
% 0.21/0.45 ( ! [Y3: A] :
% 0.21/0.45 ( ( ord_less @ A @ Y3 @ X2 )
% 0.21/0.45 => ( P @ F2 @ Y3 @ ( F2 @ Y3 ) ) )
% 0.21/0.45 => ? [X12: B] : ( P @ F2 @ X2 @ X12 ) )
% 0.21/0.45 => ? [F2: A > B] :
% 0.21/0.45 ! [X: A] : ( P @ F2 @ X @ ( F2 @ X ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dependent_wellorder_choice
% 0.21/0.45 thf(fact_23_not__less__iff__gr__or__eq,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ~ ( ord_less @ A @ X3 @ Y ) )
% 0.21/0.45 = ( ( ord_less @ A @ Y @ X3 )
% 0.21/0.45 | ( X3 = Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % not_less_iff_gr_or_eq
% 0.21/0.45 thf(fact_24_dual__order_Ostrict__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [B3: A,A2: A,C2: A] :
% 0.21/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.21/0.45 => ( ( ord_less @ A @ C2 @ B3 )
% 0.21/0.45 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.strict_trans
% 0.21/0.45 thf(fact_25_less__imp__not__less,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_imp_not_less
% 0.21/0.45 thf(fact_26_order_Ostrict__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,C2: A] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less @ A @ B3 @ C2 )
% 0.21/0.45 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.strict_trans
% 0.21/0.45 thf(fact_27_ord__eq__less__subst,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType] :
% 0.21/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 0.21/0.45 ( ( A2
% 0.21/0.45 = ( F3 @ B3 ) )
% 0.21/0.45 => ( ( ord_less @ B @ B3 @ C2 )
% 0.21/0.45 => ( ! [X2: B,Y2: B] :
% 0.21/0.45 ( ( ord_less @ B @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_eq_less_subst
% 0.21/0.45 thf(fact_28_ord__less__eq__subst,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType] :
% 0.21/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,B3: A,F3: A > B,C2: B] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ( F3 @ B3 )
% 0.21/0.45 = C2 )
% 0.21/0.45 => ( ! [X2: A,Y2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_less_eq_subst
% 0.21/0.45 thf(fact_29_order__less__subst1,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType] :
% 0.21/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.21/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ ( F3 @ B3 ) )
% 0.21/0.45 => ( ( ord_less @ B @ B3 @ C2 )
% 0.21/0.45 => ( ! [X2: B,Y2: B] :
% 0.21/0.45 ( ( ord_less @ B @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_less_subst1
% 0.21/0.45 thf(fact_30_order__less__subst2,axiom,
% 0.21/0.45 ! [A: $tType,C: $tType] :
% 0.21/0.45 ( ( ( order @ C @ ( type2 @ C ) )
% 0.21/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,B3: A,F3: A > C,C2: C] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
% 0.21/0.45 => ( ! [X2: A,Y2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_less_subst2
% 0.21/0.45 thf(fact_31_lt__ex,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A] :
% 0.21/0.45 ? [Y2: A] : ( ord_less @ A @ Y2 @ X3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % lt_ex
% 0.21/0.45 thf(fact_32_gt__ex,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( no_top @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A] :
% 0.21/0.45 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 0.21/0.45
% 0.21/0.45 % gt_ex
% 0.21/0.45 thf(fact_33_neqE,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( X3 != Y )
% 0.21/0.45 => ( ~ ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % neqE
% 0.21/0.45 thf(fact_34_neq__iff,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( X3 != Y )
% 0.21/0.45 = ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 | ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % neq_iff
% 0.21/0.45 thf(fact_35_order_Oasym,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.asym
% 0.21/0.45 thf(fact_36_dense,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ? [Z: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Z )
% 0.21/0.45 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dense
% 0.21/0.45 thf(fact_37_less__imp__neq,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( X3 != Y ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_imp_neq
% 0.21/0.45 thf(fact_38_less__asym,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_asym
% 0.21/0.45 thf(fact_39_less__asym_H,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_asym'
% 0.21/0.45 thf(fact_40_less__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,Z3: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ord_less @ A @ Y @ Z3 )
% 0.21/0.45 => ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_trans
% 0.21/0.45 thf(fact_41_less__linear,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 | ( X3 = Y )
% 0.21/0.45 | ( ord_less @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_linear
% 0.21/0.45 thf(fact_42_less__irrefl,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A] :
% 0.21/0.45 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_irrefl
% 0.21/0.45 thf(fact_43_ord__eq__less__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,C2: A] :
% 0.21/0.45 ( ( A2 = B3 )
% 0.21/0.45 => ( ( ord_less @ A @ B3 @ C2 )
% 0.21/0.45 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_eq_less_trans
% 0.21/0.45 thf(fact_44_ord__less__eq__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,C2: A] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( B3 = C2 )
% 0.21/0.45 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_less_eq_trans
% 0.21/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.21/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.21/0.45 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.21/0.45 = ( P @ A2 ) ) ).
% 0.21/0.45
% 0.21/0.45 % mem_Collect_eq
% 0.21/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.21/0.45 ! [A: $tType,A3: set @ A] :
% 0.21/0.45 ( ( collect @ A
% 0.21/0.45 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.21/0.45 = A3 ) ).
% 0.21/0.45
% 0.21/0.45 % Collect_mem_eq
% 0.21/0.45 thf(fact_47_Collect__cong,axiom,
% 0.21/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.21/0.45 ( ! [X2: A] :
% 0.21/0.45 ( ( P @ X2 )
% 0.21/0.45 = ( Q @ X2 ) )
% 0.21/0.45 => ( ( collect @ A @ P )
% 0.21/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % Collect_cong
% 0.21/0.45 thf(fact_48_ext,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
% 0.21/0.45 ( ! [X2: A] :
% 0.21/0.45 ( ( F3 @ X2 )
% 0.21/0.45 = ( G2 @ X2 ) )
% 0.21/0.45 => ( F3 = G2 ) ) ).
% 0.21/0.45
% 0.21/0.45 % ext
% 0.21/0.45 thf(fact_49_dual__order_Oasym,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [B3: A,A2: A] :
% 0.21/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.21/0.45 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.asym
% 0.21/0.45 thf(fact_50_less__imp__not__eq,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( X3 != Y ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_imp_not_eq
% 0.21/0.45 thf(fact_51_less__not__sym,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_not_sym
% 0.21/0.45 thf(fact_52_less__induct,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [P: A > $o,A2: A] :
% 0.21/0.45 ( ! [X2: A] :
% 0.21/0.45 ( ! [Y3: A] :
% 0.21/0.45 ( ( ord_less @ A @ Y3 @ X2 )
% 0.21/0.45 => ( P @ Y3 ) )
% 0.21/0.45 => ( P @ X2 ) )
% 0.21/0.45 => ( P @ A2 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_induct
% 0.21/0.45 thf(fact_53_antisym__conv3,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Y: A,X3: A] :
% 0.21/0.45 ( ~ ( ord_less @ A @ Y @ X3 )
% 0.21/0.45 => ( ( ~ ( ord_less @ A @ X3 @ Y ) )
% 0.21/0.45 = ( X3 = Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % antisym_conv3
% 0.21/0.45 thf(fact_54_less__imp__not__eq2,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( Y != X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_imp_not_eq2
% 0.21/0.45 thf(fact_55_less__imp__triv,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,P: $o] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ord_less @ A @ Y @ X3 )
% 0.21/0.45 => P ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_imp_triv
% 0.21/0.45 thf(fact_56_linorder__cases,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ~ ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ( X3 != Y )
% 0.21/0.45 => ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % linorder_cases
% 0.21/0.45 thf(fact_57_dual__order_Oirrefl,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A] :
% 0.21/0.45 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.irrefl
% 0.21/0.45 thf(fact_58_strict__monoD,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.21/0.45 => ! [F3: A > B,X3: A,Y: A] :
% 0.21/0.45 ( ( order_strict_mono @ A @ B @ F3 )
% 0.21/0.45 => ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % strict_monoD
% 0.21/0.45 thf(fact_59_strict__monoI,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.21/0.45 => ! [F3: A > B] :
% 0.21/0.45 ( ! [X2: A,Y2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( order_strict_mono @ A @ B @ F3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % strict_monoI
% 0.21/0.45 thf(fact_60_strict__mono__def,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.21/0.45 => ( ( order_strict_mono @ A @ B )
% 0.21/0.45 = ( ^ [F4: A > B] :
% 0.21/0.45 ! [X4: A,Y4: A] :
% 0.21/0.45 ( ( ord_less @ A @ X4 @ Y4 )
% 0.21/0.45 => ( ord_less @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % strict_mono_def
% 0.21/0.45 thf(fact_61_strict__mono__less,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.21/0.45 => ! [F3: A > B,X3: A,Y: A] :
% 0.21/0.45 ( ( order_strict_mono @ A @ B @ F3 )
% 0.21/0.45 => ( ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
% 0.21/0.45 = ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % strict_mono_less
% 0.21/0.45 thf(fact_62_x2__nin__l2,axiom,
% 0.21/0.45 ~ ( member @ a @ x2 @ ( set2 @ a @ l2b ) ) ).
% 0.21/0.45
% 0.21/0.45 % x2_nin_l2
% 0.21/0.45 thf(fact_63_x1__nin__l1,axiom,
% 0.21/0.45 ~ ( member @ a @ x1 @ ( set2 @ a @ l1a ) ) ).
% 0.21/0.45
% 0.21/0.45 % x1_nin_l1
% 0.21/0.45 thf(fact_64_ord_Omin__def,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( min @ A )
% 0.21/0.45 = ( ^ [Less_eq: A > A > $o,A4: A,B4: A] : ( if @ A @ ( Less_eq @ A4 @ B4 ) @ A4 @ B4 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord.min_def
% 0.21/0.45 thf(fact_65_ord_Omax__def,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( max @ A )
% 0.21/0.45 = ( ^ [Less_eq: A > A > $o,A4: A,B4: A] : ( if @ A @ ( Less_eq @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord.max_def
% 0.21/0.45 thf(fact_66_top_Oextremum__strict,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order_top @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A] :
% 0.21/0.45 ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% 0.21/0.45
% 0.21/0.45 % top.extremum_strict
% 0.21/0.45 thf(fact_67_top__apply,axiom,
% 0.21/0.45 ! [C: $tType,D: $tType] :
% 0.21/0.45 ( ( top @ C @ ( type2 @ C ) )
% 0.21/0.45 => ( ( top_top @ ( D > C ) )
% 0.21/0.45 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % top_apply
% 0.21/0.45 thf(fact_68_strict__mono__eq,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.21/0.45 => ! [F3: A > B,X3: A,Y: A] :
% 0.21/0.45 ( ( order_strict_mono @ A @ B @ F3 )
% 0.21/0.45 => ( ( ( F3 @ X3 )
% 0.21/0.45 = ( F3 @ Y ) )
% 0.21/0.45 = ( X3 = Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % strict_mono_eq
% 0.21/0.45 thf(fact_69_top_Onot__eq__extremum,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order_top @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A] :
% 0.21/0.45 ( ( A2
% 0.21/0.45 != ( top_top @ A ) )
% 0.21/0.45 = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % top.not_eq_extremum
% 0.21/0.45 thf(fact_70_x2__le,axiom,
% 0.21/0.45 ! [X3: a] :
% 0.21/0.45 ( ( member @ a @ X3 @ ( set2 @ a @ l2b ) )
% 0.21/0.45 => ( ord_less_eq @ a @ x2 @ X3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % x2_le
% 0.21/0.45 thf(fact_71_x1__le,axiom,
% 0.21/0.45 ! [X3: a] :
% 0.21/0.45 ( ( member @ a @ X3 @ ( set2 @ a @ l1a ) )
% 0.21/0.45 => ( ord_less_eq @ a @ x1 @ X3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % x1_le
% 0.21/0.45 thf(fact_72_ind__hyp__l2,axiom,
% 0.21/0.45 ( ( sorted1061247458sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b )
% 0.21/0.45 = ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ l2b ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ind_hyp_l2
% 0.21/0.45 thf(fact_73_x2__l2__props,axiom,
% 0.21/0.45 ( ( distinct @ a @ ( cons @ a @ x2 @ l2b ) )
% 0.21/0.45 & ( linorder_sorted @ a @ ( cons @ a @ x2 @ l2b ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % x2_l2_props
% 0.21/0.45 thf(fact_74_x1__l1__props,axiom,
% 0.21/0.45 ( ( distinct @ a @ ( cons @ a @ x1 @ l1a ) )
% 0.21/0.45 & ( linorder_sorted @ a @ ( cons @ a @ x1 @ l1a ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % x1_l1_props
% 0.21/0.45 thf(fact_75_Cons_Ohyps,axiom,
% 0.21/0.45 ( ( ( distinct @ a @ l2b )
% 0.21/0.45 & ( linorder_sorted @ a @ l2b ) )
% 0.21/0.45 => ( ( sorted1061247458sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b )
% 0.21/0.45 = ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ l2b ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % Cons.hyps
% 0.21/0.45 thf(fact_76_strict__mono__less__eq,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.21/0.45 => ! [F3: A > B,X3: A,Y: A] :
% 0.21/0.45 ( ( order_strict_mono @ A @ B @ F3 )
% 0.21/0.45 => ( ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
% 0.21/0.45 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % strict_mono_less_eq
% 0.21/0.45 thf(fact_77_l2__OK,axiom,
% 0.21/0.45 ( ( distinct @ a @ l2 )
% 0.21/0.45 & ( linorder_sorted @ a @ l2 ) ) ).
% 0.21/0.45
% 0.21/0.45 % l2_OK
% 0.21/0.45 thf(fact_78_l1__OK,axiom,
% 0.21/0.45 ( ( distinct @ a @ l1 )
% 0.21/0.45 & ( linorder_sorted @ a @ l1 ) ) ).
% 0.21/0.45
% 0.21/0.45 % l1_OK
% 0.21/0.45 thf(fact_79_order__refl,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_refl
% 0.21/0.45 thf(fact_80_l2__props,axiom,
% 0.21/0.45 ( ( distinct @ a @ l2b )
% 0.21/0.45 & ( linorder_sorted @ a @ l2b ) ) ).
% 0.21/0.45
% 0.21/0.45 % l2_props
% 0.21/0.45 thf(fact_81_l1__props,axiom,
% 0.21/0.45 ( ( distinct @ a @ l1a )
% 0.21/0.45 & ( linorder_sorted @ a @ l1a ) ) ).
% 0.21/0.45
% 0.21/0.45 % l1_props
% 0.21/0.45 thf(fact_82_ind__hyp__l1,axiom,
% 0.21/0.45 ! [L2: list @ a] :
% 0.21/0.45 ( ( ( distinct @ a @ L2 )
% 0.21/0.45 & ( linorder_sorted @ a @ L2 ) )
% 0.21/0.45 => ( ( sorted1061247458sorted @ a @ l1a @ L2 )
% 0.21/0.45 = ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ l1a ) @ ( set2 @ a @ L2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ind_hyp_l1
% 0.21/0.45 thf(fact_83_subset__sorted_Osimps_I3_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X13: A,L1: list @ A,X22: A,L2: list @ A] :
% 0.21/0.45 ( ( sorted1061247458sorted @ A @ ( cons @ A @ X13 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
% 0.21/0.45 = ( ~ ( ord_less @ A @ X13 @ X22 )
% 0.21/0.45 & ( ~ ( ord_less @ A @ X13 @ X22 )
% 0.21/0.45 => ( ( ( X13 = X22 )
% 0.21/0.45 => ( sorted1061247458sorted @ A @ L1 @ L2 ) )
% 0.21/0.45 & ( ( X13 != X22 )
% 0.21/0.45 => ( sorted1061247458sorted @ A @ ( cons @ A @ X13 @ L1 ) @ L2 ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % subset_sorted.simps(3)
% 0.21/0.45 thf(fact_84_le__funD,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.45 => ! [F3: A > B,G2: A > B,X3: A] :
% 0.21/0.45 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.21/0.45 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_funD
% 0.21/0.45 thf(fact_85_le__funE,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.45 => ! [F3: A > B,G2: A > B,X3: A] :
% 0.21/0.45 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.21/0.45 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_funE
% 0.21/0.45 thf(fact_86_le__funI,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.45 => ! [F3: A > B,G2: A > B] :
% 0.21/0.45 ( ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.21/0.45 => ( ord_less_eq @ ( A > B ) @ F3 @ G2 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_funI
% 0.21/0.45 thf(fact_87_le__fun__def,axiom,
% 0.21/0.45 ! [B: $tType,A: $tType] :
% 0.21/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.45 => ( ( ord_less_eq @ ( A > B ) )
% 0.21/0.45 = ( ^ [F4: A > B,G3: A > B] :
% 0.21/0.45 ! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_fun_def
% 0.21/0.45 thf(fact_88_order__subst1,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType] :
% 0.21/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.21/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B3 ) )
% 0.21/0.45 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.21/0.45 => ( ! [X2: B,Y2: B] :
% 0.21/0.45 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst1
% 0.21/0.45 thf(fact_89_order__subst2,axiom,
% 0.21/0.45 ! [A: $tType,C: $tType] :
% 0.21/0.45 ( ( ( order @ C @ ( type2 @ C ) )
% 0.21/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,B3: A,F3: A > C,C2: C] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
% 0.21/0.45 => ( ! [X2: A,Y2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst2
% 0.21/0.45 thf(fact_90_ord__eq__le__subst,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType] :
% 0.21/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 0.21/0.45 ( ( A2
% 0.21/0.45 = ( F3 @ B3 ) )
% 0.21/0.45 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.21/0.45 => ( ! [X2: B,Y2: B] :
% 0.21/0.45 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_eq_le_subst
% 0.21/0.45 thf(fact_91_ord__le__eq__subst,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType] :
% 0.21/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,B3: A,F3: A > B,C2: B] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ( F3 @ B3 )
% 0.21/0.45 = C2 )
% 0.21/0.45 => ( ! [X2: A,Y2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_le_eq_subst
% 0.21/0.45 thf(fact_92_eq__iff,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.21/0.45 = ( ^ [X4: A,Y4: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.21/0.45 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % eq_iff
% 0.21/0.45 thf(fact_93_antisym,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ord_less_eq @ A @ Y @ X3 )
% 0.21/0.45 => ( X3 = Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % antisym
% 0.21/0.45 thf(fact_94_linear,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 | ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % linear
% 0.21/0.45 thf(fact_95_eq__refl,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( X3 = Y )
% 0.21/0.45 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % eq_refl
% 0.21/0.45 thf(fact_96_le__cases,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ~ ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_cases
% 0.21/0.45 thf(fact_97_order_Otrans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,C2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.21/0.45 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.trans
% 0.21/0.45 thf(fact_98_le__cases3,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,Z3: A] :
% 0.21/0.45 ( ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.21/0.45 => ( ( ( ord_less_eq @ A @ Y @ X3 )
% 0.21/0.45 => ~ ( ord_less_eq @ A @ X3 @ Z3 ) )
% 0.21/0.45 => ( ( ( ord_less_eq @ A @ X3 @ Z3 )
% 0.21/0.45 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.21/0.45 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.21/0.45 => ~ ( ord_less_eq @ A @ Y @ X3 ) )
% 0.21/0.45 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.21/0.45 => ~ ( ord_less_eq @ A @ Z3 @ X3 ) )
% 0.21/0.45 => ~ ( ( ord_less_eq @ A @ Z3 @ X3 )
% 0.21/0.45 => ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_cases3
% 0.21/0.45 thf(fact_99_antisym__conv,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Y: A,X3: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ Y @ X3 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 = ( X3 = Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % antisym_conv
% 0.21/0.45 thf(fact_100_ord__eq__le__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,C2: A] :
% 0.21/0.45 ( ( A2 = B3 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.21/0.45 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_eq_le_trans
% 0.21/0.45 thf(fact_101_ord__le__eq__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,C2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( B3 = C2 )
% 0.21/0.45 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_le_eq_trans
% 0.21/0.45 thf(fact_102_order__class_Oorder_Oantisym,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.21/0.45 => ( A2 = B3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_class.order.antisym
% 0.21/0.45 thf(fact_103_order__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,Z3: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.21/0.45 => ( ord_less_eq @ A @ X3 @ Z3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_trans
% 0.21/0.45 thf(fact_104_dual__order_Orefl,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.refl
% 0.21/0.45 thf(fact_105_linorder__wlog,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [P: A > A > $o,A2: A,B3: A] :
% 0.21/0.45 ( ! [A5: A,B2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A5 @ B2 )
% 0.21/0.45 => ( P @ A5 @ B2 ) )
% 0.21/0.45 => ( ! [A5: A,B2: A] :
% 0.21/0.45 ( ( P @ B2 @ A5 )
% 0.21/0.45 => ( P @ A5 @ B2 ) )
% 0.21/0.45 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % linorder_wlog
% 0.21/0.45 thf(fact_106_dual__order_Otrans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [B3: A,A2: A,C2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.21/0.45 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.trans
% 0.21/0.45 thf(fact_107_dual__order_Oantisym,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [B3: A,A2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( A2 = B3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.antisym
% 0.21/0.45 thf(fact_108_leD,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Y: A,X3: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ Y @ X3 )
% 0.21/0.45 => ~ ( ord_less @ A @ X3 @ Y ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % leD
% 0.21/0.45 thf(fact_109_leI,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ~ ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % leI
% 0.21/0.45 thf(fact_110_le__less,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ( ( ord_less_eq @ A )
% 0.21/0.45 = ( ^ [X4: A,Y4: A] :
% 0.21/0.45 ( ( ord_less @ A @ X4 @ Y4 )
% 0.21/0.45 | ( X4 = Y4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_less
% 0.21/0.45 thf(fact_111_less__le,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ( ( ord_less @ A )
% 0.21/0.45 = ( ^ [X4: A,Y4: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.21/0.45 & ( X4 != Y4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_le
% 0.21/0.45 thf(fact_112_order__le__less__subst1,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType] :
% 0.21/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.21/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B3 ) )
% 0.21/0.45 => ( ( ord_less @ B @ B3 @ C2 )
% 0.21/0.45 => ( ! [X2: B,Y2: B] :
% 0.21/0.45 ( ( ord_less @ B @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_le_less_subst1
% 0.21/0.45 thf(fact_113_order__le__less__subst2,axiom,
% 0.21/0.45 ! [A: $tType,C: $tType] :
% 0.21/0.45 ( ( ( order @ C @ ( type2 @ C ) )
% 0.21/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,B3: A,F3: A > C,C2: C] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
% 0.21/0.45 => ( ! [X2: A,Y2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_le_less_subst2
% 0.21/0.45 thf(fact_114_order__less__le__subst1,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType] :
% 0.21/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.21/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ ( F3 @ B3 ) )
% 0.21/0.45 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.21/0.45 => ( ! [X2: B,Y2: B] :
% 0.21/0.45 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_less_le_subst1
% 0.21/0.45 thf(fact_115_order__less__le__subst2,axiom,
% 0.21/0.45 ! [A: $tType,C: $tType] :
% 0.21/0.45 ( ( ( order @ C @ ( type2 @ C ) )
% 0.21/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.21/0.45 => ! [A2: A,B3: A,F3: A > C,C2: C] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
% 0.21/0.45 => ( ! [X2: A,Y2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.21/0.45 => ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_less_le_subst2
% 0.21/0.45 thf(fact_116_not__le,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ~ ( ord_less_eq @ A @ X3 @ Y ) )
% 0.21/0.45 = ( ord_less @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % not_le
% 0.21/0.45 thf(fact_117_not__less,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ~ ( ord_less @ A @ X3 @ Y ) )
% 0.21/0.45 = ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % not_less
% 0.21/0.45 thf(fact_118_le__neq__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( A2 != B3 )
% 0.21/0.45 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_neq_trans
% 0.21/0.45 thf(fact_119_less__imp__le,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_imp_le
% 0.21/0.45 thf(fact_120_antisym__conv1,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ~ ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 = ( X3 = Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % antisym_conv1
% 0.21/0.45 thf(fact_121_antisym__conv2,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ~ ( ord_less @ A @ X3 @ Y ) )
% 0.21/0.45 = ( X3 = Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % antisym_conv2
% 0.21/0.45 thf(fact_122_le__less__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,Z3: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ord_less @ A @ Y @ Z3 )
% 0.21/0.45 => ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_less_trans
% 0.21/0.45 thf(fact_123_less__le__trans,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,Z3: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.21/0.45 => ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_le_trans
% 0.21/0.45 thf(fact_124_dense__ge,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Z3: A,Y: A] :
% 0.21/0.45 ( ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z3 @ X2 )
% 0.21/0.45 => ( ord_less_eq @ A @ Y @ X2 ) )
% 0.21/0.45 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dense_ge
% 0.21/0.45 thf(fact_125_dense__le,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Y: A,Z3: A] :
% 0.21/0.45 ( ! [X2: A] :
% 0.21/0.45 ( ( ord_less @ A @ X2 @ Y )
% 0.21/0.45 => ( ord_less_eq @ A @ X2 @ Z3 ) )
% 0.21/0.45 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dense_le
% 0.21/0.45 thf(fact_126_le__less__linear,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 | ( ord_less @ A @ Y @ X3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_less_linear
% 0.21/0.45 thf(fact_127_le__imp__less__or__eq,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 => ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 | ( X3 = Y ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_imp_less_or_eq
% 0.21/0.45 thf(fact_128_less__le__not__le,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ( ( ord_less @ A )
% 0.21/0.45 = ( ^ [X4: A,Y4: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.21/0.45 & ~ ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % less_le_not_le
% 0.21/0.45 thf(fact_129_not__le__imp__less,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Y: A,X3: A] :
% 0.21/0.45 ( ~ ( ord_less_eq @ A @ Y @ X3 )
% 0.21/0.45 => ( ord_less @ A @ X3 @ Y ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % not_le_imp_less
% 0.21/0.45 thf(fact_130_order_Ostrict__trans1,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,C2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less @ A @ B3 @ C2 )
% 0.21/0.45 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.strict_trans1
% 0.21/0.45 thf(fact_131_order_Ostrict__trans2,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,C2: A] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.21/0.45 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.strict_trans2
% 0.21/0.45 thf(fact_132_order_Oorder__iff__strict,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ( ( ord_less_eq @ A )
% 0.21/0.45 = ( ^ [A4: A,B4: A] :
% 0.21/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.21/0.45 | ( A4 = B4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.order_iff_strict
% 0.21/0.45 thf(fact_133_order_Ostrict__iff__order,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ( ( ord_less @ A )
% 0.21/0.45 = ( ^ [A4: A,B4: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.21/0.45 & ( A4 != B4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.strict_iff_order
% 0.21/0.45 thf(fact_134_dual__order_Ostrict__trans1,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [B3: A,A2: A,C2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.21/0.45 => ( ( ord_less @ A @ C2 @ B3 )
% 0.21/0.45 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.strict_trans1
% 0.21/0.45 thf(fact_135_dual__order_Ostrict__trans2,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [B3: A,A2: A,C2: A] :
% 0.21/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.21/0.45 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.strict_trans2
% 0.21/0.45 thf(fact_136_dense__ge__bounded,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Z3: A,X3: A,Y: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z3 @ X3 )
% 0.21/0.45 => ( ! [W: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z3 @ W )
% 0.21/0.45 => ( ( ord_less @ A @ W @ X3 )
% 0.21/0.45 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.21/0.45 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dense_ge_bounded
% 0.21/0.45 thf(fact_137_dense__le__bounded,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,Z3: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ Y )
% 0.21/0.45 => ( ! [W: A] :
% 0.21/0.45 ( ( ord_less @ A @ X3 @ W )
% 0.21/0.45 => ( ( ord_less @ A @ W @ Y )
% 0.21/0.45 => ( ord_less_eq @ A @ W @ Z3 ) ) )
% 0.21/0.45 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dense_le_bounded
% 0.21/0.45 thf(fact_138_order_Ostrict__implies__order,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.strict_implies_order
% 0.21/0.45 thf(fact_139_dual__order_Oorder__iff__strict,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ( ( ord_less_eq @ A )
% 0.21/0.45 = ( ^ [B4: A,A4: A] :
% 0.21/0.45 ( ( ord_less @ A @ B4 @ A4 )
% 0.21/0.45 | ( A4 = B4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.order_iff_strict
% 0.21/0.45 thf(fact_140_dual__order_Ostrict__iff__order,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ( ( ord_less @ A )
% 0.21/0.45 = ( ^ [B4: A,A4: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.21/0.45 & ( A4 != B4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.strict_iff_order
% 0.21/0.45 thf(fact_141_dual__order_Ostrict__implies__order,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [B3: A,A2: A] :
% 0.21/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.21/0.45 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % dual_order.strict_implies_order
% 0.21/0.45 thf(fact_142_order_Onot__eq__order__implies__strict,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A] :
% 0.21/0.45 ( ( A2 != B3 )
% 0.21/0.45 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.21/0.45 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order.not_eq_order_implies_strict
% 0.21/0.45 thf(fact_143_complete__interval,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A,B3: A,P: A > $o] :
% 0.21/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.21/0.45 => ( ( P @ A2 )
% 0.21/0.45 => ( ~ ( P @ B3 )
% 0.21/0.45 => ? [C3: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.21/0.45 & ( ord_less_eq @ A @ C3 @ B3 )
% 0.21/0.45 & ! [X: A] :
% 0.21/0.45 ( ( ( ord_less_eq @ A @ A2 @ X )
% 0.21/0.45 & ( ord_less @ A @ X @ C3 ) )
% 0.21/0.45 => ( P @ X ) )
% 0.21/0.45 & ! [D2: A] :
% 0.21/0.45 ( ! [X2: A] :
% 0.21/0.45 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.21/0.45 & ( ord_less @ A @ X2 @ D2 ) )
% 0.21/0.45 => ( P @ X2 ) )
% 0.21/0.45 => ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % complete_interval
% 0.21/0.45 thf(fact_144_minf_I8_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ X @ Z )
% 0.21/0.45 => ~ ( ord_less_eq @ A @ T @ X ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(8)
% 0.21/0.45 thf(fact_145_minf_I6_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ X @ Z )
% 0.21/0.45 => ( ord_less_eq @ A @ X @ T ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % minf(6)
% 0.21/0.45 thf(fact_146_pinf_I8_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z @ X )
% 0.21/0.45 => ( ord_less_eq @ A @ T @ X ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(8)
% 0.21/0.45 thf(fact_147_pinf_I6_J,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [T: A] :
% 0.21/0.45 ? [Z: A] :
% 0.21/0.45 ! [X: A] :
% 0.21/0.45 ( ( ord_less @ A @ Z @ X )
% 0.21/0.45 => ~ ( ord_less_eq @ A @ X @ T ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % pinf(6)
% 0.21/0.45 thf(fact_148_top__greatest,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order_top @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % top_greatest
% 0.21/0.45 thf(fact_149_top_Oextremum__unique,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order_top @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.21/0.45 = ( A2
% 0.21/0.45 = ( top_top @ A ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % top.extremum_unique
% 0.21/0.45 thf(fact_150_top_Oextremum__uniqueI,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( order_top @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [A2: A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.21/0.45 => ( A2
% 0.21/0.45 = ( top_top @ A ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % top.extremum_uniqueI
% 0.21/0.45 thf(fact_151_sorted__many__eq,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,Zs: list @ A] :
% 0.21/0.45 ( ( linorder_sorted @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs ) ) )
% 0.21/0.45 = ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.45 & ( linorder_sorted @ A @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sorted_many_eq
% 0.21/0.45 thf(fact_152_sorted_OCons,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Xs: list @ A,X3: A] :
% 0.21/0.45 ( ! [X2: A] :
% 0.21/0.45 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.21/0.45 => ( ord_less_eq @ A @ X3 @ X2 ) )
% 0.21/0.45 => ( ( linorder_sorted @ A @ Xs )
% 0.21/0.45 => ( linorder_sorted @ A @ ( cons @ A @ X3 @ Xs ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sorted.Cons
% 0.21/0.45 thf(fact_153_sorted__Cons,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Xs: list @ A] :
% 0.21/0.45 ( ( linorder_sorted @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.21/0.45 = ( ( linorder_sorted @ A @ Xs )
% 0.21/0.45 & ! [X4: A] :
% 0.21/0.45 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.21/0.45 => ( ord_less_eq @ A @ X3 @ X4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sorted_Cons
% 0.21/0.45 thf(fact_154_sorted__distinct__set__unique,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [Xs: list @ A,Ys: list @ A] :
% 0.21/0.45 ( ( linorder_sorted @ A @ Xs )
% 0.21/0.45 => ( ( distinct @ A @ Xs )
% 0.21/0.45 => ( ( linorder_sorted @ A @ Ys )
% 0.21/0.45 => ( ( distinct @ A @ Ys )
% 0.21/0.45 => ( ( ( set2 @ A @ Xs )
% 0.21/0.45 = ( set2 @ A @ Ys ) )
% 0.21/0.45 => ( Xs = Ys ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sorted_distinct_set_unique
% 0.21/0.45 thf(fact_155_distinct_Osimps_I2_J,axiom,
% 0.21/0.45 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.21/0.45 ( ( distinct @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.21/0.45 = ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.21/0.45 & ( distinct @ A @ Xs ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % distinct.simps(2)
% 0.21/0.45 thf(fact_156_sorted__many,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.45 => ! [X3: A,Y: A,Zs: list @ A] :
% 0.21/0.45 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.21/0.46 => ( ( linorder_sorted @ A @ ( cons @ A @ Y @ Zs ) )
% 0.21/0.46 => ( linorder_sorted @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % sorted_many
% 0.21/0.46 thf(fact_157_list_Oinject,axiom,
% 0.21/0.46 ! [A: $tType,X21: A,X222: list @ A,Y21: A,Y22: list @ A] :
% 0.21/0.46 ( ( ( cons @ A @ X21 @ X222 )
% 0.21/0.46 = ( cons @ A @ Y21 @ Y22 ) )
% 0.21/0.46 = ( ( X21 = Y21 )
% 0.21/0.46 & ( X222 = Y22 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list.inject
% 0.21/0.46 thf(fact_158_less__fun__def,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType] :
% 0.21/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.21/0.46 => ( ( ord_less @ ( A > B ) )
% 0.21/0.46 = ( ^ [F4: A > B,G3: A > B] :
% 0.21/0.46 ( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
% 0.21/0.46 & ~ ( ord_less_eq @ ( A > B ) @ G3 @ F4 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_fun_def
% 0.21/0.46 thf(fact_159_not__Cons__self2,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.21/0.46 ( ( cons @ A @ X3 @ Xs )
% 0.21/0.46 != Xs ) ).
% 0.21/0.46
% 0.21/0.46 % not_Cons_self2
% 0.21/0.46 thf(fact_160_list_Oset__cases,axiom,
% 0.21/0.46 ! [A: $tType,E: A,A2: list @ A] :
% 0.21/0.46 ( ( member @ A @ E @ ( set2 @ A @ A2 ) )
% 0.21/0.46 => ( ! [Z22: list @ A] :
% 0.21/0.46 ( A2
% 0.21/0.46 != ( cons @ A @ E @ Z22 ) )
% 0.21/0.46 => ~ ! [Z1: A,Z22: list @ A] :
% 0.21/0.46 ( ( A2
% 0.21/0.46 = ( cons @ A @ Z1 @ Z22 ) )
% 0.21/0.46 => ~ ( member @ A @ E @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list.set_cases
% 0.21/0.46 thf(fact_161_set__ConsD,axiom,
% 0.21/0.46 ! [A: $tType,Y: A,X3: A,Xs: list @ A] :
% 0.21/0.46 ( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
% 0.21/0.46 => ( ( Y = X3 )
% 0.21/0.46 | ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % set_ConsD
% 0.21/0.46 thf(fact_162_list_Oset__intros_I1_J,axiom,
% 0.21/0.46 ! [A: $tType,A1: A,A22: list @ A] : ( member @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list.set_intros(1)
% 0.21/0.46 thf(fact_163_list_Oset__intros_I2_J,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,A22: list @ A,A1: A] :
% 0.21/0.46 ( ( member @ A @ X3 @ ( set2 @ A @ A22 ) )
% 0.21/0.46 => ( member @ A @ X3 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list.set_intros(2)
% 0.21/0.46 thf(fact_164_subset__code_I1_J,axiom,
% 0.21/0.46 ! [A: $tType,Xs: list @ A,B5: set @ A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B5 )
% 0.21/0.46 = ( ! [X4: A] :
% 0.21/0.46 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.21/0.46 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_code(1)
% 0.21/0.46 thf(fact_165_distinct__length__2__or__more,axiom,
% 0.21/0.46 ! [A: $tType,A2: A,B3: A,Xs: list @ A] :
% 0.21/0.46 ( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B3 @ Xs ) ) )
% 0.21/0.46 = ( ( A2 != B3 )
% 0.21/0.46 & ( distinct @ A @ ( cons @ A @ A2 @ Xs ) )
% 0.21/0.46 & ( distinct @ A @ ( cons @ A @ B3 @ Xs ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % distinct_length_2_or_more
% 0.21/0.46 thf(fact_166_set__subset__Cons,axiom,
% 0.21/0.46 ! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % set_subset_Cons
% 0.21/0.46 thf(fact_167_subsetI,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ! [X2: A] :
% 0.21/0.46 ( ( member @ A @ X2 @ A3 )
% 0.21/0.46 => ( member @ A @ X2 @ B5 ) )
% 0.21/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 0.21/0.46
% 0.21/0.46 % subsetI
% 0.21/0.46 thf(fact_168_subset__antisym,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
% 0.21/0.46 => ( A3 = B5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_antisym
% 0.21/0.46 thf(fact_169_psubsetI,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( A3 != B5 )
% 0.21/0.46 => ( ord_less @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubsetI
% 0.21/0.46 thf(fact_170_subset__UNIV,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_UNIV
% 0.21/0.46 thf(fact_171_psubsetE,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubsetE
% 0.21/0.46 thf(fact_172_psubset__eq,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( ord_less @ ( set @ A ) )
% 0.21/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.21/0.46 & ( A6 != B6 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubset_eq
% 0.21/0.46 thf(fact_173_psubset__imp__subset,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubset_imp_subset
% 0.21/0.46 thf(fact_174_psubset__subset__trans,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
% 0.21/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.21/0.46 => ( ord_less @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubset_subset_trans
% 0.21/0.46 thf(fact_175_subset__not__subset__eq,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( ord_less @ ( set @ A ) )
% 0.21/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.21/0.46 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_not_subset_eq
% 0.21/0.46 thf(fact_176_subset__psubset__trans,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( ord_less @ ( set @ A ) @ B5 @ C4 )
% 0.21/0.46 => ( ord_less @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_psubset_trans
% 0.21/0.46 thf(fact_177_subset__iff__psubset__eq,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.21/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.21/0.46 ( ( ord_less @ ( set @ A ) @ A6 @ B6 )
% 0.21/0.46 | ( A6 = B6 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_iff_psubset_eq
% 0.21/0.46 thf(fact_178_Collect__mono__iff,axiom,
% 0.21/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.21/0.46 = ( ! [X4: A] :
% 0.21/0.46 ( ( P @ X4 )
% 0.21/0.46 => ( Q @ X4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Collect_mono_iff
% 0.21/0.46 thf(fact_179_contra__subsetD,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ~ ( member @ A @ C2 @ B5 )
% 0.21/0.46 => ~ ( member @ A @ C2 @ A3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % contra_subsetD
% 0.21/0.46 thf(fact_180_set__eq__subset,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( ^ [Y5: set @ A,Z4: set @ A] : ( Y5 = Z4 ) )
% 0.21/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.21/0.46 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % set_eq_subset
% 0.21/0.46 thf(fact_181_subset__trans,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.21/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_trans
% 0.21/0.46 thf(fact_182_Collect__mono,axiom,
% 0.21/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.21/0.46 ( ! [X2: A] :
% 0.21/0.46 ( ( P @ X2 )
% 0.21/0.46 => ( Q @ X2 ) )
% 0.21/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Collect_mono
% 0.21/0.46 thf(fact_183_subset__refl,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.21/0.46
% 0.21/0.46 % subset_refl
% 0.21/0.46 thf(fact_184_rev__subsetD,axiom,
% 0.21/0.46 ! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( member @ A @ C2 @ A3 )
% 0.21/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( member @ A @ C2 @ B5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rev_subsetD
% 0.21/0.46 thf(fact_185_subset__iff,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.21/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.21/0.46 ! [T2: A] :
% 0.21/0.46 ( ( member @ A @ T2 @ A6 )
% 0.21/0.46 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_iff
% 0.21/0.46 thf(fact_186_set__rev__mp,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( member @ A @ X3 @ A3 )
% 0.21/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( member @ A @ X3 @ B5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % set_rev_mp
% 0.21/0.46 thf(fact_187_equalityD2,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( A3 = B5 )
% 0.21/0.46 => ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% 0.21/0.46
% 0.21/0.46 % equalityD2
% 0.21/0.46 thf(fact_188_equalityD1,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( A3 = B5 )
% 0.21/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 0.21/0.46
% 0.21/0.46 % equalityD1
% 0.21/0.46 thf(fact_189_subset__eq,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.21/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.21/0.46 ! [X4: A] :
% 0.21/0.46 ( ( member @ A @ X4 @ A6 )
% 0.21/0.46 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_eq
% 0.21/0.46 thf(fact_190_equalityE,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.21/0.46 ( ( A3 = B5 )
% 0.21/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % equalityE
% 0.21/0.46 thf(fact_191_subsetCE,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( member @ A @ C2 @ A3 )
% 0.21/0.46 => ( member @ A @ C2 @ B5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subsetCE
% 0.21/0.46 thf(fact_192_subsetD,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( member @ A @ C2 @ A3 )
% 0.21/0.46 => ( member @ A @ C2 @ B5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subsetD
% 0.21/0.46 thf(fact_193_in__mono,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A,X3: A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( member @ A @ X3 @ A3 )
% 0.21/0.46 => ( member @ A @ X3 @ B5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % in_mono
% 0.21/0.46 thf(fact_194_set__mp,axiom,
% 0.21/0.46 ! [A: $tType,A3: set @ A,B5: set @ A,X3: A] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.21/0.46 => ( ( member @ A @ X3 @ A3 )
% 0.21/0.46 => ( member @ A @ X3 @ B5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % set_mp
% 0.21/0.46 thf(fact_195_distinct__union,axiom,
% 0.21/0.46 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.21/0.46 ( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
% 0.21/0.46 = ( distinct @ A @ Ys ) ) ).
% 0.21/0.46
% 0.21/0.46 % distinct_union
% 0.21/0.46 thf(fact_196_top_Oordering__top__axioms,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.21/0.46 => ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % top.ordering_top_axioms
% 0.21/0.46 thf(fact_197_not__in__set__insert,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.21/0.46 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.21/0.46 => ( ( insert @ A @ X3 @ Xs )
% 0.21/0.46 = ( cons @ A @ X3 @ Xs ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % not_in_set_insert
% 0.21/0.46 thf(fact_198_in__set__insert,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.21/0.46 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.21/0.46 => ( ( insert @ A @ X3 @ Xs )
% 0.21/0.46 = Xs ) ) ).
% 0.21/0.46
% 0.21/0.46 % in_set_insert
% 0.21/0.46 thf(fact_199_distinct__insert,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.21/0.46 ( ( distinct @ A @ ( insert @ A @ X3 @ Xs ) )
% 0.21/0.46 = ( distinct @ A @ Xs ) ) ).
% 0.21/0.46
% 0.21/0.46 % distinct_insert
% 0.21/0.46 thf(fact_200_ordering__top_Oextremum__uniqueI,axiom,
% 0.21/0.46 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 0.21/0.46 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 0.21/0.46 => ( ( Less_eq2 @ Top @ A2 )
% 0.21/0.46 => ( A2 = Top ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ordering_top.extremum_uniqueI
% 0.21/0.46 thf(fact_201_ordering__top_Onot__eq__extremum,axiom,
% 0.21/0.46 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 0.21/0.46 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 0.21/0.46 => ( ( A2 != Top )
% 0.21/0.46 = ( Less @ A2 @ Top ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ordering_top.not_eq_extremum
% 0.21/0.46 thf(fact_202_ordering__top_Oextremum__unique,axiom,
% 0.21/0.46 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 0.21/0.46 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 0.21/0.46 => ( ( Less_eq2 @ Top @ A2 )
% 0.21/0.46 = ( A2 = Top ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ordering_top.extremum_unique
% 0.21/0.46 thf(fact_203_ordering__top_Oextremum__strict,axiom,
% 0.21/0.46 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 0.21/0.46 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 0.21/0.46 => ~ ( Less @ Top @ A2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % ordering_top.extremum_strict
% 0.21/0.46 thf(fact_204_ordering__top_Oextremum,axiom,
% 0.21/0.46 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 0.21/0.46 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 0.21/0.46 => ( Less_eq2 @ A2 @ Top ) ) ).
% 0.21/0.46
% 0.21/0.46 % ordering_top.extremum
% 0.21/0.46 thf(fact_205_List_Oinsert__def,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( insert @ A )
% 0.21/0.46 = ( ^ [X4: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) ) @ Xs2 @ ( cons @ A @ X4 @ Xs2 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % List.insert_def
% 0.21/0.46 thf(fact_206_sorted_Oinducts,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [X3: list @ A,P: ( list @ A ) > $o] :
% 0.21/0.46 ( ( linorder_sorted @ A @ X3 )
% 0.21/0.46 => ( ( P @ ( nil @ A ) )
% 0.21/0.46 => ( ! [Xs3: list @ A,X2: A] :
% 0.21/0.46 ( ! [Xa: A] :
% 0.21/0.46 ( ( member @ A @ Xa @ ( set2 @ A @ Xs3 ) )
% 0.21/0.46 => ( ord_less_eq @ A @ X2 @ Xa ) )
% 0.21/0.46 => ( ( linorder_sorted @ A @ Xs3 )
% 0.21/0.46 => ( ( P @ Xs3 )
% 0.21/0.46 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) ) )
% 0.21/0.46 => ( P @ X3 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % sorted.inducts
% 0.21/0.46 thf(fact_207_sorted_Osimps,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ( ( linorder_sorted @ A )
% 0.21/0.46 = ( ^ [A4: list @ A] :
% 0.21/0.46 ( ( A4
% 0.21/0.46 = ( nil @ A ) )
% 0.21/0.46 | ? [Xs2: list @ A,X4: A] :
% 0.21/0.46 ( ( A4
% 0.21/0.46 = ( cons @ A @ X4 @ Xs2 ) )
% 0.21/0.46 & ! [Y4: A] :
% 0.21/0.46 ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
% 0.21/0.46 => ( ord_less_eq @ A @ X4 @ Y4 ) )
% 0.21/0.46 & ( linorder_sorted @ A @ Xs2 ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % sorted.simps
% 0.21/0.46 thf(fact_208_sorted__single,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [X3: A] : ( linorder_sorted @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % sorted_single
% 0.21/0.46 thf(fact_209_insert__Nil,axiom,
% 0.21/0.46 ! [A: $tType,X3: A] :
% 0.21/0.46 ( ( insert @ A @ X3 @ ( nil @ A ) )
% 0.21/0.46 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % insert_Nil
% 0.21/0.46 thf(fact_210_inter__sorted_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.21/0.46 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 0.21/0.46 => ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
% 0.21/0.46 => ( ! [X1: A,L12: list @ A,X23: A,L22: list @ A] :
% 0.21/0.46 ( ( ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 => ( P @ L12 @ ( cons @ A @ X23 @ L22 ) ) )
% 0.21/0.46 => ( ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 => ( ( X1 = X23 )
% 0.21/0.46 => ( P @ L12 @ L22 ) ) )
% 0.21/0.46 => ( ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 => ( ( X1 != X23 )
% 0.21/0.46 => ( P @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) )
% 0.21/0.46 => ( P @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) )
% 0.21/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % inter_sorted.induct
% 0.21/0.46 thf(fact_211_subset__sorted_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.21/0.46 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 0.21/0.46 => ( ! [X1: A,L12: list @ A] : ( P @ ( cons @ A @ X1 @ L12 ) @ ( nil @ A ) )
% 0.21/0.46 => ( ! [X1: A,L12: list @ A,X23: A,L22: list @ A] :
% 0.21/0.46 ( ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 => ( ( X1 = X23 )
% 0.21/0.46 => ( P @ L12 @ L22 ) ) )
% 0.21/0.46 => ( ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 => ( ( X1 != X23 )
% 0.21/0.46 => ( P @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) )
% 0.21/0.46 => ( P @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) )
% 0.21/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_sorted.induct
% 0.21/0.46 thf(fact_212_distinct__singleton,axiom,
% 0.21/0.46 ! [A: $tType,X3: A] : ( distinct @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % distinct_singleton
% 0.21/0.46 thf(fact_213_sorted_ONil,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ( linorder_sorted @ A @ ( nil @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % sorted.Nil
% 0.21/0.46 thf(fact_214_map__tailrec__rev_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
% 0.21/0.46 ( ! [F2: A > B,X1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X1 )
% 0.21/0.46 => ( ! [F2: A > B,A5: A,As: list @ A,Bs: list @ B] :
% 0.21/0.46 ( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A5 ) @ Bs ) )
% 0.21/0.46 => ( P @ F2 @ ( cons @ A @ A5 @ As ) @ Bs ) )
% 0.21/0.46 => ( P @ A0 @ A1 @ A22 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % map_tailrec_rev.induct
% 0.21/0.46 thf(fact_215_list__nonempty__induct,axiom,
% 0.21/0.46 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.21/0.46 ( ( Xs
% 0.21/0.46 != ( nil @ A ) )
% 0.21/0.46 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.21/0.46 => ( ! [X2: A,Xs3: list @ A] :
% 0.21/0.46 ( ( Xs3
% 0.21/0.46 != ( nil @ A ) )
% 0.21/0.46 => ( ( P @ Xs3 )
% 0.21/0.46 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) )
% 0.21/0.46 => ( P @ Xs ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list_nonempty_induct
% 0.21/0.46 thf(fact_216_remdups__adj_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 0.21/0.46 ( ( P @ ( nil @ A ) )
% 0.21/0.46 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.21/0.46 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.21/0.46 ( ( ( X2 = Y2 )
% 0.21/0.46 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 0.21/0.46 => ( ( ( X2 != Y2 )
% 0.21/0.46 => ( P @ ( cons @ A @ Y2 @ Xs3 ) ) )
% 0.21/0.46 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
% 0.21/0.46 => ( P @ A0 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % remdups_adj.induct
% 0.21/0.46 thf(fact_217_remdups__adj_Ocases,axiom,
% 0.21/0.46 ! [A: $tType,X3: list @ A] :
% 0.21/0.46 ( ( X3
% 0.21/0.46 != ( nil @ A ) )
% 0.21/0.46 => ( ! [X2: A] :
% 0.21/0.46 ( X3
% 0.21/0.46 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.21/0.46 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.21/0.46 ( X3
% 0.21/0.46 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % remdups_adj.cases
% 0.21/0.46 thf(fact_218_transpose_Ocases,axiom,
% 0.21/0.46 ! [A: $tType,X3: list @ ( list @ A )] :
% 0.21/0.46 ( ( X3
% 0.21/0.46 != ( nil @ ( list @ A ) ) )
% 0.21/0.46 => ( ! [Xss: list @ ( list @ A )] :
% 0.21/0.46 ( X3
% 0.21/0.46 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 0.21/0.46 => ~ ! [X2: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
% 0.21/0.46 ( X3
% 0.21/0.46 != ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % transpose.cases
% 0.21/0.46 thf(fact_219_splice_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.21/0.46 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 0.21/0.46 => ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
% 0.21/0.46 => ( ! [X2: A,Xs3: list @ A,Y2: A,Ys2: list @ A] :
% 0.21/0.46 ( ( P @ Xs3 @ Ys2 )
% 0.21/0.46 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
% 0.21/0.46 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % splice.induct
% 0.21/0.46 thf(fact_220_list__induct2_H,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
% 0.21/0.46 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.21/0.46 => ( ! [X2: A,Xs3: list @ A] : ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( nil @ B ) )
% 0.21/0.46 => ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys2 ) )
% 0.21/0.46 => ( ! [X2: A,Xs3: list @ A,Y2: B,Ys2: list @ B] :
% 0.21/0.46 ( ( P @ Xs3 @ Ys2 )
% 0.21/0.46 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys2 ) ) )
% 0.21/0.46 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list_induct2'
% 0.21/0.46 thf(fact_221_neq__Nil__conv,axiom,
% 0.21/0.46 ! [A: $tType,Xs: list @ A] :
% 0.21/0.46 ( ( Xs
% 0.21/0.46 != ( nil @ A ) )
% 0.21/0.46 = ( ? [Y4: A,Ys3: list @ A] :
% 0.21/0.46 ( Xs
% 0.21/0.46 = ( cons @ A @ Y4 @ Ys3 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % neq_Nil_conv
% 0.21/0.46 thf(fact_222_list_Oinducts,axiom,
% 0.21/0.46 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 0.21/0.46 ( ( P @ ( nil @ A ) )
% 0.21/0.46 => ( ! [X1: A,X23: list @ A] :
% 0.21/0.46 ( ( P @ X23 )
% 0.21/0.46 => ( P @ ( cons @ A @ X1 @ X23 ) ) )
% 0.21/0.46 => ( P @ List ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list.inducts
% 0.21/0.46 thf(fact_223_list_Oexhaust,axiom,
% 0.21/0.46 ! [A: $tType,Y: list @ A] :
% 0.21/0.46 ( ( Y
% 0.21/0.46 != ( nil @ A ) )
% 0.21/0.46 => ~ ! [X212: A,X223: list @ A] :
% 0.21/0.46 ( Y
% 0.21/0.46 != ( cons @ A @ X212 @ X223 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list.exhaust
% 0.21/0.46 thf(fact_224_list_OdiscI,axiom,
% 0.21/0.46 ! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
% 0.21/0.46 ( ( List
% 0.21/0.46 = ( cons @ A @ X21 @ X222 ) )
% 0.21/0.46 => ( List
% 0.21/0.46 != ( nil @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list.discI
% 0.21/0.46 thf(fact_225_distinct_Osimps_I1_J,axiom,
% 0.21/0.46 ! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% 0.21/0.46
% 0.21/0.46 % distinct.simps(1)
% 0.21/0.46 thf(fact_226_list_Odistinct_I1_J,axiom,
% 0.21/0.46 ! [A: $tType,X21: A,X222: list @ A] :
% 0.21/0.46 ( ( nil @ A )
% 0.21/0.46 != ( cons @ A @ X21 @ X222 ) ) ).
% 0.21/0.46
% 0.21/0.46 % list.distinct(1)
% 0.21/0.46 thf(fact_227_subset__sorted_Osimps_I1_J,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [L2: list @ A] : ( sorted1061247458sorted @ A @ ( nil @ A ) @ L2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_sorted.simps(1)
% 0.21/0.46 thf(fact_228_subset__sorted_Osimps_I2_J,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [X13: A,L1: list @ A] :
% 0.21/0.46 ~ ( sorted1061247458sorted @ A @ ( cons @ A @ X13 @ L1 ) @ ( nil @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_sorted.simps(2)
% 0.21/0.46 thf(fact_229_subset__sorted_Oelims_I3_J,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [X3: list @ A,Xa2: list @ A] :
% 0.21/0.46 ( ~ ( sorted1061247458sorted @ A @ X3 @ Xa2 )
% 0.21/0.46 => ( ( ? [X1: A,L12: list @ A] :
% 0.21/0.46 ( X3
% 0.21/0.46 = ( cons @ A @ X1 @ L12 ) )
% 0.21/0.46 => ( Xa2
% 0.21/0.46 != ( nil @ A ) ) )
% 0.21/0.46 => ~ ! [X1: A,L12: list @ A] :
% 0.21/0.46 ( ( X3
% 0.21/0.46 = ( cons @ A @ X1 @ L12 ) )
% 0.21/0.46 => ! [X23: A,L22: list @ A] :
% 0.21/0.46 ( ( Xa2
% 0.21/0.46 = ( cons @ A @ X23 @ L22 ) )
% 0.21/0.46 => ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 & ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 => ( ( ( X1 = X23 )
% 0.21/0.46 => ( sorted1061247458sorted @ A @ L12 @ L22 ) )
% 0.21/0.46 & ( ( X1 != X23 )
% 0.21/0.46 => ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_sorted.elims(3)
% 0.21/0.46 thf(fact_230_subset__sorted_Oelims_I2_J,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [X3: list @ A,Xa2: list @ A] :
% 0.21/0.46 ( ( sorted1061247458sorted @ A @ X3 @ Xa2 )
% 0.21/0.46 => ( ( ( X3
% 0.21/0.46 = ( nil @ A ) )
% 0.21/0.46 => ! [L22: list @ A] : ( Xa2 != L22 ) )
% 0.21/0.46 => ~ ! [X1: A,L12: list @ A] :
% 0.21/0.46 ( ( X3
% 0.21/0.46 = ( cons @ A @ X1 @ L12 ) )
% 0.21/0.46 => ! [X23: A,L22: list @ A] :
% 0.21/0.46 ( ( Xa2
% 0.21/0.46 = ( cons @ A @ X23 @ L22 ) )
% 0.21/0.46 => ~ ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 & ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 => ( ( ( X1 = X23 )
% 0.21/0.46 => ( sorted1061247458sorted @ A @ L12 @ L22 ) )
% 0.21/0.46 & ( ( X1 != X23 )
% 0.21/0.46 => ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_sorted.elims(2)
% 0.21/0.46 thf(fact_231_subset__sorted_Oelims_I1_J,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [X3: list @ A,Xa2: list @ A,Y: $o] :
% 0.21/0.46 ( ( ( sorted1061247458sorted @ A @ X3 @ Xa2 )
% 0.21/0.46 = Y )
% 0.21/0.46 => ( ( ( X3
% 0.21/0.46 = ( nil @ A ) )
% 0.21/0.46 => ~ Y )
% 0.21/0.46 => ( ( ? [X1: A,L12: list @ A] :
% 0.21/0.46 ( X3
% 0.21/0.46 = ( cons @ A @ X1 @ L12 ) )
% 0.21/0.46 => ( ( Xa2
% 0.21/0.46 = ( nil @ A ) )
% 0.21/0.46 => Y ) )
% 0.21/0.46 => ~ ! [X1: A,L12: list @ A] :
% 0.21/0.46 ( ( X3
% 0.21/0.46 = ( cons @ A @ X1 @ L12 ) )
% 0.21/0.46 => ! [X23: A,L22: list @ A] :
% 0.21/0.46 ( ( Xa2
% 0.21/0.46 = ( cons @ A @ X23 @ L22 ) )
% 0.21/0.46 => ( Y
% 0.21/0.46 = ( ~ ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 & ( ~ ( ord_less @ A @ X1 @ X23 )
% 0.21/0.46 => ( ( ( X1 = X23 )
% 0.21/0.46 => ( sorted1061247458sorted @ A @ L12 @ L22 ) )
% 0.21/0.46 & ( ( X1 != X23 )
% 0.21/0.46 => ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_sorted.elims(1)
% 0.21/0.46 thf(fact_232_sorted_Ocases,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [A2: list @ A] :
% 0.21/0.46 ( ( linorder_sorted @ A @ A2 )
% 0.21/0.46 => ( ( A2
% 0.21/0.46 != ( nil @ A ) )
% 0.21/0.46 => ~ ! [Xs3: list @ A,X2: A] :
% 0.21/0.46 ( ( A2
% 0.21/0.46 = ( cons @ A @ X2 @ Xs3 ) )
% 0.21/0.46 => ( ! [Xa: A] :
% 0.21/0.46 ( ( member @ A @ Xa @ ( set2 @ A @ Xs3 ) )
% 0.21/0.46 => ( ord_less_eq @ A @ X2 @ Xa ) )
% 0.21/0.46 => ~ ( linorder_sorted @ A @ Xs3 ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % sorted.cases
% 0.21/0.46 thf(fact_233_the__elem__set,axiom,
% 0.21/0.46 ! [A: $tType,X3: A] :
% 0.21/0.46 ( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
% 0.21/0.46 = X3 ) ).
% 0.21/0.46
% 0.21/0.46 % the_elem_set
% 0.21/0.46 thf(fact_234_quicksort_Ocases,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [X3: list @ A] :
% 0.21/0.46 ( ( X3
% 0.21/0.46 != ( nil @ A ) )
% 0.21/0.46 => ~ ! [X2: A,Xs3: list @ A] :
% 0.21/0.46 ( X3
% 0.21/0.46 != ( cons @ A @ X2 @ Xs3 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % quicksort.cases
% 0.21/0.46 thf(fact_235_sublists_Osimps_I1_J,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( sublists @ A @ ( nil @ A ) )
% 0.21/0.46 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % sublists.simps(1)
% 0.21/0.46 thf(fact_236_revg_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.21/0.46 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 0.21/0.46 => ( ! [A5: A,As: list @ A,B2: list @ A] :
% 0.21/0.46 ( ( P @ As @ ( cons @ A @ A5 @ B2 ) )
% 0.21/0.46 => ( P @ ( cons @ A @ A5 @ As ) @ B2 ) )
% 0.21/0.46 => ( P @ A0 @ A1 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % revg.induct
% 0.21/0.46 thf(fact_237_ord__eq__le__eq__trans,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [A2: A,B3: A,C2: A,D3: A] :
% 0.21/0.46 ( ( A2 = B3 )
% 0.21/0.46 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.21/0.46 => ( ( C2 = D3 )
% 0.21/0.46 => ( ord_less_eq @ A @ A2 @ D3 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_eq_le_eq_trans
% 0.21/0.46 thf(fact_238_list__tail__coinc,axiom,
% 0.21/0.46 ! [A: $tType,N1: A,R1: list @ A,N2: A,R2: list @ A] :
% 0.21/0.46 ( ( ( cons @ A @ N1 @ R1 )
% 0.21/0.46 = ( cons @ A @ N2 @ R2 ) )
% 0.21/0.46 => ( ( N1 = N2 )
% 0.21/0.46 & ( R1 = R2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list_tail_coinc
% 0.21/0.46 thf(fact_239_subset__Collect__conv,axiom,
% 0.21/0.46 ! [A: $tType,S: set @ A,P: A > $o] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 0.21/0.46 = ( ! [X4: A] :
% 0.21/0.46 ( ( member @ A @ X4 @ S )
% 0.21/0.46 => ( P @ X4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_Collect_conv
% 0.21/0.46 thf(fact_240_mergesort__by__rel__merge__induct,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R3: A > B > $o,Xs: list @ A,Ys: list @ B] :
% 0.21/0.46 ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ B ) )
% 0.21/0.46 => ( ! [X1: list @ B] : ( P @ ( nil @ A ) @ X1 )
% 0.21/0.46 => ( ! [X2: A,Xs3: list @ A,Y2: B,Ys2: list @ B] :
% 0.21/0.46 ( ( R3 @ X2 @ Y2 )
% 0.21/0.46 => ( ( P @ Xs3 @ ( cons @ B @ Y2 @ Ys2 ) )
% 0.21/0.46 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
% 0.21/0.46 => ( ! [X2: A,Xs3: list @ A,Y2: B,Ys2: list @ B] :
% 0.21/0.46 ( ~ ( R3 @ X2 @ Y2 )
% 0.21/0.46 => ( ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys2 )
% 0.21/0.46 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
% 0.21/0.46 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % mergesort_by_rel_merge_induct
% 0.21/0.46 thf(fact_241_mergesort__by__rel__merge_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A,A22: list @ A] :
% 0.21/0.46 ( ! [R4: A > A > $o,X2: A,Xs3: list @ A,Y2: A,Ys2: list @ A] :
% 0.21/0.46 ( ( ( R4 @ X2 @ Y2 )
% 0.21/0.46 => ( P @ R4 @ Xs3 @ ( cons @ A @ Y2 @ Ys2 ) ) )
% 0.21/0.46 => ( ( ~ ( R4 @ X2 @ Y2 )
% 0.21/0.46 => ( P @ R4 @ ( cons @ A @ X2 @ Xs3 ) @ Ys2 ) )
% 0.21/0.46 => ( P @ R4 @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) )
% 0.21/0.46 => ( ! [R4: A > A > $o,Xs3: list @ A] : ( P @ R4 @ Xs3 @ ( nil @ A ) )
% 0.21/0.46 => ( ! [R4: A > A > $o,V: A,Va: list @ A] : ( P @ R4 @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
% 0.21/0.46 => ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % mergesort_by_rel_merge.induct
% 0.21/0.46 thf(fact_242_list__all__zip_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,P: ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > $o,A1: list @ A,A22: list @ B] :
% 0.21/0.46 ( ! [P3: A > B > $o] : ( P @ P3 @ ( nil @ A ) @ ( nil @ B ) )
% 0.21/0.46 => ( ! [P3: A > B > $o,A5: A,As: list @ A,B2: B,Bs: list @ B] :
% 0.21/0.46 ( ( P @ P3 @ As @ Bs )
% 0.21/0.46 => ( P @ P3 @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B2 @ Bs ) ) )
% 0.21/0.46 => ( ! [P3: A > B > $o,V: A,Va: list @ A] : ( P @ P3 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
% 0.21/0.46 => ( ! [P3: A > B > $o,V: B,Va: list @ B] : ( P @ P3 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
% 0.21/0.46 => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list_all_zip.induct
% 0.21/0.46 thf(fact_243_list__induct__first2,axiom,
% 0.21/0.46 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.21/0.46 ( ( P @ ( nil @ A ) )
% 0.21/0.46 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.21/0.46 => ( ! [X1: A,X23: A,Xs3: list @ A] :
% 0.21/0.46 ( ( P @ Xs3 )
% 0.21/0.46 => ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs3 ) ) ) )
% 0.21/0.46 => ( P @ Xs ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list_induct_first2
% 0.21/0.46 thf(fact_244_list__2pre__induct,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W2: list @ B] :
% 0.21/0.46 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.21/0.46 => ( ! [E2: A,W12: list @ A,W22: list @ B] :
% 0.21/0.46 ( ( P @ W12 @ W22 )
% 0.21/0.46 => ( P @ ( cons @ A @ E2 @ W12 ) @ W22 ) )
% 0.21/0.46 => ( ! [E2: B,W13: list @ A,W23: list @ B] :
% 0.21/0.46 ( ( P @ W13 @ W23 )
% 0.21/0.46 => ( P @ W13 @ ( cons @ B @ E2 @ W23 ) ) )
% 0.21/0.46 => ( P @ W1 @ W2 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % list_2pre_induct
% 0.21/0.46 thf(fact_245_zipf_Oinduct,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,B: $tType,P: ( A > B > C ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > C,A1: list @ A,A22: list @ B] :
% 0.21/0.46 ( ! [F2: A > B > C] : ( P @ F2 @ ( nil @ A ) @ ( nil @ B ) )
% 0.21/0.46 => ( ! [F2: A > B > C,A5: A,As: list @ A,B2: B,Bs: list @ B] :
% 0.21/0.46 ( ( P @ F2 @ As @ Bs )
% 0.21/0.46 => ( P @ F2 @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B2 @ Bs ) ) )
% 0.21/0.46 => ( ! [A5: A > B > C,V: A,Va: list @ A] : ( P @ A5 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
% 0.21/0.46 => ( ! [A5: A > B > C,V: B,Va: list @ B] : ( P @ A5 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
% 0.21/0.46 => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % zipf.induct
% 0.21/0.46 thf(fact_246_mergesort__remdups__correct,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [L: list @ A] :
% 0.21/0.46 ( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
% 0.21/0.46 & ( linorder_sorted @ A @ ( mergesort_remdups @ A @ L ) )
% 0.21/0.46 & ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
% 0.21/0.46 = ( set2 @ A @ L ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % mergesort_remdups_correct
% 0.21/0.46 thf(fact_247_product__lists_Osimps_I1_J,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
% 0.21/0.46 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % product_lists.simps(1)
% 0.21/0.46 thf(fact_248_distinct__product__lists,axiom,
% 0.21/0.46 ! [A: $tType,Xss2: list @ ( list @ A )] :
% 0.21/0.46 ( ! [X2: list @ A] :
% 0.21/0.46 ( ( member @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xss2 ) )
% 0.21/0.46 => ( distinct @ A @ X2 ) )
% 0.21/0.46 => ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % distinct_product_lists
% 0.21/0.46 thf(fact_249_subset__code_I3_J,axiom,
% 0.21/0.46 ! [C: $tType] :
% 0.21/0.46 ~ ( ord_less_eq @ ( set @ C ) @ ( coset @ C @ ( nil @ C ) ) @ ( set2 @ C @ ( nil @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_code(3)
% 0.21/0.46 thf(fact_250_combinatorial__product_Osimps_I1_J,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( list_c659805718roduct @ A @ ( nil @ ( list @ A ) ) )
% 0.21/0.46 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % combinatorial_product.simps(1)
% 0.21/0.46 thf(fact_251_UNIV__coset,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( top_top @ ( set @ A ) )
% 0.21/0.46 = ( coset @ A @ ( nil @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % UNIV_coset
% 0.21/0.46 thf(fact_252_subset__code_I2_J,axiom,
% 0.21/0.46 ! [B: $tType,A3: set @ B,Ys: list @ B] :
% 0.21/0.46 ( ( ord_less_eq @ ( set @ B ) @ A3 @ ( coset @ B @ Ys ) )
% 0.21/0.46 = ( ! [X4: B] :
% 0.21/0.46 ( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
% 0.21/0.46 => ~ ( member @ B @ X4 @ A3 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_code(2)
% 0.21/0.46 thf(fact_253_remove__code_I2_J,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.21/0.46 ( ( remove @ A @ X3 @ ( coset @ A @ Xs ) )
% 0.21/0.46 = ( coset @ A @ ( insert @ A @ X3 @ Xs ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % remove_code(2)
% 0.21/0.46 thf(fact_254_sorted__append__bigger,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ! [Xs: list @ A,Y: A] :
% 0.21/0.46 ( ( linorder_sorted @ A @ Xs )
% 0.21/0.46 => ( ! [X2: A] :
% 0.21/0.46 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.21/0.46 => ( ord_less_eq @ A @ X2 @ Y ) )
% 0.21/0.46 => ( linorder_sorted @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % sorted_append_bigger
% 0.21/0.46
% 0.21/0.46 %----Subclasses (4)
% 0.21/0.46 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ( type @ A @ ( type2 @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ( ord @ A @ ( type2 @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ( order @ A @ ( type2 @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.21/0.46 => ( preorder @ A @ ( type2 @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 %----Type constructors (16)
% 0.21/0.46 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.21/0.46 ! [A7: $tType,A8: $tType] :
% 0.21/0.46 ( ( order_top @ A8 @ ( type2 @ A8 ) )
% 0.21/0.46 => ( order_top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.21/0.46 ! [A7: $tType,A8: $tType] :
% 0.21/0.46 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.21/0.46 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.21/0.46 ! [A7: $tType,A8: $tType] :
% 0.21/0.46 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.21/0.46 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.21/0.46 ! [A7: $tType,A8: $tType] :
% 0.21/0.46 ( ( top @ A8 @ ( type2 @ A8 ) )
% 0.21/0.46 => ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.21/0.46 ! [A7: $tType,A8: $tType] :
% 0.21/0.46 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.21/0.46 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
% 0.21/0.46 ! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 0.21/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.21/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_Set_Oset___Orderings_Otop_4,axiom,
% 0.21/0.46 ! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
% 0.21/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
% 0.21/0.46 order_top @ $o @ ( type2 @ $o ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.21/0.46 preorder @ $o @ ( type2 @ $o ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.21/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 0.21/0.46 order @ $o @ ( type2 @ $o ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_HOL_Obool___Orderings_Otop_9,axiom,
% 0.21/0.46 top @ $o @ ( type2 @ $o ) ).
% 0.21/0.46
% 0.21/0.46 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.21/0.46 ord @ $o @ ( type2 @ $o ) ).
% 0.21/0.46
% 0.21/0.46 %----Helper facts (3)
% 0.21/0.46 thf(help_If_3_1_T,axiom,
% 0.21/0.46 ! [P: $o] :
% 0.21/0.46 ( ( P = $true )
% 0.21/0.46 | ( P = $false ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(help_If_2_1_T,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,Y: A] :
% 0.21/0.46 ( ( if @ A @ $false @ X3 @ Y )
% 0.21/0.46 = Y ) ).
% 0.21/0.46
% 0.21/0.46 thf(help_If_1_1_T,axiom,
% 0.21/0.46 ! [A: $tType,X3: A,Y: A] :
% 0.21/0.46 ( ( if @ A @ $true @ X3 @ Y )
% 0.21/0.46 = X3 ) ).
% 0.21/0.46
% 0.21/0.46 %----Free types (1)
% 0.21/0.46 thf(tfree_0,hypothesis,
% 0.21/0.46 linorder @ a @ ( type2 @ a ) ).
% 0.21/0.46
% 0.21/0.46 %----Conjectures (1)
% 0.21/0.46 thf(conj_0,conjecture,
% 0.21/0.46 ord_less @ a @ x2 @ x1 ).
% 0.21/0.46
% 0.21/0.46 %------------------------------------------------------------------------------
% 0.21/0.46 ------------------------
% 2.60/1.49 ---- Embedded in THF ---
% 2.60/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.60/1.49 %%% Generated on Tue Feb 24 06:10:33 EST 2026
% 2.60/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.60/1.49 %%% Logic specification used:
% 2.60/1.49 %%% thf(spec, logic, $$dhol).
% 2.60/1.49
% 2.60/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.TYv64KhOhS/DTF2DTF19011.p
% 2.60/1.49 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.60/1.49 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.60/1.49 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.60/1.49 thf(ty_tf_a, type, a: $tType).
% 2.60/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.60/1.49 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.60/1.49 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.60/1.49 thf(sy_c_List_Ocoset, type, coset: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.60/1.49 thf(sy_c_List_Odistinct, type, distinct: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.60/1.49 thf(sy_c_List_Oinsert, type, insert: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 2.60/1.49 thf(sy_c_List_Olinorder__class_Osorted, type, linorder_sorted: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.60/1.49 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 2.60/1.49 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 2.60/1.49 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.60/1.49 thf(sy_c_List_Oproduct__lists, type, product_lists: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 2.60/1.49 thf(sy_c_List_Osublists, type, sublists: (!> [A:$tType]: (((list @ A) > (list @ (list @ A)))))).
% 2.60/1.49 thf(sy_c_List_Ounion, type, union: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.60/1.49 thf(sy_c_List__More_Ocombinatorial__product, type, list_c659805718roduct: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 2.60/1.49 thf(sy_c_Misc_Omergesort__remdups, type, mergesort_remdups: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.60/1.49 thf(sy_c_Orderings_Oord_Omax, type, max: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 2.60/1.49 thf(sy_c_Orderings_Oord_Omin, type, min: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 2.60/1.49 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.60/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.60/1.49 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.60/1.49 thf(sy_c_Orderings_Oordering__top, type, ordering_top: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > (A > $o)))))).
% 2.60/1.49 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.60/1.49 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.60/1.49 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.60/1.49 thf(sy_c_Set_Oremove, type, remove: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.60/1.49 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.60/1.49 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Osubset__sorted, type, sorted1061247458sorted: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 2.60/1.49 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.60/1.49 thf(sy_v_l1, type, l1: (list @ a)).
% 2.60/1.49 thf(sy_v_l1a____, type, l1a: (list @ a)).
% 2.60/1.49 thf(sy_v_l2, type, l2: (list @ a)).
% 2.60/1.49 thf(sy_v_l2b____, type, l2b: (list @ a)).
% 2.60/1.49 thf(sy_v_x1____, type, x1: a).
% 2.60/1.49 thf(sy_v_x2____, type, x2: a).
% 2.60/1.49 thf(fact_0_x1__neq__x2, axiom, (x1 != x2)).
% 2.60/1.49 thf(fact_1_x2__le__x1, axiom, (~ ((((ord_less @ a) @ x1) @ x2)))).
% 2.60/1.49 thf(fact_2_minf_I11_J, axiom, (! [C:$tType,D:$tType]: ((((ord @ C) @ (type2 @ C)) => (! [F:D]: ((? [Z:C]: ((! [X:C]: (((((ord_less @ C) @ X) @ Z) => (F = F)))))))))))).
% 2.60/1.49 thf(fact_3_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z) => (~ ((((ord_less @ A) @ T) @ X)))))))))))))).
% 2.60/1.49 thf(fact_4_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z) => (((ord_less @ A) @ X) @ T)))))))))))).
% 2.60/1.49 thf(fact_5_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z) => (X != T)))))))))))).
% 2.60/1.49 thf(fact_6_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z) => (X != T)))))))))))).
% 2.60/1.49 thf(fact_7_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z2:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z2:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z) => (((P @ X) | (Q @ X)) = ((P2 @ X) | (Q2 @ X)))))))))))))))).
% 2.60/1.49 thf(fact_8_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z2:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z2:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z) => (((P @ X) & (Q @ X)) = ((P2 @ X) & (Q2 @ X)))))))))))))))).
% 2.60/1.49 thf(fact_9_pinf_I11_J, axiom, (! [C:$tType,D:$tType]: ((((ord @ C) @ (type2 @ C)) => (! [F:D]: ((? [Z:C]: ((! [X:C]: (((((ord_less @ C) @ Z) @ X) => (F = F)))))))))))).
% 2.60/1.49 thf(fact_10_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (((ord_less @ A) @ T) @ X)))))))))))).
% 2.60/1.49 thf(fact_11_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (~ ((((ord_less @ A) @ X) @ T)))))))))))))).
% 2.60/1.49 thf(fact_12_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (X != T)))))))))))).
% 2.60/1.49 thf(fact_13_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (X != T)))))))))))).
% 2.60/1.49 thf(fact_14_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z2:A]: ((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z2:A]: ((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (((P @ X) & (Q @ X)) = ((P2 @ X) & (Q2 @ X)))))))))))))))).
% 2.60/1.49 thf(fact_15_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z2:A]: ((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z2:A]: ((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (((P @ X) | (Q @ X)) = ((P2 @ X) | (Q2 @ X)))))))))))))))).
% 2.60/1.49 thf(fact_16_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B2:A]: (((((ord_less @ A) @ A2) @ B2) | (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.60/1.49 thf(fact_17_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((X3 != Y) => ((~ ((((ord_less @ A) @ X3) @ Y))) => (((ord_less @ A) @ Y) @ X3))))))))).
% 2.60/1.49 thf(fact_18_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (A2 != B3)))))))).
% 2.60/1.49 thf(fact_19_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.60/1.49 thf(fact_20_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y2:A]: ((((ord_less @ A) @ Y2) @ X))))))))).
% 2.60/1.49 thf(fact_21_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (A2 != B3)))))))).
% 2.60/1.49 thf(fact_22_dependent__wellorder__choice, axiom, (! [B:$tType,A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:((A > B) > (A > (B > $o)))]: (((! [R:B,F2:(A > B),G:(A > B),X2:A]: (((! [Y3:A]: (((((ord_less @ A) @ Y3) @ X2) => ((F2 @ Y3) = (G @ Y3))))) => ((((P @ F2) @ X2) @ R) = (((P @ G) @ X2) @ R))))) => ((! [X2:A,F2:(A > B)]: (((! [Y3:A]: (((((ord_less @ A) @ Y3) @ X2) => (((P @ F2) @ Y3) @ (F2 @ Y3))))) => (? [X12:B]: ((((P @ F2) @ X2) @ X12)))))) => (? [F2:(A > B)]: ((! [X:A]: ((((P @ F2) @ X) @ (F2 @ X)))))))))))))).
% 2.60/1.49 thf(fact_23_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) = ((((ord_less @ A) @ Y) @ X3) | (X3 = Y))))))))).
% 2.60/1.49 thf(fact_24_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C2:A]: (((((ord_less @ A) @ B3) @ A2) => ((((ord_less @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.60/1.49 thf(fact_25_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (~ ((((ord_less @ A) @ Y) @ X3)))))))))).
% 2.60/1.49 thf(fact_26_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.60/1.49 thf(fact_27_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B3:B,C2:B]: (((A2 = (F3 @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less @ B) @ X2) @ Y2) => (((ord_less @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ A) @ A2) @ (F3 @ C2))))))))))).
% 2.60/1.49 thf(fact_28_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F3:(A > B),C2:B]: (((((ord_less @ A) @ A2) @ B3) => (((F3 @ B3) = C2) => ((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ B) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ B) @ (F3 @ A2)) @ C2)))))))))).
% 2.60/1.49 thf(fact_29_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B3:B,C2:B]: (((((ord_less @ A) @ A2) @ (F3 @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less @ B) @ X2) @ Y2) => (((ord_less @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ A) @ A2) @ (F3 @ C2))))))))))).
% 2.60/1.49 thf(fact_30_order__less__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F3:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ C) @ (F3 @ B3)) @ C2) => ((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ C) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ C) @ (F3 @ A2)) @ C2)))))))))).
% 2.60/1.49 thf(fact_31_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X3:A]: ((? [Y2:A]: ((((ord_less @ A) @ Y2) @ X3))))))))).
% 2.60/1.49 thf(fact_32_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X3:A]: ((? [X1:A]: ((((ord_less @ A) @ X3) @ X1))))))))).
% 2.60/1.49 thf(fact_33_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((X3 != Y) => ((~ ((((ord_less @ A) @ X3) @ Y))) => (((ord_less @ A) @ Y) @ X3))))))))).
% 2.60/1.49 thf(fact_34_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((X3 != Y) = ((((ord_less @ A) @ X3) @ Y) | (((ord_less @ A) @ Y) @ X3))))))))).
% 2.60/1.49 thf(fact_35_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.60/1.49 thf(fact_36_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (? [Z:A]: (((((ord_less @ A) @ X3) @ Z) & (((ord_less @ A) @ Z) @ Y))))))))))).
% 2.60/1.49 thf(fact_37_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (X3 != Y)))))))).
% 2.60/1.49 thf(fact_38_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (~ ((((ord_less @ A) @ Y) @ X3)))))))))).
% 2.60/1.49 thf(fact_39_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.60/1.49 thf(fact_40_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,Z3:A]: (((((ord_less @ A) @ X3) @ Y) => ((((ord_less @ A) @ Y) @ Z3) => (((ord_less @ A) @ X3) @ Z3))))))))).
% 2.60/1.49 thf(fact_41_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) | ((X3 = Y) | (((ord_less @ A) @ Y) @ X3))))))))).
% 2.60/1.49 thf(fact_42_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A]: ((~ ((((ord_less @ A) @ X3) @ X3))))))))).
% 2.60/1.49 thf(fact_43_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A]: (((A2 = B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.60/1.49 thf(fact_44_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A]: (((((ord_less @ A) @ A2) @ B3) => ((B3 = C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.60/1.49 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.60/1.49 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3)))) = A3)))).
% 2.60/1.49 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.60/1.49 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G2:(A > B)]: (((! [X2:A]: (((F3 @ X2) = (G2 @ X2)))) => (F3 = G2))))).
% 2.60/1.49 thf(fact_49_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (~ ((((ord_less @ A) @ A2) @ B3)))))))))).
% 2.60/1.49 thf(fact_50_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (X3 != Y)))))))).
% 2.60/1.49 thf(fact_51_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (~ ((((ord_less @ A) @ Y) @ X3)))))))))).
% 2.60/1.49 thf(fact_52_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y3:A]: (((((ord_less @ A) @ Y3) @ X2) => (P @ Y3)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.60/1.49 thf(fact_53_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X3:A]: (((~ ((((ord_less @ A) @ Y) @ X3))) => ((~ ((((ord_less @ A) @ X3) @ Y))) = (X3 = Y))))))))).
% 2.60/1.49 thf(fact_54_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (Y != X3)))))))).
% 2.60/1.49 thf(fact_55_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,P:$o]: (((((ord_less @ A) @ X3) @ Y) => ((((ord_less @ A) @ Y) @ X3) => P)))))))).
% 2.60/1.49 thf(fact_56_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) => ((X3 != Y) => (((ord_less @ A) @ Y) @ X3))))))))).
% 2.60/1.49 thf(fact_57_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.60/1.49 thf(fact_58_strict__monoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F3:(A > B),X3:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F3) => ((((ord_less @ A) @ X3) @ Y) => (((ord_less @ B) @ (F3 @ X3)) @ (F3 @ Y)))))))))).
% 2.60/1.49 thf(fact_59_strict__monoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F3:(A > B)]: (((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ B) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((order_strict_mono @ A) @ B) @ F3)))))))).
% 2.60/1.49 thf(fact_60_strict__mono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_strict_mono @ A) @ B) = (^ [F4:(A > B)]: ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ B) @ (F4 @ X4)) @ (F4 @ Y4)))))))))))).
% 2.60/1.49 thf(fact_61_strict__mono__less, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F3:(A > B),X3:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F3) => ((((ord_less @ B) @ (F3 @ X3)) @ (F3 @ Y)) = (((ord_less @ A) @ X3) @ Y))))))))).
% 2.60/1.49 thf(fact_62_x2__nin__l2, axiom, (~ ((((member @ a) @ x2) @ ((set2 @ a) @ l2b))))).
% 2.60/1.49 thf(fact_63_x1__nin__l1, axiom, (~ ((((member @ a) @ x1) @ ((set2 @ a) @ l1a))))).
% 2.60/1.49 thf(fact_64_ord_Omin__def, axiom, (! [A:$tType]: (((min @ A) = (^ [Less_eq:(A > (A > $o)),A4:A,B4:A]: (((((if @ A) @ ((Less_eq @ A4) @ B4)) @ A4) @ B4))))))).
% 2.60/1.49 thf(fact_65_ord_Omax__def, axiom, (! [A:$tType]: (((max @ A) = (^ [Less_eq:(A > (A > $o)),A4:A,B4:A]: (((((if @ A) @ ((Less_eq @ A4) @ B4)) @ B4) @ A4))))))).
% 2.60/1.49 thf(fact_66_top_Oextremum__strict, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ (top_top @ A)) @ A2))))))))).
% 2.60/1.49 thf(fact_67_top__apply, axiom, (! [C:$tType,D:$tType]: ((((top @ C) @ (type2 @ C)) => ((top_top @ (D > C)) = (^ [X4:D]: ((top_top @ C)))))))).
% 2.60/1.49 thf(fact_68_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F3:(A > B),X3:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F3) => (((F3 @ X3) = (F3 @ Y)) = (X3 = Y))))))))).
% 2.60/1.49 thf(fact_69_top_Onot__eq__extremum, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: (((A2 != (top_top @ A)) = (((ord_less @ A) @ A2) @ (top_top @ A))))))))).
% 2.60/1.49 thf(fact_70_x2__le, axiom, (! [X3:a]: (((((member @ a) @ X3) @ ((set2 @ a) @ l2b)) => (((ord_less_eq @ a) @ x2) @ X3))))).
% 2.60/1.49 thf(fact_71_x1__le, axiom, (! [X3:a]: (((((member @ a) @ X3) @ ((set2 @ a) @ l1a)) => (((ord_less_eq @ a) @ x1) @ X3))))).
% 2.60/1.49 thf(fact_72_ind__hyp__l2, axiom, ((((sorted1061247458sorted @ a) @ (((cons @ a) @ x1) @ l1a)) @ l2b) = (((ord_less_eq @ (set @ a)) @ ((set2 @ a) @ (((cons @ a) @ x1) @ l1a))) @ ((set2 @ a) @ l2b)))).
% 2.60/1.49 thf(fact_73_x2__l2__props, axiom, (((distinct @ a) @ (((cons @ a) @ x2) @ l2b)) & ((linorder_sorted @ a) @ (((cons @ a) @ x2) @ l2b)))).
% 2.60/1.49 thf(fact_74_x1__l1__props, axiom, (((distinct @ a) @ (((cons @ a) @ x1) @ l1a)) & ((linorder_sorted @ a) @ (((cons @ a) @ x1) @ l1a)))).
% 2.60/1.49 thf(fact_75_Cons_Ohyps, axiom, ((((distinct @ a) @ l2b) & ((linorder_sorted @ a) @ l2b)) => ((((sorted1061247458sorted @ a) @ (((cons @ a) @ x1) @ l1a)) @ l2b) = (((ord_less_eq @ (set @ a)) @ ((set2 @ a) @ (((cons @ a) @ x1) @ l1a))) @ ((set2 @ a) @ l2b))))).
% 2.60/1.49 thf(fact_76_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F3:(A > B),X3:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F3) => ((((ord_less_eq @ B) @ (F3 @ X3)) @ (F3 @ Y)) = (((ord_less_eq @ A) @ X3) @ Y))))))))).
% 2.60/1.49 thf(fact_77_l2__OK, axiom, (((distinct @ a) @ l2) & ((linorder_sorted @ a) @ l2))).
% 2.60/1.49 thf(fact_78_l1__OK, axiom, (((distinct @ a) @ l1) & ((linorder_sorted @ a) @ l1))).
% 2.60/1.49 thf(fact_79_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.60/1.49 thf(fact_80_l2__props, axiom, (((distinct @ a) @ l2b) & ((linorder_sorted @ a) @ l2b))).
% 2.60/1.49 thf(fact_81_l1__props, axiom, (((distinct @ a) @ l1a) & ((linorder_sorted @ a) @ l1a))).
% 2.60/1.49 thf(fact_82_ind__hyp__l1, axiom, (! [L2:(list @ a)]: (((((distinct @ a) @ L2) & ((linorder_sorted @ a) @ L2)) => ((((sorted1061247458sorted @ a) @ l1a) @ L2) = (((ord_less_eq @ (set @ a)) @ ((set2 @ a) @ l1a)) @ ((set2 @ a) @ L2))))))).
% 2.60/1.49 thf(fact_83_subset__sorted_Osimps_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X13:A,L1:(list @ A),X22:A,L2:(list @ A)]: (((((sorted1061247458sorted @ A) @ (((cons @ A) @ X13) @ L1)) @ (((cons @ A) @ X22) @ L2)) = ((~ ((((ord_less @ A) @ X13) @ X22))) & ((~ ((((ord_less @ A) @ X13) @ X22))) => (((X13 = X22) => (((sorted1061247458sorted @ A) @ L1) @ L2)) & ((X13 != X22) => (((sorted1061247458sorted @ A) @ (((cons @ A) @ X13) @ L1)) @ L2)))))))))))).
% 2.60/1.49 thf(fact_84_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G2:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) => (((ord_less_eq @ B) @ (F3 @ X3)) @ (G2 @ X3))))))))).
% 2.60/1.49 thf(fact_85_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G2:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) => (((ord_less_eq @ B) @ (F3 @ X3)) @ (G2 @ X3))))))))).
% 2.60/1.49 thf(fact_86_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G2:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F3 @ X2)) @ (G2 @ X2)))) => (((ord_less_eq @ (A > B)) @ F3) @ G2)))))))).
% 2.60/1.49 thf(fact_87_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G3:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F4 @ X4)) @ (G3 @ X4))))))))))).
% 2.60/1.49 thf(fact_88_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C2))))))))))).
% 2.60/1.49 thf(fact_89_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F3:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ C) @ (F3 @ B3)) @ C2) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ C) @ (F3 @ A2)) @ C2)))))))))).
% 2.60/1.49 thf(fact_90_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B3:B,C2:B]: (((A2 = (F3 @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C2))))))))))).
% 2.60/1.49 thf(fact_91_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F3:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B3) => (((F3 @ B3) = C2) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ B) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ B) @ (F3 @ A2)) @ C2)))))))))).
% 2.60/1.49 thf(fact_92_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X4))))))))).
% 2.60/1.49 thf(fact_93_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ A) @ Y) @ X3) => (X3 = Y))))))))).
% 2.60/1.49 thf(fact_94_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) | (((ord_less_eq @ A) @ Y) @ X3)))))))).
% 2.60/1.49 thf(fact_95_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((X3 = Y) => (((ord_less_eq @ A) @ X3) @ Y)))))))).
% 2.60/1.49 thf(fact_96_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y))) => (((ord_less_eq @ A) @ Y) @ X3)))))))).
% 2.60/1.49 thf(fact_97_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.60/1.49 thf(fact_98_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z3)))) => (((((ord_less_eq @ A) @ X3) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X3)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y))))))))))))))))).
% 2.60/1.49 thf(fact_99_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y) = (X3 = Y))))))))).
% 2.60/1.49 thf(fact_100_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A]: (((A2 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.60/1.49 thf(fact_101_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.60/1.49 thf(fact_102_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ A2) => (A2 = B3))))))))).
% 2.60/1.49 thf(fact_103_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X3) @ Z3))))))))).
% 2.60/1.49 thf(fact_104_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.60/1.49 thf(fact_105_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B3:A]: (((! [A5:A,B2:A]: (((((ord_less_eq @ A) @ A5) @ B2) => ((P @ A5) @ B2)))) => ((! [A5:A,B2:A]: ((((P @ B2) @ A5) => ((P @ A5) @ B2)))) => ((P @ A2) @ B3))))))))).
% 2.60/1.49 thf(fact_106_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ 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))))))))).
% 2.60/1.49 thf(fact_107_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ A2) @ B3) => (A2 = B3))))))))).
% 2.60/1.49 thf(fact_108_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => (~ ((((ord_less @ A) @ X3) @ Y)))))))))).
% 2.60/1.49 thf(fact_109_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) => (((ord_less_eq @ A) @ Y) @ X3)))))))).
% 2.60/1.49 thf(fact_110_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) | (X4 = Y4))))))))).
% 2.60/1.49 thf(fact_111_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (X4 != Y4))))))))).
% 2.60/1.49 thf(fact_112_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less @ B) @ X2) @ Y2) => (((ord_less @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ A) @ A2) @ (F3 @ C2))))))))))).
% 2.60/1.49 thf(fact_113_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F3:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less @ C) @ (F3 @ B3)) @ C2) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ C) @ (F3 @ A2)) @ C2)))))))))).
% 2.60/1.49 thf(fact_114_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B3:B,C2:B]: (((((ord_less @ A) @ A2) @ (F3 @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ A) @ A2) @ (F3 @ C2))))))))))).
% 2.60/1.49 thf(fact_115_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F3:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less_eq @ C) @ (F3 @ B3)) @ C2) => ((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ C) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ C) @ (F3 @ A2)) @ C2)))))))))).
% 2.60/1.49 thf(fact_116_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y))) = (((ord_less @ A) @ Y) @ X3)))))))).
% 2.60/1.49 thf(fact_117_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) = (((ord_less_eq @ A) @ Y) @ X3)))))))).
% 2.60/1.49 thf(fact_118_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((A2 != B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.60/1.49 thf(fact_119_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((ord_less_eq @ A) @ X3) @ Y)))))))).
% 2.60/1.49 thf(fact_120_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) => ((((ord_less_eq @ A) @ X3) @ Y) = (X3 = Y))))))))).
% 2.60/1.49 thf(fact_121_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((~ ((((ord_less @ A) @ X3) @ Y))) = (X3 = Y))))))))).
% 2.60/1.49 thf(fact_122_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less @ A) @ Y) @ Z3) => (((ord_less @ A) @ X3) @ Z3))))))))).
% 2.60/1.49 thf(fact_123_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,Z3:A]: (((((ord_less @ A) @ X3) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less @ A) @ X3) @ Z3))))))))).
% 2.60/1.49 thf(fact_124_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z3:A,Y:A]: (((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (((ord_less_eq @ A) @ Y) @ X2)))) => (((ord_less_eq @ A) @ Y) @ Z3)))))))).
% 2.60/1.49 thf(fact_125_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y:A,Z3:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ X2) @ Z3)))) => (((ord_less_eq @ A) @ Y) @ Z3)))))))).
% 2.60/1.49 thf(fact_126_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) | (((ord_less @ A) @ Y) @ X3)))))))).
% 2.60/1.49 thf(fact_127_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less @ A) @ X3) @ Y) | (X3 = Y))))))))).
% 2.60/1.49 thf(fact_128_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (~ ((((ord_less_eq @ A) @ Y4) @ X4))))))))))).
% 2.60/1.49 thf(fact_129_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X3:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X3))) => (((ord_less @ A) @ X3) @ Y)))))))).
% 2.60/1.49 thf(fact_130_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.60/1.49 thf(fact_131_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.60/1.49 thf(fact_132_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) | (A4 = B4))))))))).
% 2.60/1.49 thf(fact_133_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (A4 != B4))))))))).
% 2.60/1.49 thf(fact_134_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.60/1.49 thf(fact_135_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C2:A]: (((((ord_less @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.60/1.49 thf(fact_136_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z3:A,X3:A,Y:A]: (((((ord_less @ A) @ Z3) @ X3) => ((! [W:A]: (((((ord_less @ A) @ Z3) @ W) => ((((ord_less @ A) @ W) @ X3) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z3))))))))).
% 2.60/1.49 thf(fact_137_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,Z3:A]: (((((ord_less @ A) @ X3) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X3) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z3))))) => (((ord_less_eq @ A) @ Y) @ Z3))))))))).
% 2.60/1.49 thf(fact_138_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (((ord_less_eq @ A) @ A2) @ B3)))))))).
% 2.60/1.49 thf(fact_139_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) | (A4 = B4))))))))).
% 2.60/1.49 thf(fact_140_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (A4 != B4))))))))).
% 2.60/1.49 thf(fact_141_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (((ord_less_eq @ A) @ B3) @ A2)))))))).
% 2.60/1.49 thf(fact_142_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((A2 != B3) => ((((ord_less_eq @ A) @ A2) @ B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.60/1.49 thf(fact_143_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B3:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B3) => ((P @ A2) => ((~ ((P @ B3))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B3) & ((! [X:A]: ((((((ord_less_eq @ A) @ A2) @ X) & (((ord_less @ A) @ X) @ C3)) => (P @ X)))) & (! [D2:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ D2)) => (P @ X2)))) => (((ord_less_eq @ A) @ D2) @ C3)))))))))))))))))).
% 2.60/1.49 thf(fact_144_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ T) @ X)))))))))))))).
% 2.60/1.49 thf(fact_145_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z) => (((ord_less_eq @ A) @ X) @ T)))))))))))).
% 2.60/1.49 thf(fact_146_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (((ord_less_eq @ A) @ T) @ X)))))))))))).
% 2.60/1.49 thf(fact_147_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ T)))))))))))))).
% 2.60/1.49 thf(fact_148_top__greatest, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))).
% 2.60/1.49 thf(fact_149_top_Oextremum__unique, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))).
% 2.60/1.49 thf(fact_150_top_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))).
% 2.60/1.49 thf(fact_151_sorted__many__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,Zs:(list @ A)]: ((((linorder_sorted @ A) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Zs))) = ((((ord_less_eq @ A) @ X3) @ Y) & ((linorder_sorted @ A) @ (((cons @ A) @ Y) @ Zs)))))))))).
% 2.60/1.49 thf(fact_152_sorted_OCons, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A),X3:A]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X3) @ X2)))) => (((linorder_sorted @ A) @ Xs) => ((linorder_sorted @ A) @ (((cons @ A) @ X3) @ Xs)))))))))).
% 2.60/1.49 thf(fact_153_sorted__Cons, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Xs:(list @ A)]: ((((linorder_sorted @ A) @ (((cons @ A) @ X3) @ Xs)) = (((linorder_sorted @ A) @ Xs) & (! [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X3) @ X4)))))))))))).
% 2.60/1.49 thf(fact_154_sorted__distinct__set__unique, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A),Ys:(list @ A)]: ((((linorder_sorted @ A) @ Xs) => (((distinct @ A) @ Xs) => (((linorder_sorted @ A) @ Ys) => (((distinct @ A) @ Ys) => ((((set2 @ A) @ Xs) = ((set2 @ A) @ Ys)) => (Xs = Ys)))))))))))).
% 2.60/1.49 thf(fact_155_distinct_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((distinct @ A) @ (((cons @ A) @ X3) @ Xs)) = ((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) & ((distinct @ A) @ Xs)))))).
% 2.60/1.49 thf(fact_156_sorted__many, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y:A,Zs:(list @ A)]: (((((ord_less_eq @ A) @ X3) @ Y) => (((linorder_sorted @ A) @ (((cons @ A) @ Y) @ Zs)) => ((linorder_sorted @ A) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Zs))))))))))).
% 2.60/1.49 thf(fact_157_list_Oinject, axiom, (! [A:$tType,X21:A,X222:(list @ A),Y21:A,Y22:(list @ A)]: ((((((cons @ A) @ X21) @ X222) = (((cons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X222 = Y22)))))).
% 2.60/1.49 thf(fact_158_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F4:(A > B),G3:(A > B)]: (((((ord_less_eq @ (A > B)) @ F4) @ G3) & (~ ((((ord_less_eq @ (A > B)) @ G3) @ F4))))))))))).
% 2.60/1.49 thf(fact_159_not__Cons__self2, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((cons @ A) @ X3) @ Xs) != Xs)))).
% 2.60/1.49 thf(fact_160_list_Oset__cases, axiom, (! [A:$tType,E:A,A2:(list @ A)]: (((((member @ A) @ E) @ ((set2 @ A) @ A2)) => ((! [Z22:(list @ A)]: ((A2 != (((cons @ A) @ E) @ Z22)))) => (~ ((! [Z1:A,Z22:(list @ A)]: (((A2 = (((cons @ A) @ Z1) @ Z22)) => (~ ((((member @ A) @ E) @ ((set2 @ A) @ Z22)))))))))))))).
% 2.60/1.49 thf(fact_161_set__ConsD, axiom, (! [A:$tType,Y:A,X3:A,Xs:(list @ A)]: (((((member @ A) @ Y) @ ((set2 @ A) @ (((cons @ A) @ X3) @ Xs))) => ((Y = X3) | (((member @ A) @ Y) @ ((set2 @ A) @ Xs))))))).
% 2.60/1.49 thf(fact_162_list_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A22:(list @ A)]: ((((member @ A) @ A1) @ ((set2 @ A) @ (((cons @ A) @ A1) @ A22)))))).
% 2.60/1.49 thf(fact_163_list_Oset__intros_I2_J, axiom, (! [A:$tType,X3:A,A22:(list @ A),A1:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ A22)) => (((member @ A) @ X3) @ ((set2 @ A) @ (((cons @ A) @ A1) @ A22))))))).
% 2.60/1.49 thf(fact_164_subset__code_I1_J, axiom, (! [A:$tType,Xs:(list @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ B5) = (! [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) => (((member @ A) @ X4) @ B5)))))))).
% 2.60/1.49 thf(fact_165_distinct__length__2__or__more, axiom, (! [A:$tType,A2:A,B3:A,Xs:(list @ A)]: ((((distinct @ A) @ (((cons @ A) @ A2) @ (((cons @ A) @ B3) @ Xs))) = ((A2 != B3) & (((distinct @ A) @ (((cons @ A) @ A2) @ Xs)) & ((distinct @ A) @ (((cons @ A) @ B3) @ Xs)))))))).
% 2.60/1.49 thf(fact_166_set__subset__Cons, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ (((cons @ A) @ X3) @ Xs)))))).
% 2.60/1.49 thf(fact_167_subsetI, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B5)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B5))))).
% 2.60/1.49 thf(fact_168_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A3) => (A3 = B5)))))).
% 2.60/1.49 thf(fact_169_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((A3 != B5) => (((ord_less @ (set @ A)) @ A3) @ B5)))))).
% 2.60/1.49 thf(fact_170_subset__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ (top_top @ (set @ A)))))).
% 2.60/1.49 thf(fact_171_psubsetE, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B5) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A3)))))))).
% 2.60/1.49 thf(fact_172_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (A6 != B6)))))))).
% 2.60/1.49 thf(fact_173_psubset__imp__subset, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B5) => (((ord_less_eq @ (set @ A)) @ A3) @ B5))))).
% 2.60/1.49 thf(fact_174_psubset__subset__trans, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => (((ord_less @ (set @ A)) @ A3) @ C4)))))).
% 2.60/1.49 thf(fact_175_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))))).
% 2.60/1.49 thf(fact_176_subset__psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((ord_less @ (set @ A)) @ B5) @ C4) => (((ord_less @ (set @ A)) @ A3) @ C4)))))).
% 2.60/1.49 thf(fact_177_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B6) | (A6 = B6)))))))).
% 2.60/1.49 thf(fact_178_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 2.60/1.49 thf(fact_179_contra__subsetD, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((~ ((((member @ A) @ C2) @ B5))) => (~ ((((member @ A) @ C2) @ A3)))))))).
% 2.60/1.49 thf(fact_180_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z4:(set @ A)]: ((Y5 = Z4))) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))).
% 2.60/1.49 thf(fact_181_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => (((ord_less_eq @ (set @ A)) @ A3) @ C4)))))).
% 2.60/1.49 thf(fact_182_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.60/1.49 thf(fact_183_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.60/1.49 thf(fact_184_rev__subsetD, axiom, (! [A:$tType,C2:A,A3:(set @ A),B5:(set @ A)]: (((((member @ A) @ C2) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B5) => (((member @ A) @ C2) @ B5)))))).
% 2.60/1.49 thf(fact_185_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A6) => (((member @ A) @ T2) @ B6)))))))))).
% 2.60/1.49 thf(fact_186_set__rev__mp, axiom, (! [A:$tType,X3:A,A3:(set @ A),B5:(set @ A)]: (((((member @ A) @ X3) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B5) => (((member @ A) @ X3) @ B5)))))).
% 2.60/1.49 thf(fact_187_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((A3 = B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A3))))).
% 2.60/1.49 thf(fact_188_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((A3 = B5) => (((ord_less_eq @ (set @ A)) @ A3) @ B5))))).
% 2.60/1.49 thf(fact_189_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A6) => (((member @ A) @ X4) @ B6)))))))))).
% 2.60/1.49 thf(fact_190_equalityE, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((A3 = B5) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A3)))))))))).
% 2.60/1.49 thf(fact_191_subsetCE, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B5)))))).
% 2.60/1.49 thf(fact_192_subsetD, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B5)))))).
% 2.60/1.49 thf(fact_193_in__mono, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B5)))))).
% 2.60/1.49 thf(fact_194_set__mp, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B5)))))).
% 2.60/1.49 thf(fact_195_distinct__union, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((distinct @ A) @ (((union @ A) @ Xs) @ Ys)) = ((distinct @ A) @ Ys))))).
% 2.60/1.49 thf(fact_196_top_Oordering__top__axioms, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => ((((ordering_top @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)) @ (top_top @ A)))))).
% 2.60/1.49 thf(fact_197_not__in__set__insert, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => ((((insert @ A) @ X3) @ Xs) = (((cons @ A) @ X3) @ Xs)))))).
% 2.60/1.49 thf(fact_198_in__set__insert, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((((insert @ A) @ X3) @ Xs) = Xs))))).
% 2.60/1.49 thf(fact_199_distinct__insert, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((distinct @ A) @ (((insert @ A) @ X3) @ Xs)) = ((distinct @ A) @ Xs))))).
% 2.60/1.49 thf(fact_200_ordering__top_Oextremum__uniqueI, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A2:A]: ((((((ordering_top @ A) @ Less_eq2) @ Less) @ Top) => (((Less_eq2 @ Top) @ A2) => (A2 = Top)))))).
% 2.60/1.49 thf(fact_201_ordering__top_Onot__eq__extremum, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A2:A]: ((((((ordering_top @ A) @ Less_eq2) @ Less) @ Top) => ((A2 != Top) = ((Less @ A2) @ Top)))))).
% 2.60/1.49 thf(fact_202_ordering__top_Oextremum__unique, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A2:A]: ((((((ordering_top @ A) @ Less_eq2) @ Less) @ Top) => (((Less_eq2 @ Top) @ A2) = (A2 = Top)))))).
% 2.60/1.49 thf(fact_203_ordering__top_Oextremum__strict, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A2:A]: ((((((ordering_top @ A) @ Less_eq2) @ Less) @ Top) => (~ (((Less @ Top) @ A2))))))).
% 2.60/1.49 thf(fact_204_ordering__top_Oextremum, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A2:A]: ((((((ordering_top @ A) @ Less_eq2) @ Less) @ Top) => ((Less_eq2 @ A2) @ Top))))).
% 2.60/1.49 thf(fact_205_List_Oinsert__def, axiom, (! [A:$tType]: (((insert @ A) = (^ [X4:A,Xs2:(list @ A)]: (((((if @ (list @ A)) @ (((member @ A) @ X4) @ ((set2 @ A) @ Xs2))) @ Xs2) @ (((cons @ A) @ X4) @ Xs2)))))))).
% 2.60/1.49 thf(fact_206_sorted_Oinducts, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:(list @ A),P:((list @ A) > $o)]: ((((linorder_sorted @ A) @ X3) => ((P @ (nil @ A)) => ((! [Xs3:(list @ A),X2:A]: (((! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Xs3)) => (((ord_less_eq @ A) @ X2) @ Xa)))) => (((linorder_sorted @ A) @ Xs3) => ((P @ Xs3) => (P @ (((cons @ A) @ X2) @ Xs3))))))) => (P @ X3)))))))))).
% 2.60/1.49 thf(fact_207_sorted_Osimps, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((linorder_sorted @ A) = (^ [A4:(list @ A)]: (((A4 = (nil @ A)) | (? [Xs2:(list @ A),X4:A]: (((A4 = (((cons @ A) @ X4) @ Xs2)) & ((! [Y4:A]: (((((member @ A) @ Y4) @ ((set2 @ A) @ Xs2)) => (((ord_less_eq @ A) @ X4) @ Y4)))) & ((linorder_sorted @ A) @ Xs2))))))))))))).
% 2.60/1.49 thf(fact_208_sorted__single, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A]: (((linorder_sorted @ A) @ (((cons @ A) @ X3) @ (nil @ A))))))))).
% 2.60/1.49 thf(fact_209_insert__Nil, axiom, (! [A:$tType,X3:A]: (((((insert @ A) @ X3) @ (nil @ A)) = (((cons @ A) @ X3) @ (nil @ A)))))).
% 2.60/1.49 thf(fact_210_inter__sorted_Oinduct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X1:(list @ A)]: (((P @ (nil @ A)) @ X1))) => ((! [V:A,Va:(list @ A)]: (((P @ (((cons @ A) @ V) @ Va)) @ (nil @ A)))) => ((! [X1:A,L12:(list @ A),X23:A,L22:(list @ A)]: ((((((ord_less @ A) @ X1) @ X23) => ((P @ L12) @ (((cons @ A) @ X23) @ L22))) => (((~ ((((ord_less @ A) @ X1) @ X23))) => ((X1 = X23) => ((P @ L12) @ L22))) => (((~ ((((ord_less @ A) @ X1) @ X23))) => ((X1 != X23) => ((P @ (((cons @ A) @ X1) @ L12)) @ L22))) => ((P @ (((cons @ A) @ X1) @ L12)) @ (((cons @ A) @ X23) @ L22))))))) => ((P @ A0) @ A1)))))))))).
% 2.60/1.49 thf(fact_211_subset__sorted_Oinduct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X1:(list @ A)]: (((P @ (nil @ A)) @ X1))) => ((! [X1:A,L12:(list @ A)]: (((P @ (((cons @ A) @ X1) @ L12)) @ (nil @ A)))) => ((! [X1:A,L12:(list @ A),X23:A,L22:(list @ A)]: ((((~ ((((ord_less @ A) @ X1) @ X23))) => ((X1 = X23) => ((P @ L12) @ L22))) => (((~ ((((ord_less @ A) @ X1) @ X23))) => ((X1 != X23) => ((P @ (((cons @ A) @ X1) @ L12)) @ L22))) => ((P @ (((cons @ A) @ X1) @ L12)) @ (((cons @ A) @ X23) @ L22)))))) => ((P @ A0) @ A1)))))))))).
% 2.60/1.49 thf(fact_212_distinct__singleton, axiom, (! [A:$tType,X3:A]: (((distinct @ A) @ (((cons @ A) @ X3) @ (nil @ A)))))).
% 2.60/1.49 thf(fact_213_sorted_ONil, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((linorder_sorted @ A) @ (nil @ A)))))).
% 2.60/1.49 thf(fact_214_map__tailrec__rev_Oinduct, axiom, (! [A:$tType,B:$tType,P:((A > B) > ((list @ A) > ((list @ B) > $o))),A0:(A > B),A1:(list @ A),A22:(list @ B)]: (((! [F2:(A > B),X1:(list @ B)]: ((((P @ F2) @ (nil @ A)) @ X1))) => ((! [F2:(A > B),A5:A,As:(list @ A),Bs:(list @ B)]: (((((P @ F2) @ As) @ (((cons @ B) @ (F2 @ A5)) @ Bs)) => (((P @ F2) @ (((cons @ A) @ A5) @ As)) @ Bs)))) => (((P @ A0) @ A1) @ A22)))))).
% 2.60/1.49 thf(fact_215_list__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Xs3:(list @ A)]: (((Xs3 != (nil @ A)) => ((P @ Xs3) => (P @ (((cons @ A) @ X2) @ Xs3)))))) => (P @ Xs))))))).
% 2.60/1.49 thf(fact_216_remdups__adj_Oinduct, axiom, (! [A:$tType,P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Y2:A,Xs3:(list @ A)]: ((((X2 = Y2) => (P @ (((cons @ A) @ X2) @ Xs3))) => (((X2 != Y2) => (P @ (((cons @ A) @ Y2) @ Xs3))) => (P @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3))))))) => (P @ A0))))))).
% 2.60/1.49 thf(fact_217_remdups__adj_Ocases, axiom, (! [A:$tType,X3:(list @ A)]: (((X3 != (nil @ A)) => ((! [X2:A]: ((X3 != (((cons @ A) @ X2) @ (nil @ A))))) => (~ ((! [X2:A,Y2:A,Xs3:(list @ A)]: ((X3 != (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3)))))))))))).
% 2.60/1.49 thf(fact_218_transpose_Ocases, axiom, (! [A:$tType,X3:(list @ (list @ A))]: (((X3 != (nil @ (list @ A))) => ((! [Xss:(list @ (list @ A))]: ((X3 != (((cons @ (list @ A)) @ (nil @ A)) @ Xss)))) => (~ ((! [X2:A,Xs3:(list @ A),Xss:(list @ (list @ A))]: ((X3 != (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ Xs3)) @ Xss))))))))))).
% 2.60/1.49 thf(fact_219_splice_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X1:(list @ A)]: (((P @ (nil @ A)) @ X1))) => ((! [V:A,Va:(list @ A)]: (((P @ (((cons @ A) @ V) @ Va)) @ (nil @ A)))) => ((! [X2:A,Xs3:(list @ A),Y2:A,Ys2:(list @ A)]: ((((P @ Xs3) @ Ys2) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys2))))) => ((P @ A0) @ A1))))))).
% 2.60/1.49 thf(fact_220_list__induct2_H, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),Xs:(list @ A),Ys:(list @ B)]: ((((P @ (nil @ A)) @ (nil @ B)) => ((! [X2:A,Xs3:(list @ A)]: (((P @ (((cons @ A) @ X2) @ Xs3)) @ (nil @ B)))) => ((! [Y2:B,Ys2:(list @ B)]: (((P @ (nil @ A)) @ (((cons @ B) @ Y2) @ Ys2)))) => ((! [X2:A,Xs3:(list @ A),Y2:B,Ys2:(list @ B)]: ((((P @ Xs3) @ Ys2) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ B) @ Y2) @ Ys2))))) => ((P @ Xs) @ Ys)))))))).
% 2.60/1.49 thf(fact_221_neq__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) = (? [Y4:A,Ys3:(list @ A)]: ((Xs = (((cons @ A) @ Y4) @ Ys3)))))))).
% 2.60/1.49 thf(fact_222_list_Oinducts, axiom, (! [A:$tType,P:((list @ A) > $o),List:(list @ A)]: (((P @ (nil @ A)) => ((! [X1:A,X23:(list @ A)]: (((P @ X23) => (P @ (((cons @ A) @ X1) @ X23))))) => (P @ List)))))).
% 2.60/1.49 thf(fact_223_list_Oexhaust, axiom, (! [A:$tType,Y:(list @ A)]: (((Y != (nil @ A)) => (~ ((! [X212:A,X223:(list @ A)]: ((Y != (((cons @ A) @ X212) @ X223)))))))))).
% 2.60/1.49 thf(fact_224_list_OdiscI, axiom, (! [A:$tType,List:(list @ A),X21:A,X222:(list @ A)]: (((List = (((cons @ A) @ X21) @ X222)) => (List != (nil @ A)))))).
% 2.60/1.49 thf(fact_225_distinct_Osimps_I1_J, axiom, (! [A:$tType]: (((distinct @ A) @ (nil @ A))))).
% 2.60/1.49 thf(fact_226_list_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: (((nil @ A) != (((cons @ A) @ X21) @ X222))))).
% 2.60/1.49 thf(fact_227_subset__sorted_Osimps_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [L2:(list @ A)]: ((((sorted1061247458sorted @ A) @ (nil @ A)) @ L2))))))).
% 2.60/1.49 thf(fact_228_subset__sorted_Osimps_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X13:A,L1:(list @ A)]: ((~ ((((sorted1061247458sorted @ A) @ (((cons @ A) @ X13) @ L1)) @ (nil @ A)))))))))).
% 2.60/1.49 thf(fact_229_subset__sorted_Oelims_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:(list @ A),Xa2:(list @ A)]: (((~ ((((sorted1061247458sorted @ A) @ X3) @ Xa2))) => (((? [X1:A,L12:(list @ A)]: ((X3 = (((cons @ A) @ X1) @ L12)))) => (Xa2 != (nil @ A))) => (~ ((! [X1:A,L12:(list @ A)]: (((X3 = (((cons @ A) @ X1) @ L12)) => (! [X23:A,L22:(list @ A)]: (((Xa2 = (((cons @ A) @ X23) @ L22)) => ((~ ((((ord_less @ A) @ X1) @ X23))) & ((~ ((((ord_less @ A) @ X1) @ X23))) => (((X1 = X23) => (((sorted1061247458sorted @ A) @ L12) @ L22)) & ((X1 != X23) => (((sorted1061247458sorted @ A) @ (((cons @ A) @ X1) @ L12)) @ L22))))))))))))))))))))).
% 2.60/1.49 thf(fact_230_subset__sorted_Oelims_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:(list @ A),Xa2:(list @ A)]: (((((sorted1061247458sorted @ A) @ X3) @ Xa2) => (((X3 = (nil @ A)) => (! [L22:(list @ A)]: ((Xa2 != L22)))) => (~ ((! [X1:A,L12:(list @ A)]: (((X3 = (((cons @ A) @ X1) @ L12)) => (! [X23:A,L22:(list @ A)]: (((Xa2 = (((cons @ A) @ X23) @ L22)) => (~ (((~ ((((ord_less @ A) @ X1) @ X23))) & ((~ ((((ord_less @ A) @ X1) @ X23))) => (((X1 = X23) => (((sorted1061247458sorted @ A) @ L12) @ L22)) & ((X1 != X23) => (((sorted1061247458sorted @ A) @ (((cons @ A) @ X1) @ L12)) @ L22))))))))))))))))))))))).
% 2.60/1.49 thf(fact_231_subset__sorted_Oelims_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:(list @ A),Xa2:(list @ A),Y:$o]: ((((((sorted1061247458sorted @ A) @ X3) @ Xa2) = Y) => (((X3 = (nil @ A)) => (~ (Y))) => (((? [X1:A,L12:(list @ A)]: ((X3 = (((cons @ A) @ X1) @ L12)))) => ((Xa2 = (nil @ A)) => Y)) => (~ ((! [X1:A,L12:(list @ A)]: (((X3 = (((cons @ A) @ X1) @ L12)) => (! [X23:A,L22:(list @ A)]: (((Xa2 = (((cons @ A) @ X23) @ L22)) => (Y = (~ (((~ ((((ord_less @ A) @ X1) @ X23))) & ((~ ((((ord_less @ A) @ X1) @ X23))) => (((X1 = X23) => (((sorted1061247458sorted @ A) @ L12) @ L22)) & ((X1 != X23) => (((sorted1061247458sorted @ A) @ (((cons @ A) @ X1) @ L12)) @ L22))))))))))))))))))))))))).
% 2.60/1.49 thf(fact_232_sorted_Ocases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A2:(list @ A)]: ((((linorder_sorted @ A) @ A2) => ((A2 != (nil @ A)) => (~ ((! [Xs3:(list @ A),X2:A]: (((A2 = (((cons @ A) @ X2) @ Xs3)) => ((! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Xs3)) => (((ord_less_eq @ A) @ X2) @ Xa)))) => (~ (((linorder_sorted @ A) @ Xs3))))))))))))))))).
% 2.60/1.49 thf(fact_233_the__elem__set, axiom, (! [A:$tType,X3:A]: ((((the_elem @ A) @ ((set2 @ A) @ (((cons @ A) @ X3) @ (nil @ A)))) = X3)))).
% 2.60/1.49 thf(fact_234_quicksort_Ocases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:(list @ A)]: (((X3 != (nil @ A)) => (~ ((! [X2:A,Xs3:(list @ A)]: ((X3 != (((cons @ A) @ X2) @ Xs3))))))))))))).
% 2.60/1.49 thf(fact_235_sublists_Osimps_I1_J, axiom, (! [A:$tType]: ((((sublists @ A) @ (nil @ A)) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 2.60/1.49 thf(fact_236_revg_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X1:(list @ A)]: (((P @ (nil @ A)) @ X1))) => ((! [A5:A,As:(list @ A),B2:(list @ A)]: ((((P @ As) @ (((cons @ A) @ A5) @ B2)) => ((P @ (((cons @ A) @ A5) @ As)) @ B2)))) => ((P @ A0) @ A1)))))).
% 2.60/1.49 thf(fact_237_ord__eq__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C2:A,D3:A]: (((A2 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => ((C2 = D3) => (((ord_less_eq @ A) @ A2) @ D3)))))))))).
% 2.60/1.49 thf(fact_238_list__tail__coinc, axiom, (! [A:$tType,N1:A,R1:(list @ A),N2:A,R2:(list @ A)]: ((((((cons @ A) @ N1) @ R1) = (((cons @ A) @ N2) @ R2)) => ((N1 = N2) & (R1 = R2)))))).
% 2.60/1.49 thf(fact_239_subset__Collect__conv, axiom, (! [A:$tType,S:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ S) @ ((collect @ A) @ P)) = (! [X4:A]: (((((member @ A) @ X4) @ S) => (P @ X4)))))))).
% 2.60/1.49 thf(fact_240_mergesort__by__rel__merge__induct, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),R3:(A > (B > $o)),Xs:(list @ A),Ys:(list @ B)]: (((! [Xs3:(list @ A)]: (((P @ Xs3) @ (nil @ B)))) => ((! [X1:(list @ B)]: (((P @ (nil @ A)) @ X1))) => ((! [X2:A,Xs3:(list @ A),Y2:B,Ys2:(list @ B)]: ((((R3 @ X2) @ Y2) => (((P @ Xs3) @ (((cons @ B) @ Y2) @ Ys2)) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ B) @ Y2) @ Ys2)))))) => ((! [X2:A,Xs3:(list @ A),Y2:B,Ys2:(list @ B)]: (((~ (((R3 @ X2) @ Y2))) => (((P @ (((cons @ A) @ X2) @ Xs3)) @ Ys2) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ B) @ Y2) @ Ys2)))))) => ((P @ Xs) @ Ys)))))))).
% 2.60/1.49 thf(fact_241_mergesort__by__rel__merge_Oinduct, axiom, (! [A:$tType,P:((A > (A > $o)) > ((list @ A) > ((list @ A) > $o))),A0:(A > (A > $o)),A1:(list @ A),A22:(list @ A)]: (((! [R4:(A > (A > $o)),X2:A,Xs3:(list @ A),Y2:A,Ys2:(list @ A)]: (((((R4 @ X2) @ Y2) => (((P @ R4) @ Xs3) @ (((cons @ A) @ Y2) @ Ys2))) => (((~ (((R4 @ X2) @ Y2))) => (((P @ R4) @ (((cons @ A) @ X2) @ Xs3)) @ Ys2)) => (((P @ R4) @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys2)))))) => ((! [R4:(A > (A > $o)),Xs3:(list @ A)]: ((((P @ R4) @ Xs3) @ (nil @ A)))) => ((! [R4:(A > (A > $o)),V:A,Va:(list @ A)]: ((((P @ R4) @ (nil @ A)) @ (((cons @ A) @ V) @ Va)))) => (((P @ A0) @ A1) @ A22))))))).
% 2.60/1.49 thf(fact_242_list__all__zip_Oinduct, axiom, (! [A:$tType,B:$tType,P:((A > (B > $o)) > ((list @ A) > ((list @ B) > $o))),A0:(A > (B > $o)),A1:(list @ A),A22:(list @ B)]: (((! [P3:(A > (B > $o))]: ((((P @ P3) @ (nil @ A)) @ (nil @ B)))) => ((! [P3:(A > (B > $o)),A5:A,As:(list @ A),B2:B,Bs:(list @ B)]: (((((P @ P3) @ As) @ Bs) => (((P @ P3) @ (((cons @ A) @ A5) @ As)) @ (((cons @ B) @ B2) @ Bs))))) => ((! [P3:(A > (B > $o)),V:A,Va:(list @ A)]: ((((P @ P3) @ (((cons @ A) @ V) @ Va)) @ (nil @ B)))) => ((! [P3:(A > (B > $o)),V:B,Va:(list @ B)]: ((((P @ P3) @ (nil @ A)) @ (((cons @ B) @ V) @ Va)))) => (((P @ A0) @ A1) @ A22)))))))).
% 2.60/1.49 thf(fact_243_list__induct__first2, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X1:A,X23:A,Xs3:(list @ A)]: (((P @ Xs3) => (P @ (((cons @ A) @ X1) @ (((cons @ A) @ X23) @ Xs3)))))) => (P @ Xs))))))).
% 2.60/1.49 thf(fact_244_list__2pre__induct, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),W1:(list @ A),W2:(list @ B)]: ((((P @ (nil @ A)) @ (nil @ B)) => ((! [E2:A,W12:(list @ A),W22:(list @ B)]: ((((P @ W12) @ W22) => ((P @ (((cons @ A) @ E2) @ W12)) @ W22)))) => ((! [E2:B,W13:(list @ A),W23:(list @ B)]: ((((P @ W13) @ W23) => ((P @ W13) @ (((cons @ B) @ E2) @ W23))))) => ((P @ W1) @ W2))))))).
% 2.60/1.49 thf(fact_245_zipf_Oinduct, axiom, (! [A:$tType,C:$tType,B:$tType,P:((A > (B > C)) > ((list @ A) > ((list @ B) > $o))),A0:(A > (B > C)),A1:(list @ A),A22:(list @ B)]: (((! [F2:(A > (B > C))]: ((((P @ F2) @ (nil @ A)) @ (nil @ B)))) => ((! [F2:(A > (B > C)),A5:A,As:(list @ A),B2:B,Bs:(list @ B)]: (((((P @ F2) @ As) @ Bs) => (((P @ F2) @ (((cons @ A) @ A5) @ As)) @ (((cons @ B) @ B2) @ Bs))))) => ((! [A5:(A > (B > C)),V:A,Va:(list @ A)]: ((((P @ A5) @ (((cons @ A) @ V) @ Va)) @ (nil @ B)))) => ((! [A5:(A > (B > C)),V:B,Va:(list @ B)]: ((((P @ A5) @ (nil @ A)) @ (((cons @ B) @ V) @ Va)))) => (((P @ A0) @ A1) @ A22)))))))).
% 2.60/1.49 thf(fact_246_mergesort__remdups__correct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [L:(list @ A)]: ((((distinct @ A) @ ((mergesort_remdups @ A) @ L)) & (((linorder_sorted @ A) @ ((mergesort_remdups @ A) @ L)) & (((set2 @ A) @ ((mergesort_remdups @ A) @ L)) = ((set2 @ A) @ L)))))))))).
% 2.60/1.49 thf(fact_247_product__lists_Osimps_I1_J, axiom, (! [A:$tType]: ((((product_lists @ A) @ (nil @ (list @ A))) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 2.60/1.49 thf(fact_248_distinct__product__lists, axiom, (! [A:$tType,Xss2:(list @ (list @ A))]: (((! [X2:(list @ A)]: (((((member @ (list @ A)) @ X2) @ ((set2 @ (list @ A)) @ Xss2)) => ((distinct @ A) @ X2)))) => ((distinct @ (list @ A)) @ ((product_lists @ A) @ Xss2)))))).
% 2.60/1.49 thf(fact_249_subset__code_I3_J, axiom, (! [C:$tType]: ((~ ((((ord_less_eq @ (set @ C)) @ ((coset @ C) @ (nil @ C))) @ ((set2 @ C) @ (nil @ C)))))))).
% 2.60/1.49 thf(fact_250_combinatorial__product_Osimps_I1_J, axiom, (! [A:$tType]: ((((list_c659805718roduct @ A) @ (nil @ (list @ A))) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 2.60/1.49 thf(fact_251_UNIV__coset, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((coset @ A) @ (nil @ A)))))).
% 2.60/1.49 thf(fact_252_subset__code_I2_J, axiom, (! [B:$tType,A3:(set @ B),Ys:(list @ B)]: (((((ord_less_eq @ (set @ B)) @ A3) @ ((coset @ B) @ Ys)) = (! [X4:B]: (((((member @ B) @ X4) @ ((set2 @ B) @ Ys)) => (~ ((((member @ B) @ X4) @ A3)))))))))).
% 2.60/1.49 thf(fact_253_remove__code_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((remove @ A) @ X3) @ ((coset @ A) @ Xs)) = ((coset @ A) @ (((insert @ A) @ X3) @ Xs)))))).
% 2.60/1.49 thf(fact_254_sorted__append__bigger, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A),Y:A]: ((((linorder_sorted @ A) @ Xs) => ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X2) @ Y)))) => ((linorder_sorted @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ Y) @ (nil @ A)))))))))))).
% 2.60/1.49 thf(subcl_Orderings_Olinorder___HOL_Otype, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((type @ A) @ (type2 @ A)))))).
% 2.60/1.49 thf(subcl_Orderings_Olinorder___Orderings_Oord, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((ord @ A) @ (type2 @ A)))))).
% 2.60/1.49 thf(subcl_Orderings_Olinorder___Orderings_Oorder, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((order @ A) @ (type2 @ A)))))).
% 2.60/1.49 thf(subcl_Orderings_Olinorder___Orderings_Opreorder, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((preorder @ A) @ (type2 @ A)))))).
% 2.60/1.49 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A7:$tType,A8:$tType]: ((((order_top @ A8) @ (type2 @ A8)) => ((order_top @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.60/1.49 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.60/1.49 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.60/1.49 thf(tcon_fun___Orderings_Otop, axiom, (! [A7:$tType,A8:$tType]: ((((top @ A8) @ (type2 @ A8)) => ((top @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.60/1.49 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.60/1.49 thf(tcon_Set_Oset___Orderings_Oorder__top_1, axiom, (! [A7:$tType]: (((order_top @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.60/1.49 thf(tcon_Set_Oset___Orderings_Opreorder_2, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.60/1.49 thf(tcon_Set_Oset___Orderings_Oorder_3, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.60/1.49 thf(tcon_Set_Oset___Orderings_Otop_4, axiom, (! [A7:$tType]: (((top @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.60/1.49 thf(tcon_Set_Oset___Orderings_Oord_5, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.60/1.49 thf(tcon_HOL_Obool___Orderings_Oorder__top_6, axiom, ((order_top @ $o) @ (type2 @ $o))).
% 2.60/1.49 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.60/1.49 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.60/1.49 thf(tcon_HOL_Obool___Orderings_Oorder_8, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.60/1.49 thf(tcon_HOL_Obool___Orderings_Otop_9, axiom, ((top @ $o) @ (type2 @ $o))).
% 2.60/1.49 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.60/1.49 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.60/1.49 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y:A]: ((((((if @ A) @ $false) @ X3) @ Y) = Y)))).
% 2.60/1.49 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y:A]: ((((((if @ A) @ $true) @ X3) @ Y) = X3)))).
% 2.60/1.49 thf(tfree_0, hypothesis, ((linorder @ a) @ (type2 @ a))).
% 2.60/1.49 thf(conj_0, conjecture, (((ord_less @ a) @ x2) @ x1)).
% 2.60/1.49 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.TYv64KhOhS/DTF2DTF19011.p
% 2.60/1.49 ------------------------
% 2.97/1.54 ---- Cleaned THF ---
% 2.97/1.54 thf(ty_t_List_Olist,type,
% 2.97/1.54 list: $tType > $tType ).
% 2.97/1.54
% 2.97/1.54 thf(ty_t_Set_Oset,type,
% 2.97/1.54 set: $tType > $tType ).
% 2.97/1.54
% 2.97/1.54 thf(ty_t_itself,type,
% 2.97/1.54 itself: $tType > $tType ).
% 2.97/1.54
% 2.97/1.54 thf(ty_tf_a,type,
% 2.97/1.54 a: $tType ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_HOL_Otype,type,
% 2.97/1.54 type:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Oord,type,
% 2.97/1.54 ord:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Otop,type,
% 2.97/1.54 top:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Oorder,type,
% 2.97/1.54 order:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Ono__bot,type,
% 2.97/1.54 no_bot:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Ono__top,type,
% 2.97/1.54 no_top:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Olinorder,type,
% 2.97/1.54 linorder:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Opreorder,type,
% 2.97/1.54 preorder:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Oorder__top,type,
% 2.97/1.54 order_top:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Owellorder,type,
% 2.97/1.54 wellorder:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Odense__order,type,
% 2.97/1.54 dense_order:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.97/1.54 linordered_idom:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Fields_Olinordered__field,type,
% 2.97/1.54 linordered_field:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.97/1.54 dense_linorder:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.97/1.54 condit1656338222tinuum:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.97/1.54 condit1037483654norder:
% 2.97/1.54 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_If,type,
% 2.97/1.54 if:
% 2.97/1.54 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Oappend,type,
% 2.97/1.54 append:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Ocoset,type,
% 2.97/1.54 coset:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Odistinct,type,
% 2.97/1.54 distinct:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Oinsert,type,
% 2.97/1.54 insert:
% 2.97/1.54 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Olinorder__class_Osorted,type,
% 2.97/1.54 linorder_sorted:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Olist_OCons,type,
% 2.97/1.54 cons:
% 2.97/1.54 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Olist_ONil,type,
% 2.97/1.54 nil:
% 2.97/1.54 !>[A: $tType] : ( list @ A ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Olist_Oset,type,
% 2.97/1.54 set2:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Oproduct__lists,type,
% 2.97/1.54 product_lists:
% 2.97/1.54 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Osublists,type,
% 2.97/1.54 sublists:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List_Ounion,type,
% 2.97/1.54 union:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_List__More_Ocombinatorial__product,type,
% 2.97/1.54 list_c659805718roduct:
% 2.97/1.54 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Misc_Omergesort__remdups,type,
% 2.97/1.54 mergesort_remdups:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Orderings_Oord_Omax,type,
% 2.97/1.54 max:
% 2.97/1.54 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Orderings_Oord_Omin,type,
% 2.97/1.54 min:
% 2.97/1.54 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.97/1.54 ord_less:
% 2.97/1.54 !>[A: $tType] : ( A > A > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.97/1.54 ord_less_eq:
% 2.97/1.54 !>[A: $tType] : ( A > A > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.97/1.54 order_strict_mono:
% 2.97/1.54 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Orderings_Oordering__top,type,
% 2.97/1.54 ordering_top:
% 2.97/1.54 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.97/1.54 top_top:
% 2.97/1.54 !>[A: $tType] : A ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Pure_Otype,type,
% 2.97/1.54 type2:
% 2.97/1.54 !>[A: $tType] : ( itself @ A ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Set_OCollect,type,
% 2.97/1.54 collect:
% 2.97/1.54 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Set_Oremove,type,
% 2.97/1.54 remove:
% 2.97/1.54 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Set_Othe__elem,type,
% 2.97/1.54 the_elem:
% 2.97/1.54 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Osubset__sorted,type,
% 2.97/1.54 sorted1061247458sorted:
% 2.97/1.54 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_c_member,type,
% 2.97/1.54 member:
% 2.97/1.54 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(sy_v_l1,type,
% 2.97/1.54 l1: list @ a ).
% 2.97/1.54
% 2.97/1.54 thf(sy_v_l1a____,type,
% 2.97/1.54 l1a: list @ a ).
% 2.97/1.54
% 2.97/1.54 thf(sy_v_l2,type,
% 2.97/1.54 l2: list @ a ).
% 2.97/1.54
% 2.97/1.54 thf(sy_v_l2b____,type,
% 2.97/1.54 l2b: list @ a ).
% 2.97/1.54
% 2.97/1.54 thf(sy_v_x1____,type,
% 2.97/1.54 x1: a ).
% 2.97/1.54
% 2.97/1.54 thf(sy_v_x2____,type,
% 2.97/1.54 x2: a ).
% 2.97/1.54
% 2.97/1.54 thf(fact_0_x1__neq__x2,axiom,
% 2.97/1.54 x1 != x2 ).
% 2.97/1.54
% 2.97/1.54 thf(fact_1_x2__le__x1,axiom,
% 2.97/1.54 ~ ( ord_less @ a @ x1 @ x2 ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_2_minf_I11_J,axiom,
% 2.97/1.54 ! [C: $tType,D: $tType] :
% 2.97/1.54 ( ( ord @ C @ ( type2 @ C ) )
% 2.97/1.54 => ! [F: D] :
% 2.97/1.54 ? [Z: C] :
% 2.97/1.54 ! [X: C] :
% 2.97/1.54 ( ( ord_less @ C @ X @ Z )
% 2.97/1.54 => ( F = F ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_3_minf_I7_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ X @ Z )
% 2.97/1.54 => ~ ( ord_less @ A @ T @ X ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_4_minf_I5_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ X @ Z )
% 2.97/1.54 => ( ord_less @ A @ X @ T ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_5_minf_I4_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ X @ Z )
% 2.97/1.54 => ( X != T ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_6_minf_I3_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ X @ Z )
% 2.97/1.54 => ( X != T ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_7_minf_I2_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.97/1.54 ( ? [Z2: A] :
% 2.97/1.54 ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Z2 )
% 2.97/1.54 => ( ( P @ X2 )
% 2.97/1.54 = ( P2 @ X2 ) ) )
% 2.97/1.54 => ( ? [Z2: A] :
% 2.97/1.54 ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Z2 )
% 2.97/1.54 => ( ( Q @ X2 )
% 2.97/1.54 = ( Q2 @ X2 ) ) )
% 2.97/1.54 => ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ X @ Z )
% 2.97/1.54 => ( ( ( P @ X )
% 2.97/1.54 | ( Q @ X ) )
% 2.97/1.54 = ( ( P2 @ X )
% 2.97/1.54 | ( Q2 @ X ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_8_minf_I1_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.97/1.54 ( ? [Z2: A] :
% 2.97/1.54 ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Z2 )
% 2.97/1.54 => ( ( P @ X2 )
% 2.97/1.54 = ( P2 @ X2 ) ) )
% 2.97/1.54 => ( ? [Z2: A] :
% 2.97/1.54 ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Z2 )
% 2.97/1.54 => ( ( Q @ X2 )
% 2.97/1.54 = ( Q2 @ X2 ) ) )
% 2.97/1.54 => ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ X @ Z )
% 2.97/1.54 => ( ( ( P @ X )
% 2.97/1.54 & ( Q @ X ) )
% 2.97/1.54 = ( ( P2 @ X )
% 2.97/1.54 & ( Q2 @ X ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_9_pinf_I11_J,axiom,
% 2.97/1.54 ! [C: $tType,D: $tType] :
% 2.97/1.54 ( ( ord @ C @ ( type2 @ C ) )
% 2.97/1.54 => ! [F: D] :
% 2.97/1.54 ? [Z: C] :
% 2.97/1.54 ! [X: C] :
% 2.97/1.54 ( ( ord_less @ C @ Z @ X )
% 2.97/1.54 => ( F = F ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_10_pinf_I7_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z @ X )
% 2.97/1.54 => ( ord_less @ A @ T @ X ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_11_pinf_I5_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z @ X )
% 2.97/1.54 => ~ ( ord_less @ A @ X @ T ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_12_pinf_I4_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z @ X )
% 2.97/1.54 => ( X != T ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_13_pinf_I3_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z @ X )
% 2.97/1.54 => ( X != T ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_14_pinf_I1_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.97/1.54 ( ? [Z2: A] :
% 2.97/1.54 ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z2 @ X2 )
% 2.97/1.54 => ( ( P @ X2 )
% 2.97/1.54 = ( P2 @ X2 ) ) )
% 2.97/1.54 => ( ? [Z2: A] :
% 2.97/1.54 ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z2 @ X2 )
% 2.97/1.54 => ( ( Q @ X2 )
% 2.97/1.54 = ( Q2 @ X2 ) ) )
% 2.97/1.54 => ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z @ X )
% 2.97/1.54 => ( ( ( P @ X )
% 2.97/1.54 & ( Q @ X ) )
% 2.97/1.54 = ( ( P2 @ X )
% 2.97/1.54 & ( Q2 @ X ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_15_pinf_I2_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.97/1.54 ( ? [Z2: A] :
% 2.97/1.54 ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z2 @ X2 )
% 2.97/1.54 => ( ( P @ X2 )
% 2.97/1.54 = ( P2 @ X2 ) ) )
% 2.97/1.54 => ( ? [Z2: A] :
% 2.97/1.54 ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z2 @ X2 )
% 2.97/1.54 => ( ( Q @ X2 )
% 2.97/1.54 = ( Q2 @ X2 ) ) )
% 2.97/1.54 => ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z @ X )
% 2.97/1.54 => ( ( ( P @ X )
% 2.97/1.54 | ( Q @ X ) )
% 2.97/1.54 = ( ( P2 @ X )
% 2.97/1.54 | ( Q2 @ X ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_16_ex__gt__or__lt,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A] :
% 2.97/1.54 ? [B2: A] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B2 )
% 2.97/1.54 | ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_17_linorder__neqE__linordered__idom,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( X3 != Y )
% 2.97/1.54 => ( ~ ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_18_dual__order_Ostrict__implies__not__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [B3: A,A2: A] :
% 2.97/1.54 ( ( ord_less @ A @ B3 @ A2 )
% 2.97/1.54 => ( A2 != B3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_19_linordered__field__no__ub,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X: A] :
% 2.97/1.54 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_20_linordered__field__no__lb,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X: A] :
% 2.97/1.54 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_21_order_Ostrict__implies__not__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( A2 != B3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_22_dependent__wellorder__choice,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: ( A > B ) > A > B > $o] :
% 2.97/1.54 ( ! [R: B,F2: A > B,G: A > B,X2: A] :
% 2.97/1.54 ( ! [Y3: A] :
% 2.97/1.54 ( ( ord_less @ A @ Y3 @ X2 )
% 2.97/1.54 => ( ( F2 @ Y3 )
% 2.97/1.54 = ( G @ Y3 ) ) )
% 2.97/1.54 => ( ( P @ F2 @ X2 @ R )
% 2.97/1.54 = ( P @ G @ X2 @ R ) ) )
% 2.97/1.54 => ( ! [X2: A,F2: A > B] :
% 2.97/1.54 ( ! [Y3: A] :
% 2.97/1.54 ( ( ord_less @ A @ Y3 @ X2 )
% 2.97/1.54 => ( P @ F2 @ Y3 @ ( F2 @ Y3 ) ) )
% 2.97/1.54 => ? [X12: B] : ( P @ F2 @ X2 @ X12 ) )
% 2.97/1.54 => ? [F2: A > B] :
% 2.97/1.54 ! [X: A] : ( P @ F2 @ X @ ( F2 @ X ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_23_not__less__iff__gr__or__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ~ ( ord_less @ A @ X3 @ Y ) )
% 2.97/1.54 = ( ( ord_less @ A @ Y @ X3 )
% 2.97/1.54 | ( X3 = Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_24_dual__order_Ostrict__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [B3: A,A2: A,C2: A] :
% 2.97/1.54 ( ( ord_less @ A @ B3 @ A2 )
% 2.97/1.54 => ( ( ord_less @ A @ C2 @ B3 )
% 2.97/1.54 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_25_less__imp__not__less,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_26_order_Ostrict__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less @ A @ B3 @ C2 )
% 2.97/1.54 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_27_ord__eq__less__subst,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType] :
% 2.97/1.54 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 & ( ord @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 2.97/1.54 ( ( A2
% 2.97/1.54 = ( F3 @ B3 ) )
% 2.97/1.54 => ( ( ord_less @ B @ B3 @ C2 )
% 2.97/1.54 => ( ! [X2: B,Y2: B] :
% 2.97/1.54 ( ( ord_less @ B @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_28_ord__less__eq__subst,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType] :
% 2.97/1.54 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 & ( ord @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,B3: A,F3: A > B,C2: B] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ( F3 @ B3 )
% 2.97/1.54 = C2 )
% 2.97/1.54 => ( ! [X2: A,Y2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_29_order__less__subst1,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType] :
% 2.97/1.54 ( ( ( order @ B @ ( type2 @ B ) )
% 2.97/1.54 & ( order @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ ( F3 @ B3 ) )
% 2.97/1.54 => ( ( ord_less @ B @ B3 @ C2 )
% 2.97/1.54 => ( ! [X2: B,Y2: B] :
% 2.97/1.54 ( ( ord_less @ B @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_30_order__less__subst2,axiom,
% 2.97/1.54 ! [A: $tType,C: $tType] :
% 2.97/1.54 ( ( ( order @ C @ ( type2 @ C ) )
% 2.97/1.54 & ( order @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,B3: A,F3: A > C,C2: C] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
% 2.97/1.54 => ( ! [X2: A,Y2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_31_lt__ex,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A] :
% 2.97/1.54 ? [Y2: A] : ( ord_less @ A @ Y2 @ X3 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_32_gt__ex,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( no_top @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A] :
% 2.97/1.54 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_33_neqE,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( X3 != Y )
% 2.97/1.54 => ( ~ ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_34_neq__iff,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( X3 != Y )
% 2.97/1.54 = ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 | ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_35_order_Oasym,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_36_dense,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ? [Z: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Z )
% 2.97/1.54 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_37_less__imp__neq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( X3 != Y ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_38_less__asym,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_39_less__asym_H,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_40_less__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,Z3: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ord_less @ A @ Y @ Z3 )
% 2.97/1.54 => ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_41_less__linear,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 | ( X3 = Y )
% 2.97/1.54 | ( ord_less @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_42_less__irrefl,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A] :
% 2.97/1.54 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_43_ord__eq__less__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A] :
% 2.97/1.54 ( ( A2 = B3 )
% 2.97/1.54 => ( ( ord_less @ A @ B3 @ C2 )
% 2.97/1.54 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_44_ord__less__eq__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( B3 = C2 )
% 2.97/1.54 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_45_mem__Collect__eq,axiom,
% 2.97/1.54 ! [A: $tType,A2: A,P: A > $o] :
% 2.97/1.54 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.97/1.54 = ( P @ A2 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_46_Collect__mem__eq,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A] :
% 2.97/1.54 ( ( collect @ A
% 2.97/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.97/1.54 = A3 ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_47_Collect__cong,axiom,
% 2.97/1.54 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ( P @ X2 )
% 2.97/1.54 = ( Q @ X2 ) )
% 2.97/1.54 => ( ( collect @ A @ P )
% 2.97/1.54 = ( collect @ A @ Q ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_48_ext,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ( F3 @ X2 )
% 2.97/1.54 = ( G2 @ X2 ) )
% 2.97/1.54 => ( F3 = G2 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_49_dual__order_Oasym,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [B3: A,A2: A] :
% 2.97/1.54 ( ( ord_less @ A @ B3 @ A2 )
% 2.97/1.54 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_50_less__imp__not__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( X3 != Y ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_51_less__not__sym,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_52_less__induct,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: A > $o,A2: A] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ! [Y3: A] :
% 2.97/1.54 ( ( ord_less @ A @ Y3 @ X2 )
% 2.97/1.54 => ( P @ Y3 ) )
% 2.97/1.54 => ( P @ X2 ) )
% 2.97/1.54 => ( P @ A2 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_53_antisym__conv3,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Y: A,X3: A] :
% 2.97/1.54 ( ~ ( ord_less @ A @ Y @ X3 )
% 2.97/1.54 => ( ( ~ ( ord_less @ A @ X3 @ Y ) )
% 2.97/1.54 = ( X3 = Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_54_less__imp__not__eq2,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( Y != X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_55_less__imp__triv,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,P: $o] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ord_less @ A @ Y @ X3 )
% 2.97/1.54 => P ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_56_linorder__cases,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ~ ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ( X3 != Y )
% 2.97/1.54 => ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_57_dual__order_Oirrefl,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A] :
% 2.97/1.54 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_58_strict__monoD,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 & ( order @ B @ ( type2 @ B ) ) )
% 2.97/1.54 => ! [F3: A > B,X3: A,Y: A] :
% 2.97/1.54 ( ( order_strict_mono @ A @ B @ F3 )
% 2.97/1.54 => ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_59_strict__monoI,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 & ( order @ B @ ( type2 @ B ) ) )
% 2.97/1.54 => ! [F3: A > B] :
% 2.97/1.54 ( ! [X2: A,Y2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( order_strict_mono @ A @ B @ F3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_60_strict__mono__def,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 & ( order @ B @ ( type2 @ B ) ) )
% 2.97/1.54 => ( ( order_strict_mono @ A @ B )
% 2.97/1.54 = ( ^ [F4: A > B] :
% 2.97/1.54 ! [X4: A,Y4: A] :
% 2.97/1.54 ( ( ord_less @ A @ X4 @ Y4 )
% 2.97/1.54 => ( ord_less @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_61_strict__mono__less,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 & ( order @ B @ ( type2 @ B ) ) )
% 2.97/1.54 => ! [F3: A > B,X3: A,Y: A] :
% 2.97/1.54 ( ( order_strict_mono @ A @ B @ F3 )
% 2.97/1.54 => ( ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
% 2.97/1.54 = ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_62_x2__nin__l2,axiom,
% 2.97/1.54 ~ ( member @ a @ x2 @ ( set2 @ a @ l2b ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_63_x1__nin__l1,axiom,
% 2.97/1.54 ~ ( member @ a @ x1 @ ( set2 @ a @ l1a ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_64_ord_Omin__def,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( min @ A )
% 2.97/1.54 = ( ^ [Less_eq: A > A > $o,A4: A,B4: A] : ( if @ A @ ( Less_eq @ A4 @ B4 ) @ A4 @ B4 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_65_ord_Omax__def,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( max @ A )
% 2.97/1.54 = ( ^ [Less_eq: A > A > $o,A4: A,B4: A] : ( if @ A @ ( Less_eq @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_66_top_Oextremum__strict,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order_top @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A] :
% 2.97/1.54 ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_67_top__apply,axiom,
% 2.97/1.54 ! [C: $tType,D: $tType] :
% 2.97/1.54 ( ( top @ C @ ( type2 @ C ) )
% 2.97/1.54 => ( ( top_top @ ( D > C ) )
% 2.97/1.54 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_68_strict__mono__eq,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 & ( order @ B @ ( type2 @ B ) ) )
% 2.97/1.54 => ! [F3: A > B,X3: A,Y: A] :
% 2.97/1.54 ( ( order_strict_mono @ A @ B @ F3 )
% 2.97/1.54 => ( ( ( F3 @ X3 )
% 2.97/1.54 = ( F3 @ Y ) )
% 2.97/1.54 = ( X3 = Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_69_top_Onot__eq__extremum,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order_top @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A] :
% 2.97/1.54 ( ( A2
% 2.97/1.54 != ( top_top @ A ) )
% 2.97/1.54 = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_70_x2__le,axiom,
% 2.97/1.54 ! [X3: a] :
% 2.97/1.54 ( ( member @ a @ X3 @ ( set2 @ a @ l2b ) )
% 2.97/1.54 => ( ord_less_eq @ a @ x2 @ X3 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_71_x1__le,axiom,
% 2.97/1.54 ! [X3: a] :
% 2.97/1.54 ( ( member @ a @ X3 @ ( set2 @ a @ l1a ) )
% 2.97/1.54 => ( ord_less_eq @ a @ x1 @ X3 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_72_ind__hyp__l2,axiom,
% 2.97/1.54 ( ( sorted1061247458sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b )
% 2.97/1.54 = ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ l2b ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_73_x2__l2__props,axiom,
% 2.97/1.54 ( ( distinct @ a @ ( cons @ a @ x2 @ l2b ) )
% 2.97/1.54 & ( linorder_sorted @ a @ ( cons @ a @ x2 @ l2b ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_74_x1__l1__props,axiom,
% 2.97/1.54 ( ( distinct @ a @ ( cons @ a @ x1 @ l1a ) )
% 2.97/1.54 & ( linorder_sorted @ a @ ( cons @ a @ x1 @ l1a ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_75_Cons_Ohyps,axiom,
% 2.97/1.54 ( ( ( distinct @ a @ l2b )
% 2.97/1.54 & ( linorder_sorted @ a @ l2b ) )
% 2.97/1.54 => ( ( sorted1061247458sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b )
% 2.97/1.54 = ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ l2b ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_76_strict__mono__less__eq,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 & ( order @ B @ ( type2 @ B ) ) )
% 2.97/1.54 => ! [F3: A > B,X3: A,Y: A] :
% 2.97/1.54 ( ( order_strict_mono @ A @ B @ F3 )
% 2.97/1.54 => ( ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
% 2.97/1.54 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_77_l2__OK,axiom,
% 2.97/1.54 ( ( distinct @ a @ l2 )
% 2.97/1.54 & ( linorder_sorted @ a @ l2 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_78_l1__OK,axiom,
% 2.97/1.54 ( ( distinct @ a @ l1 )
% 2.97/1.54 & ( linorder_sorted @ a @ l1 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_79_order__refl,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_80_l2__props,axiom,
% 2.97/1.54 ( ( distinct @ a @ l2b )
% 2.97/1.54 & ( linorder_sorted @ a @ l2b ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_81_l1__props,axiom,
% 2.97/1.54 ( ( distinct @ a @ l1a )
% 2.97/1.54 & ( linorder_sorted @ a @ l1a ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_82_ind__hyp__l1,axiom,
% 2.97/1.54 ! [L2: list @ a] :
% 2.97/1.54 ( ( ( distinct @ a @ L2 )
% 2.97/1.54 & ( linorder_sorted @ a @ L2 ) )
% 2.97/1.54 => ( ( sorted1061247458sorted @ a @ l1a @ L2 )
% 2.97/1.54 = ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ l1a ) @ ( set2 @ a @ L2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_83_subset__sorted_Osimps_I3_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X13: A,L1: list @ A,X22: A,L2: list @ A] :
% 2.97/1.54 ( ( sorted1061247458sorted @ A @ ( cons @ A @ X13 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
% 2.97/1.54 = ( ~ ( ord_less @ A @ X13 @ X22 )
% 2.97/1.54 & ( ~ ( ord_less @ A @ X13 @ X22 )
% 2.97/1.54 => ( ( ( X13 = X22 )
% 2.97/1.54 => ( sorted1061247458sorted @ A @ L1 @ L2 ) )
% 2.97/1.54 & ( ( X13 != X22 )
% 2.97/1.54 => ( sorted1061247458sorted @ A @ ( cons @ A @ X13 @ L1 ) @ L2 ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_84_le__funD,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 => ! [F3: A > B,G2: A > B,X3: A] :
% 2.97/1.54 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 2.97/1.54 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_85_le__funE,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 => ! [F3: A > B,G2: A > B,X3: A] :
% 2.97/1.54 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 2.97/1.54 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_86_le__funI,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 => ! [F3: A > B,G2: A > B] :
% 2.97/1.54 ( ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 2.97/1.54 => ( ord_less_eq @ ( A > B ) @ F3 @ G2 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_87_le__fun__def,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 => ( ( ord_less_eq @ ( A > B ) )
% 2.97/1.54 = ( ^ [F4: A > B,G3: A > B] :
% 2.97/1.54 ! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_88_order__subst1,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType] :
% 2.97/1.54 ( ( ( order @ B @ ( type2 @ B ) )
% 2.97/1.54 & ( order @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B3 ) )
% 2.97/1.54 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.97/1.54 => ( ! [X2: B,Y2: B] :
% 2.97/1.54 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_89_order__subst2,axiom,
% 2.97/1.54 ! [A: $tType,C: $tType] :
% 2.97/1.54 ( ( ( order @ C @ ( type2 @ C ) )
% 2.97/1.54 & ( order @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,B3: A,F3: A > C,C2: C] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
% 2.97/1.54 => ( ! [X2: A,Y2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less_eq @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_90_ord__eq__le__subst,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType] :
% 2.97/1.54 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 & ( ord @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 2.97/1.54 ( ( A2
% 2.97/1.54 = ( F3 @ B3 ) )
% 2.97/1.54 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.97/1.54 => ( ! [X2: B,Y2: B] :
% 2.97/1.54 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_91_ord__le__eq__subst,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType] :
% 2.97/1.54 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 & ( ord @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,B3: A,F3: A > B,C2: B] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ( F3 @ B3 )
% 2.97/1.54 = C2 )
% 2.97/1.54 => ( ! [X2: A,Y2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_92_eq__iff,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 2.97/1.54 = ( ^ [X4: A,Y4: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.97/1.54 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_93_antisym,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ord_less_eq @ A @ Y @ X3 )
% 2.97/1.54 => ( X3 = Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_94_linear,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 | ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_95_eq__refl,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( X3 = Y )
% 2.97/1.54 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_96_le__cases,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ~ ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_97_order_Otrans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.97/1.54 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_98_le__cases3,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,Z3: A] :
% 2.97/1.54 ( ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 2.97/1.54 => ( ( ( ord_less_eq @ A @ Y @ X3 )
% 2.97/1.54 => ~ ( ord_less_eq @ A @ X3 @ Z3 ) )
% 2.97/1.54 => ( ( ( ord_less_eq @ A @ X3 @ Z3 )
% 2.97/1.54 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 2.97/1.54 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 2.97/1.54 => ~ ( ord_less_eq @ A @ Y @ X3 ) )
% 2.97/1.54 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.97/1.54 => ~ ( ord_less_eq @ A @ Z3 @ X3 ) )
% 2.97/1.54 => ~ ( ( ord_less_eq @ A @ Z3 @ X3 )
% 2.97/1.54 => ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_99_antisym__conv,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Y: A,X3: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ Y @ X3 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 = ( X3 = Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_100_ord__eq__le__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A] :
% 2.97/1.54 ( ( A2 = B3 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.97/1.54 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_101_ord__le__eq__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( B3 = C2 )
% 2.97/1.54 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_102_order__class_Oorder_Oantisym,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.97/1.54 => ( A2 = B3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_103_order__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,Z3: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.97/1.54 => ( ord_less_eq @ A @ X3 @ Z3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_104_dual__order_Orefl,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_105_linorder__wlog,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: A > A > $o,A2: A,B3: A] :
% 2.97/1.54 ( ! [A5: A,B2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A5 @ B2 )
% 2.97/1.54 => ( P @ A5 @ B2 ) )
% 2.97/1.54 => ( ! [A5: A,B2: A] :
% 2.97/1.54 ( ( P @ B2 @ A5 )
% 2.97/1.54 => ( P @ A5 @ B2 ) )
% 2.97/1.54 => ( P @ A2 @ B3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_106_dual__order_Otrans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [B3: A,A2: A,C2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.97/1.54 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_107_dual__order_Oantisym,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [B3: A,A2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( A2 = B3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_108_leD,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Y: A,X3: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ Y @ X3 )
% 2.97/1.54 => ~ ( ord_less @ A @ X3 @ Y ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_109_leI,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ~ ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_110_le__less,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( ord_less_eq @ A )
% 2.97/1.54 = ( ^ [X4: A,Y4: A] :
% 2.97/1.54 ( ( ord_less @ A @ X4 @ Y4 )
% 2.97/1.54 | ( X4 = Y4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_111_less__le,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( ord_less @ A )
% 2.97/1.54 = ( ^ [X4: A,Y4: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.97/1.54 & ( X4 != Y4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_112_order__le__less__subst1,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType] :
% 2.97/1.54 ( ( ( order @ B @ ( type2 @ B ) )
% 2.97/1.54 & ( order @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B3 ) )
% 2.97/1.54 => ( ( ord_less @ B @ B3 @ C2 )
% 2.97/1.54 => ( ! [X2: B,Y2: B] :
% 2.97/1.54 ( ( ord_less @ B @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_113_order__le__less__subst2,axiom,
% 2.97/1.54 ! [A: $tType,C: $tType] :
% 2.97/1.54 ( ( ( order @ C @ ( type2 @ C ) )
% 2.97/1.54 & ( order @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,B3: A,F3: A > C,C2: C] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
% 2.97/1.54 => ( ! [X2: A,Y2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less_eq @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_114_order__less__le__subst1,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType] :
% 2.97/1.54 ( ( ( order @ B @ ( type2 @ B ) )
% 2.97/1.54 & ( order @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,F3: B > A,B3: B,C2: B] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ ( F3 @ B3 ) )
% 2.97/1.54 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.97/1.54 => ( ! [X2: B,Y2: B] :
% 2.97/1.54 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_115_order__less__le__subst2,axiom,
% 2.97/1.54 ! [A: $tType,C: $tType] :
% 2.97/1.54 ( ( ( order @ C @ ( type2 @ C ) )
% 2.97/1.54 & ( order @ A @ ( type2 @ A ) ) )
% 2.97/1.54 => ! [A2: A,B3: A,F3: A > C,C2: C] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
% 2.97/1.54 => ( ! [X2: A,Y2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Y2 )
% 2.97/1.54 => ( ord_less @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.97/1.54 => ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_116_not__le,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ~ ( ord_less_eq @ A @ X3 @ Y ) )
% 2.97/1.54 = ( ord_less @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_117_not__less,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ~ ( ord_less @ A @ X3 @ Y ) )
% 2.97/1.54 = ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_118_le__neq__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( A2 != B3 )
% 2.97/1.54 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_119_less__imp__le,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_120_antisym__conv1,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ~ ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 = ( X3 = Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_121_antisym__conv2,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ~ ( ord_less @ A @ X3 @ Y ) )
% 2.97/1.54 = ( X3 = Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_122_le__less__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,Z3: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ord_less @ A @ Y @ Z3 )
% 2.97/1.54 => ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_123_less__le__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,Z3: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.97/1.54 => ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_124_dense__ge,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Z3: A,Y: A] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z3 @ X2 )
% 2.97/1.54 => ( ord_less_eq @ A @ Y @ X2 ) )
% 2.97/1.54 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_125_dense__le,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Y: A,Z3: A] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ( ord_less @ A @ X2 @ Y )
% 2.97/1.54 => ( ord_less_eq @ A @ X2 @ Z3 ) )
% 2.97/1.54 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_126_le__less__linear,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 | ( ord_less @ A @ Y @ X3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_127_le__imp__less__or__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 => ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 | ( X3 = Y ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_128_less__le__not__le,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( preorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( ord_less @ A )
% 2.97/1.54 = ( ^ [X4: A,Y4: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.97/1.54 & ~ ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_129_not__le__imp__less,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Y: A,X3: A] :
% 2.97/1.54 ( ~ ( ord_less_eq @ A @ Y @ X3 )
% 2.97/1.54 => ( ord_less @ A @ X3 @ Y ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_130_order_Ostrict__trans1,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less @ A @ B3 @ C2 )
% 2.97/1.54 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_131_order_Ostrict__trans2,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.97/1.54 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_132_order_Oorder__iff__strict,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( ord_less_eq @ A )
% 2.97/1.54 = ( ^ [A4: A,B4: A] :
% 2.97/1.54 ( ( ord_less @ A @ A4 @ B4 )
% 2.97/1.54 | ( A4 = B4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_133_order_Ostrict__iff__order,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( ord_less @ A )
% 2.97/1.54 = ( ^ [A4: A,B4: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.97/1.54 & ( A4 != B4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_134_dual__order_Ostrict__trans1,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [B3: A,A2: A,C2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.97/1.54 => ( ( ord_less @ A @ C2 @ B3 )
% 2.97/1.54 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_135_dual__order_Ostrict__trans2,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [B3: A,A2: A,C2: A] :
% 2.97/1.54 ( ( ord_less @ A @ B3 @ A2 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.97/1.54 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_136_dense__ge__bounded,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Z3: A,X3: A,Y: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z3 @ X3 )
% 2.97/1.54 => ( ! [W: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z3 @ W )
% 2.97/1.54 => ( ( ord_less @ A @ W @ X3 )
% 2.97/1.54 => ( ord_less_eq @ A @ Y @ W ) ) )
% 2.97/1.54 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_137_dense__le__bounded,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,Z3: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ Y )
% 2.97/1.54 => ( ! [W: A] :
% 2.97/1.54 ( ( ord_less @ A @ X3 @ W )
% 2.97/1.54 => ( ( ord_less @ A @ W @ Y )
% 2.97/1.54 => ( ord_less_eq @ A @ W @ Z3 ) ) )
% 2.97/1.54 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_138_order_Ostrict__implies__order,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_139_dual__order_Oorder__iff__strict,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( ord_less_eq @ A )
% 2.97/1.54 = ( ^ [B4: A,A4: A] :
% 2.97/1.54 ( ( ord_less @ A @ B4 @ A4 )
% 2.97/1.54 | ( A4 = B4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_140_dual__order_Ostrict__iff__order,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( ord_less @ A )
% 2.97/1.54 = ( ^ [B4: A,A4: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.97/1.54 & ( A4 != B4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_141_dual__order_Ostrict__implies__order,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [B3: A,A2: A] :
% 2.97/1.54 ( ( ord_less @ A @ B3 @ A2 )
% 2.97/1.54 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_142_order_Onot__eq__order__implies__strict,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A] :
% 2.97/1.54 ( ( A2 != B3 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.97/1.54 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_143_complete__interval,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,P: A > $o] :
% 2.97/1.54 ( ( ord_less @ A @ A2 @ B3 )
% 2.97/1.54 => ( ( P @ A2 )
% 2.97/1.54 => ( ~ ( P @ B3 )
% 2.97/1.54 => ? [C3: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.97/1.54 & ( ord_less_eq @ A @ C3 @ B3 )
% 2.97/1.54 & ! [X: A] :
% 2.97/1.54 ( ( ( ord_less_eq @ A @ A2 @ X )
% 2.97/1.54 & ( ord_less @ A @ X @ C3 ) )
% 2.97/1.54 => ( P @ X ) )
% 2.97/1.54 & ! [D2: A] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 2.97/1.54 & ( ord_less @ A @ X2 @ D2 ) )
% 2.97/1.54 => ( P @ X2 ) )
% 2.97/1.54 => ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_144_minf_I8_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ X @ Z )
% 2.97/1.54 => ~ ( ord_less_eq @ A @ T @ X ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_145_minf_I6_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ X @ Z )
% 2.97/1.54 => ( ord_less_eq @ A @ X @ T ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_146_pinf_I8_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z @ X )
% 2.97/1.54 => ( ord_less_eq @ A @ T @ X ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_147_pinf_I6_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [T: A] :
% 2.97/1.54 ? [Z: A] :
% 2.97/1.54 ! [X: A] :
% 2.97/1.54 ( ( ord_less @ A @ Z @ X )
% 2.97/1.54 => ~ ( ord_less_eq @ A @ X @ T ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_148_top__greatest,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order_top @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_149_top_Oextremum__unique,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order_top @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.97/1.54 = ( A2
% 2.97/1.54 = ( top_top @ A ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_150_top_Oextremum__uniqueI,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order_top @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.97/1.54 => ( A2
% 2.97/1.54 = ( top_top @ A ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_151_sorted__many__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,Zs: list @ A] :
% 2.97/1.54 ( ( linorder_sorted @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs ) ) )
% 2.97/1.54 = ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 & ( linorder_sorted @ A @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_152_sorted_OCons,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Xs: list @ A,X3: A] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 2.97/1.54 => ( ord_less_eq @ A @ X3 @ X2 ) )
% 2.97/1.54 => ( ( linorder_sorted @ A @ Xs )
% 2.97/1.54 => ( linorder_sorted @ A @ ( cons @ A @ X3 @ Xs ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_153_sorted__Cons,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Xs: list @ A] :
% 2.97/1.54 ( ( linorder_sorted @ A @ ( cons @ A @ X3 @ Xs ) )
% 2.97/1.54 = ( ( linorder_sorted @ A @ Xs )
% 2.97/1.54 & ! [X4: A] :
% 2.97/1.54 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.97/1.54 => ( ord_less_eq @ A @ X3 @ X4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_154_sorted__distinct__set__unique,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Xs: list @ A,Ys: list @ A] :
% 2.97/1.54 ( ( linorder_sorted @ A @ Xs )
% 2.97/1.54 => ( ( distinct @ A @ Xs )
% 2.97/1.54 => ( ( linorder_sorted @ A @ Ys )
% 2.97/1.54 => ( ( distinct @ A @ Ys )
% 2.97/1.54 => ( ( ( set2 @ A @ Xs )
% 2.97/1.54 = ( set2 @ A @ Ys ) )
% 2.97/1.54 => ( Xs = Ys ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_155_distinct_Osimps_I2_J,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,Xs: list @ A] :
% 2.97/1.54 ( ( distinct @ A @ ( cons @ A @ X3 @ Xs ) )
% 2.97/1.54 = ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.97/1.54 & ( distinct @ A @ Xs ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_156_sorted__many,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A,Y: A,Zs: list @ A] :
% 2.97/1.54 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.97/1.54 => ( ( linorder_sorted @ A @ ( cons @ A @ Y @ Zs ) )
% 2.97/1.54 => ( linorder_sorted @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_157_list_Oinject,axiom,
% 2.97/1.54 ! [A: $tType,X21: A,X222: list @ A,Y21: A,Y22: list @ A] :
% 2.97/1.54 ( ( ( cons @ A @ X21 @ X222 )
% 2.97/1.54 = ( cons @ A @ Y21 @ Y22 ) )
% 2.97/1.54 = ( ( X21 = Y21 )
% 2.97/1.54 & ( X222 = Y22 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_158_less__fun__def,axiom,
% 2.97/1.54 ! [B: $tType,A: $tType] :
% 2.97/1.54 ( ( ord @ B @ ( type2 @ B ) )
% 2.97/1.54 => ( ( ord_less @ ( A > B ) )
% 2.97/1.54 = ( ^ [F4: A > B,G3: A > B] :
% 2.97/1.54 ( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
% 2.97/1.54 & ~ ( ord_less_eq @ ( A > B ) @ G3 @ F4 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_159_not__Cons__self2,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,Xs: list @ A] :
% 2.97/1.54 ( ( cons @ A @ X3 @ Xs )
% 2.97/1.54 != Xs ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_160_list_Oset__cases,axiom,
% 2.97/1.54 ! [A: $tType,E: A,A2: list @ A] :
% 2.97/1.54 ( ( member @ A @ E @ ( set2 @ A @ A2 ) )
% 2.97/1.54 => ( ! [Z22: list @ A] :
% 2.97/1.54 ( A2
% 2.97/1.54 != ( cons @ A @ E @ Z22 ) )
% 2.97/1.54 => ~ ! [Z1: A,Z22: list @ A] :
% 2.97/1.54 ( ( A2
% 2.97/1.54 = ( cons @ A @ Z1 @ Z22 ) )
% 2.97/1.54 => ~ ( member @ A @ E @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_161_set__ConsD,axiom,
% 2.97/1.54 ! [A: $tType,Y: A,X3: A,Xs: list @ A] :
% 2.97/1.54 ( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
% 2.97/1.54 => ( ( Y = X3 )
% 2.97/1.54 | ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_162_list_Oset__intros_I1_J,axiom,
% 2.97/1.54 ! [A: $tType,A1: A,A22: list @ A] : ( member @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_163_list_Oset__intros_I2_J,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,A22: list @ A,A1: A] :
% 2.97/1.54 ( ( member @ A @ X3 @ ( set2 @ A @ A22 ) )
% 2.97/1.54 => ( member @ A @ X3 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_164_subset__code_I1_J,axiom,
% 2.97/1.54 ! [A: $tType,Xs: list @ A,B5: set @ A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B5 )
% 2.97/1.54 = ( ! [X4: A] :
% 2.97/1.54 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.97/1.54 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_165_distinct__length__2__or__more,axiom,
% 2.97/1.54 ! [A: $tType,A2: A,B3: A,Xs: list @ A] :
% 2.97/1.54 ( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B3 @ Xs ) ) )
% 2.97/1.54 = ( ( A2 != B3 )
% 2.97/1.54 & ( distinct @ A @ ( cons @ A @ A2 @ Xs ) )
% 2.97/1.54 & ( distinct @ A @ ( cons @ A @ B3 @ Xs ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_166_set__subset__Cons,axiom,
% 2.97/1.54 ! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_167_subsetI,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ( member @ A @ X2 @ A3 )
% 2.97/1.54 => ( member @ A @ X2 @ B5 ) )
% 2.97/1.54 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_168_subset__antisym,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
% 2.97/1.54 => ( A3 = B5 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_169_psubsetI,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( A3 != B5 )
% 2.97/1.54 => ( ord_less @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_170_subset__UNIV,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_171_psubsetE,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( ord_less @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_172_psubset__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord_less @ ( set @ A ) )
% 2.97/1.54 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.97/1.54 & ( A6 != B6 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_173_psubset__imp__subset,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( ord_less @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_174_psubset__subset__trans,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
% 2.97/1.54 ( ( ord_less @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 2.97/1.54 => ( ord_less @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_175_subset__not__subset__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord_less @ ( set @ A ) )
% 2.97/1.54 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.97/1.54 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_176_subset__psubset__trans,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( ord_less @ ( set @ A ) @ B5 @ C4 )
% 2.97/1.54 => ( ord_less @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_177_subset__iff__psubset__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) )
% 2.97/1.54 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.97/1.54 ( ( ord_less @ ( set @ A ) @ A6 @ B6 )
% 2.97/1.54 | ( A6 = B6 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_178_Collect__mono__iff,axiom,
% 2.97/1.54 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.97/1.54 = ( ! [X4: A] :
% 2.97/1.54 ( ( P @ X4 )
% 2.97/1.54 => ( Q @ X4 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_179_contra__subsetD,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ~ ( member @ A @ C2 @ B5 )
% 2.97/1.54 => ~ ( member @ A @ C2 @ A3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_180_set__eq__subset,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ^ [Y5: set @ A,Z4: set @ A] : ( Y5 = Z4 ) )
% 2.97/1.54 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.97/1.54 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_181_subset__trans,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 2.97/1.54 => ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_182_Collect__mono,axiom,
% 2.97/1.54 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.97/1.54 ( ! [X2: A] :
% 2.97/1.54 ( ( P @ X2 )
% 2.97/1.54 => ( Q @ X2 ) )
% 2.97/1.54 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_183_subset__refl,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_184_rev__subsetD,axiom,
% 2.97/1.54 ! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( member @ A @ C2 @ A3 )
% 2.97/1.54 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( member @ A @ C2 @ B5 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_185_subset__iff,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) )
% 2.97/1.54 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.97/1.54 ! [T2: A] :
% 2.97/1.54 ( ( member @ A @ T2 @ A6 )
% 2.97/1.54 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_186_set__rev__mp,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( member @ A @ X3 @ A3 )
% 2.97/1.54 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( member @ A @ X3 @ B5 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_187_equalityD2,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( A3 = B5 )
% 2.97/1.54 => ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_188_equalityD1,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( A3 = B5 )
% 2.97/1.54 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_189_subset__eq,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) )
% 2.97/1.54 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.97/1.54 ! [X4: A] :
% 2.97/1.54 ( ( member @ A @ X4 @ A6 )
% 2.97/1.54 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_190_equalityE,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 2.97/1.54 ( ( A3 = B5 )
% 2.97/1.54 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_191_subsetCE,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( member @ A @ C2 @ A3 )
% 2.97/1.54 => ( member @ A @ C2 @ B5 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_192_subsetD,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( member @ A @ C2 @ A3 )
% 2.97/1.54 => ( member @ A @ C2 @ B5 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_193_in__mono,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A,X3: A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( member @ A @ X3 @ A3 )
% 2.97/1.54 => ( member @ A @ X3 @ B5 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_194_set__mp,axiom,
% 2.97/1.54 ! [A: $tType,A3: set @ A,B5: set @ A,X3: A] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 2.97/1.54 => ( ( member @ A @ X3 @ A3 )
% 2.97/1.54 => ( member @ A @ X3 @ B5 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_195_distinct__union,axiom,
% 2.97/1.54 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.97/1.54 ( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
% 2.97/1.54 = ( distinct @ A @ Ys ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_196_top_Oordering__top__axioms,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( order_top @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_197_not__in__set__insert,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,Xs: list @ A] :
% 2.97/1.54 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.97/1.54 => ( ( insert @ A @ X3 @ Xs )
% 2.97/1.54 = ( cons @ A @ X3 @ Xs ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_198_in__set__insert,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,Xs: list @ A] :
% 2.97/1.54 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.97/1.54 => ( ( insert @ A @ X3 @ Xs )
% 2.97/1.54 = Xs ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_199_distinct__insert,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,Xs: list @ A] :
% 2.97/1.54 ( ( distinct @ A @ ( insert @ A @ X3 @ Xs ) )
% 2.97/1.54 = ( distinct @ A @ Xs ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_200_ordering__top_Oextremum__uniqueI,axiom,
% 2.97/1.54 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 2.97/1.54 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 2.97/1.54 => ( ( Less_eq2 @ Top @ A2 )
% 2.97/1.54 => ( A2 = Top ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_201_ordering__top_Onot__eq__extremum,axiom,
% 2.97/1.54 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 2.97/1.54 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 2.97/1.54 => ( ( A2 != Top )
% 2.97/1.54 = ( Less @ A2 @ Top ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_202_ordering__top_Oextremum__unique,axiom,
% 2.97/1.54 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 2.97/1.54 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 2.97/1.54 => ( ( Less_eq2 @ Top @ A2 )
% 2.97/1.54 = ( A2 = Top ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_203_ordering__top_Oextremum__strict,axiom,
% 2.97/1.54 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 2.97/1.54 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 2.97/1.54 => ~ ( Less @ Top @ A2 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_204_ordering__top_Oextremum,axiom,
% 2.97/1.54 ! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
% 2.97/1.54 ( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
% 2.97/1.54 => ( Less_eq2 @ A2 @ Top ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_205_List_Oinsert__def,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( insert @ A )
% 2.97/1.54 = ( ^ [X4: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) ) @ Xs2 @ ( cons @ A @ X4 @ Xs2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_206_sorted_Oinducts,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: list @ A,P: ( list @ A ) > $o] :
% 2.97/1.54 ( ( linorder_sorted @ A @ X3 )
% 2.97/1.54 => ( ( P @ ( nil @ A ) )
% 2.97/1.54 => ( ! [Xs3: list @ A,X2: A] :
% 2.97/1.54 ( ! [Xa: A] :
% 2.97/1.54 ( ( member @ A @ Xa @ ( set2 @ A @ Xs3 ) )
% 2.97/1.54 => ( ord_less_eq @ A @ X2 @ Xa ) )
% 2.97/1.54 => ( ( linorder_sorted @ A @ Xs3 )
% 2.97/1.54 => ( ( P @ Xs3 )
% 2.97/1.54 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) ) )
% 2.97/1.54 => ( P @ X3 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_207_sorted_Osimps,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ( linorder_sorted @ A )
% 2.97/1.54 = ( ^ [A4: list @ A] :
% 2.97/1.54 ( ( A4
% 2.97/1.54 = ( nil @ A ) )
% 2.97/1.54 | ? [Xs2: list @ A,X4: A] :
% 2.97/1.54 ( ( A4
% 2.97/1.54 = ( cons @ A @ X4 @ Xs2 ) )
% 2.97/1.54 & ! [Y4: A] :
% 2.97/1.54 ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
% 2.97/1.54 => ( ord_less_eq @ A @ X4 @ Y4 ) )
% 2.97/1.54 & ( linorder_sorted @ A @ Xs2 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_208_sorted__single,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: A] : ( linorder_sorted @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_209_insert__Nil,axiom,
% 2.97/1.54 ! [A: $tType,X3: A] :
% 2.97/1.54 ( ( insert @ A @ X3 @ ( nil @ A ) )
% 2.97/1.54 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_210_inter__sorted_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.97/1.54 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 2.97/1.54 => ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
% 2.97/1.54 => ( ! [X1: A,L12: list @ A,X23: A,L22: list @ A] :
% 2.97/1.54 ( ( ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 => ( P @ L12 @ ( cons @ A @ X23 @ L22 ) ) )
% 2.97/1.54 => ( ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 => ( ( X1 = X23 )
% 2.97/1.54 => ( P @ L12 @ L22 ) ) )
% 2.97/1.54 => ( ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 => ( ( X1 != X23 )
% 2.97/1.54 => ( P @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) )
% 2.97/1.54 => ( P @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) )
% 2.97/1.54 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_211_subset__sorted_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.97/1.54 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 2.97/1.54 => ( ! [X1: A,L12: list @ A] : ( P @ ( cons @ A @ X1 @ L12 ) @ ( nil @ A ) )
% 2.97/1.54 => ( ! [X1: A,L12: list @ A,X23: A,L22: list @ A] :
% 2.97/1.54 ( ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 => ( ( X1 = X23 )
% 2.97/1.54 => ( P @ L12 @ L22 ) ) )
% 2.97/1.54 => ( ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 => ( ( X1 != X23 )
% 2.97/1.54 => ( P @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) )
% 2.97/1.54 => ( P @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) )
% 2.97/1.54 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_212_distinct__singleton,axiom,
% 2.97/1.54 ! [A: $tType,X3: A] : ( distinct @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_213_sorted_ONil,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( linorder_sorted @ A @ ( nil @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_214_map__tailrec__rev_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
% 2.97/1.54 ( ! [F2: A > B,X1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X1 )
% 2.97/1.54 => ( ! [F2: A > B,A5: A,As: list @ A,Bs: list @ B] :
% 2.97/1.54 ( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A5 ) @ Bs ) )
% 2.97/1.54 => ( P @ F2 @ ( cons @ A @ A5 @ As ) @ Bs ) )
% 2.97/1.54 => ( P @ A0 @ A1 @ A22 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_215_list__nonempty__induct,axiom,
% 2.97/1.54 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 2.97/1.54 ( ( Xs
% 2.97/1.54 != ( nil @ A ) )
% 2.97/1.54 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.97/1.54 => ( ! [X2: A,Xs3: list @ A] :
% 2.97/1.54 ( ( Xs3
% 2.97/1.54 != ( nil @ A ) )
% 2.97/1.54 => ( ( P @ Xs3 )
% 2.97/1.54 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) )
% 2.97/1.54 => ( P @ Xs ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_216_remdups__adj_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 2.97/1.54 ( ( P @ ( nil @ A ) )
% 2.97/1.54 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.97/1.54 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.97/1.54 ( ( ( X2 = Y2 )
% 2.97/1.54 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 2.97/1.54 => ( ( ( X2 != Y2 )
% 2.97/1.54 => ( P @ ( cons @ A @ Y2 @ Xs3 ) ) )
% 2.97/1.54 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
% 2.97/1.54 => ( P @ A0 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_217_remdups__adj_Ocases,axiom,
% 2.97/1.54 ! [A: $tType,X3: list @ A] :
% 2.97/1.54 ( ( X3
% 2.97/1.54 != ( nil @ A ) )
% 2.97/1.54 => ( ! [X2: A] :
% 2.97/1.54 ( X3
% 2.97/1.54 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.97/1.54 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.97/1.54 ( X3
% 2.97/1.54 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_218_transpose_Ocases,axiom,
% 2.97/1.54 ! [A: $tType,X3: list @ ( list @ A )] :
% 2.97/1.54 ( ( X3
% 2.97/1.54 != ( nil @ ( list @ A ) ) )
% 2.97/1.54 => ( ! [Xss: list @ ( list @ A )] :
% 2.97/1.54 ( X3
% 2.97/1.54 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 2.97/1.54 => ~ ! [X2: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
% 2.97/1.54 ( X3
% 2.97/1.54 != ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_219_splice_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.97/1.54 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 2.97/1.54 => ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
% 2.97/1.54 => ( ! [X2: A,Xs3: list @ A,Y2: A,Ys2: list @ A] :
% 2.97/1.54 ( ( P @ Xs3 @ Ys2 )
% 2.97/1.54 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
% 2.97/1.54 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_220_list__induct2_H,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
% 2.97/1.54 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 2.97/1.54 => ( ! [X2: A,Xs3: list @ A] : ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( nil @ B ) )
% 2.97/1.54 => ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys2 ) )
% 2.97/1.54 => ( ! [X2: A,Xs3: list @ A,Y2: B,Ys2: list @ B] :
% 2.97/1.54 ( ( P @ Xs3 @ Ys2 )
% 2.97/1.54 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys2 ) ) )
% 2.97/1.54 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_221_neq__Nil__conv,axiom,
% 2.97/1.54 ! [A: $tType,Xs: list @ A] :
% 2.97/1.54 ( ( Xs
% 2.97/1.54 != ( nil @ A ) )
% 2.97/1.54 = ( ? [Y4: A,Ys3: list @ A] :
% 2.97/1.54 ( Xs
% 2.97/1.54 = ( cons @ A @ Y4 @ Ys3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_222_list_Oinducts,axiom,
% 2.97/1.54 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 2.97/1.54 ( ( P @ ( nil @ A ) )
% 2.97/1.54 => ( ! [X1: A,X23: list @ A] :
% 2.97/1.54 ( ( P @ X23 )
% 2.97/1.54 => ( P @ ( cons @ A @ X1 @ X23 ) ) )
% 2.97/1.54 => ( P @ List ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_223_list_Oexhaust,axiom,
% 2.97/1.54 ! [A: $tType,Y: list @ A] :
% 2.97/1.54 ( ( Y
% 2.97/1.54 != ( nil @ A ) )
% 2.97/1.54 => ~ ! [X212: A,X223: list @ A] :
% 2.97/1.54 ( Y
% 2.97/1.54 != ( cons @ A @ X212 @ X223 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_224_list_OdiscI,axiom,
% 2.97/1.54 ! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
% 2.97/1.54 ( ( List
% 2.97/1.54 = ( cons @ A @ X21 @ X222 ) )
% 2.97/1.54 => ( List
% 2.97/1.54 != ( nil @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_225_distinct_Osimps_I1_J,axiom,
% 2.97/1.54 ! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_226_list_Odistinct_I1_J,axiom,
% 2.97/1.54 ! [A: $tType,X21: A,X222: list @ A] :
% 2.97/1.54 ( ( nil @ A )
% 2.97/1.54 != ( cons @ A @ X21 @ X222 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_227_subset__sorted_Osimps_I1_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [L2: list @ A] : ( sorted1061247458sorted @ A @ ( nil @ A ) @ L2 ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_228_subset__sorted_Osimps_I2_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X13: A,L1: list @ A] :
% 2.97/1.54 ~ ( sorted1061247458sorted @ A @ ( cons @ A @ X13 @ L1 ) @ ( nil @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_229_subset__sorted_Oelims_I3_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: list @ A,Xa2: list @ A] :
% 2.97/1.54 ( ~ ( sorted1061247458sorted @ A @ X3 @ Xa2 )
% 2.97/1.54 => ( ( ? [X1: A,L12: list @ A] :
% 2.97/1.54 ( X3
% 2.97/1.54 = ( cons @ A @ X1 @ L12 ) )
% 2.97/1.54 => ( Xa2
% 2.97/1.54 != ( nil @ A ) ) )
% 2.97/1.54 => ~ ! [X1: A,L12: list @ A] :
% 2.97/1.54 ( ( X3
% 2.97/1.54 = ( cons @ A @ X1 @ L12 ) )
% 2.97/1.54 => ! [X23: A,L22: list @ A] :
% 2.97/1.54 ( ( Xa2
% 2.97/1.54 = ( cons @ A @ X23 @ L22 ) )
% 2.97/1.54 => ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 & ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 => ( ( ( X1 = X23 )
% 2.97/1.54 => ( sorted1061247458sorted @ A @ L12 @ L22 ) )
% 2.97/1.54 & ( ( X1 != X23 )
% 2.97/1.54 => ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_230_subset__sorted_Oelims_I2_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: list @ A,Xa2: list @ A] :
% 2.97/1.54 ( ( sorted1061247458sorted @ A @ X3 @ Xa2 )
% 2.97/1.54 => ( ( ( X3
% 2.97/1.54 = ( nil @ A ) )
% 2.97/1.54 => ! [L22: list @ A] : ( Xa2 != L22 ) )
% 2.97/1.54 => ~ ! [X1: A,L12: list @ A] :
% 2.97/1.54 ( ( X3
% 2.97/1.54 = ( cons @ A @ X1 @ L12 ) )
% 2.97/1.54 => ! [X23: A,L22: list @ A] :
% 2.97/1.54 ( ( Xa2
% 2.97/1.54 = ( cons @ A @ X23 @ L22 ) )
% 2.97/1.54 => ~ ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 & ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 => ( ( ( X1 = X23 )
% 2.97/1.54 => ( sorted1061247458sorted @ A @ L12 @ L22 ) )
% 2.97/1.54 & ( ( X1 != X23 )
% 2.97/1.54 => ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_231_subset__sorted_Oelims_I1_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: list @ A,Xa2: list @ A,Y: $o] :
% 2.97/1.54 ( ( ( sorted1061247458sorted @ A @ X3 @ Xa2 )
% 2.97/1.54 = Y )
% 2.97/1.54 => ( ( ( X3
% 2.97/1.54 = ( nil @ A ) )
% 2.97/1.54 => ~ Y )
% 2.97/1.54 => ( ( ? [X1: A,L12: list @ A] :
% 2.97/1.54 ( X3
% 2.97/1.54 = ( cons @ A @ X1 @ L12 ) )
% 2.97/1.54 => ( ( Xa2
% 2.97/1.54 = ( nil @ A ) )
% 2.97/1.54 => Y ) )
% 2.97/1.54 => ~ ! [X1: A,L12: list @ A] :
% 2.97/1.54 ( ( X3
% 2.97/1.54 = ( cons @ A @ X1 @ L12 ) )
% 2.97/1.54 => ! [X23: A,L22: list @ A] :
% 2.97/1.54 ( ( Xa2
% 2.97/1.54 = ( cons @ A @ X23 @ L22 ) )
% 2.97/1.54 => ( Y
% 2.97/1.54 = ( ~ ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 & ( ~ ( ord_less @ A @ X1 @ X23 )
% 2.97/1.54 => ( ( ( X1 = X23 )
% 2.97/1.54 => ( sorted1061247458sorted @ A @ L12 @ L22 ) )
% 2.97/1.54 & ( ( X1 != X23 )
% 2.97/1.54 => ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_232_sorted_Ocases,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: list @ A] :
% 2.97/1.54 ( ( linorder_sorted @ A @ A2 )
% 2.97/1.54 => ( ( A2
% 2.97/1.54 != ( nil @ A ) )
% 2.97/1.54 => ~ ! [Xs3: list @ A,X2: A] :
% 2.97/1.54 ( ( A2
% 2.97/1.54 = ( cons @ A @ X2 @ Xs3 ) )
% 2.97/1.54 => ( ! [Xa: A] :
% 2.97/1.54 ( ( member @ A @ Xa @ ( set2 @ A @ Xs3 ) )
% 2.97/1.54 => ( ord_less_eq @ A @ X2 @ Xa ) )
% 2.97/1.54 => ~ ( linorder_sorted @ A @ Xs3 ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_233_the__elem__set,axiom,
% 2.97/1.54 ! [A: $tType,X3: A] :
% 2.97/1.54 ( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
% 2.97/1.54 = X3 ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_234_quicksort_Ocases,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [X3: list @ A] :
% 2.97/1.54 ( ( X3
% 2.97/1.54 != ( nil @ A ) )
% 2.97/1.54 => ~ ! [X2: A,Xs3: list @ A] :
% 2.97/1.54 ( X3
% 2.97/1.54 != ( cons @ A @ X2 @ Xs3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_235_sublists_Osimps_I1_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( sublists @ A @ ( nil @ A ) )
% 2.97/1.54 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_236_revg_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.97/1.54 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 2.97/1.54 => ( ! [A5: A,As: list @ A,B2: list @ A] :
% 2.97/1.54 ( ( P @ As @ ( cons @ A @ A5 @ B2 ) )
% 2.97/1.54 => ( P @ ( cons @ A @ A5 @ As ) @ B2 ) )
% 2.97/1.54 => ( P @ A0 @ A1 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_237_ord__eq__le__eq__trans,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( ord @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [A2: A,B3: A,C2: A,D3: A] :
% 2.97/1.54 ( ( A2 = B3 )
% 2.97/1.54 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.97/1.54 => ( ( C2 = D3 )
% 2.97/1.54 => ( ord_less_eq @ A @ A2 @ D3 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_238_list__tail__coinc,axiom,
% 2.97/1.54 ! [A: $tType,N1: A,R1: list @ A,N2: A,R2: list @ A] :
% 2.97/1.54 ( ( ( cons @ A @ N1 @ R1 )
% 2.97/1.54 = ( cons @ A @ N2 @ R2 ) )
% 2.97/1.54 => ( ( N1 = N2 )
% 2.97/1.54 & ( R1 = R2 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_239_subset__Collect__conv,axiom,
% 2.97/1.54 ! [A: $tType,S: set @ A,P: A > $o] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 2.97/1.54 = ( ! [X4: A] :
% 2.97/1.54 ( ( member @ A @ X4 @ S )
% 2.97/1.54 => ( P @ X4 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_240_mergesort__by__rel__merge__induct,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R3: A > B > $o,Xs: list @ A,Ys: list @ B] :
% 2.97/1.54 ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ B ) )
% 2.97/1.54 => ( ! [X1: list @ B] : ( P @ ( nil @ A ) @ X1 )
% 2.97/1.54 => ( ! [X2: A,Xs3: list @ A,Y2: B,Ys2: list @ B] :
% 2.97/1.54 ( ( R3 @ X2 @ Y2 )
% 2.97/1.54 => ( ( P @ Xs3 @ ( cons @ B @ Y2 @ Ys2 ) )
% 2.97/1.54 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
% 2.97/1.54 => ( ! [X2: A,Xs3: list @ A,Y2: B,Ys2: list @ B] :
% 2.97/1.54 ( ~ ( R3 @ X2 @ Y2 )
% 2.97/1.54 => ( ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys2 )
% 2.97/1.54 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
% 2.97/1.54 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_241_mergesort__by__rel__merge_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A,A22: list @ A] :
% 2.97/1.54 ( ! [R4: A > A > $o,X2: A,Xs3: list @ A,Y2: A,Ys2: list @ A] :
% 2.97/1.54 ( ( ( R4 @ X2 @ Y2 )
% 2.97/1.54 => ( P @ R4 @ Xs3 @ ( cons @ A @ Y2 @ Ys2 ) ) )
% 2.97/1.54 => ( ( ~ ( R4 @ X2 @ Y2 )
% 2.97/1.54 => ( P @ R4 @ ( cons @ A @ X2 @ Xs3 ) @ Ys2 ) )
% 2.97/1.54 => ( P @ R4 @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) )
% 2.97/1.54 => ( ! [R4: A > A > $o,Xs3: list @ A] : ( P @ R4 @ Xs3 @ ( nil @ A ) )
% 2.97/1.54 => ( ! [R4: A > A > $o,V: A,Va: list @ A] : ( P @ R4 @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
% 2.97/1.54 => ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_242_list__all__zip_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType,P: ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > $o,A1: list @ A,A22: list @ B] :
% 2.97/1.54 ( ! [P3: A > B > $o] : ( P @ P3 @ ( nil @ A ) @ ( nil @ B ) )
% 2.97/1.54 => ( ! [P3: A > B > $o,A5: A,As: list @ A,B2: B,Bs: list @ B] :
% 2.97/1.54 ( ( P @ P3 @ As @ Bs )
% 2.97/1.54 => ( P @ P3 @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B2 @ Bs ) ) )
% 2.97/1.54 => ( ! [P3: A > B > $o,V: A,Va: list @ A] : ( P @ P3 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
% 2.97/1.54 => ( ! [P3: A > B > $o,V: B,Va: list @ B] : ( P @ P3 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
% 2.97/1.54 => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_243_list__induct__first2,axiom,
% 2.97/1.54 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.97/1.54 ( ( P @ ( nil @ A ) )
% 2.97/1.54 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.97/1.54 => ( ! [X1: A,X23: A,Xs3: list @ A] :
% 2.97/1.54 ( ( P @ Xs3 )
% 2.97/1.54 => ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs3 ) ) ) )
% 2.97/1.54 => ( P @ Xs ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_244_list__2pre__induct,axiom,
% 2.97/1.54 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W2: list @ B] :
% 2.97/1.54 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 2.97/1.54 => ( ! [E2: A,W12: list @ A,W22: list @ B] :
% 2.97/1.54 ( ( P @ W12 @ W22 )
% 2.97/1.54 => ( P @ ( cons @ A @ E2 @ W12 ) @ W22 ) )
% 2.97/1.54 => ( ! [E2: B,W13: list @ A,W23: list @ B] :
% 2.97/1.54 ( ( P @ W13 @ W23 )
% 2.97/1.54 => ( P @ W13 @ ( cons @ B @ E2 @ W23 ) ) )
% 2.97/1.54 => ( P @ W1 @ W2 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_245_zipf_Oinduct,axiom,
% 2.97/1.54 ! [A: $tType,C: $tType,B: $tType,P: ( A > B > C ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > C,A1: list @ A,A22: list @ B] :
% 2.97/1.54 ( ! [F2: A > B > C] : ( P @ F2 @ ( nil @ A ) @ ( nil @ B ) )
% 2.97/1.54 => ( ! [F2: A > B > C,A5: A,As: list @ A,B2: B,Bs: list @ B] :
% 2.97/1.54 ( ( P @ F2 @ As @ Bs )
% 2.97/1.54 => ( P @ F2 @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B2 @ Bs ) ) )
% 2.97/1.54 => ( ! [A5: A > B > C,V: A,Va: list @ A] : ( P @ A5 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
% 2.97/1.54 => ( ! [A5: A > B > C,V: B,Va: list @ B] : ( P @ A5 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
% 2.97/1.54 => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_246_mergesort__remdups__correct,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [L: list @ A] :
% 2.97/1.54 ( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
% 2.97/1.54 & ( linorder_sorted @ A @ ( mergesort_remdups @ A @ L ) )
% 2.97/1.54 & ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
% 2.97/1.54 = ( set2 @ A @ L ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_247_product__lists_Osimps_I1_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
% 2.97/1.54 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_248_distinct__product__lists,axiom,
% 2.97/1.54 ! [A: $tType,Xss2: list @ ( list @ A )] :
% 2.97/1.54 ( ! [X2: list @ A] :
% 2.97/1.54 ( ( member @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xss2 ) )
% 2.97/1.54 => ( distinct @ A @ X2 ) )
% 2.97/1.54 => ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_249_subset__code_I3_J,axiom,
% 2.97/1.54 ! [C: $tType] :
% 2.97/1.54 ~ ( ord_less_eq @ ( set @ C ) @ ( coset @ C @ ( nil @ C ) ) @ ( set2 @ C @ ( nil @ C ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_250_combinatorial__product_Osimps_I1_J,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( list_c659805718roduct @ A @ ( nil @ ( list @ A ) ) )
% 2.97/1.54 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_251_UNIV__coset,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( top_top @ ( set @ A ) )
% 2.97/1.54 = ( coset @ A @ ( nil @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_252_subset__code_I2_J,axiom,
% 2.97/1.54 ! [B: $tType,A3: set @ B,Ys: list @ B] :
% 2.97/1.54 ( ( ord_less_eq @ ( set @ B ) @ A3 @ ( coset @ B @ Ys ) )
% 2.97/1.54 = ( ! [X4: B] :
% 2.97/1.54 ( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
% 2.97/1.54 => ~ ( member @ B @ X4 @ A3 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_253_remove__code_I2_J,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,Xs: list @ A] :
% 2.97/1.54 ( ( remove @ A @ X3 @ ( coset @ A @ Xs ) )
% 2.97/1.54 = ( coset @ A @ ( insert @ A @ X3 @ Xs ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(fact_254_sorted__append__bigger,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ! [Xs: list @ A,Y: A] :
% 2.97/1.54 ( ( linorder_sorted @ A @ Xs )
% 2.97/1.54 => ( ! [X2: A] :
% 2.97/1.54 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 2.97/1.54 => ( ord_less_eq @ A @ X2 @ Y ) )
% 2.97/1.54 => ( linorder_sorted @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( type @ A @ ( type2 @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( ord @ A @ ( type2 @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( order @ A @ ( type2 @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
% 2.97/1.54 ! [A: $tType] :
% 2.97/1.54 ( ( linorder @ A @ ( type2 @ A ) )
% 2.97/1.54 => ( preorder @ A @ ( type2 @ A ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.97/1.54 ! [A7: $tType,A8: $tType] :
% 2.97/1.54 ( ( order_top @ A8 @ ( type2 @ A8 ) )
% 2.97/1.54 => ( order_top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.97/1.54 ! [A7: $tType,A8: $tType] :
% 2.97/1.54 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.97/1.54 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.97/1.54 ! [A7: $tType,A8: $tType] :
% 2.97/1.54 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.97/1.54 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_fun___Orderings_Otop,axiom,
% 2.97/1.54 ! [A7: $tType,A8: $tType] :
% 2.97/1.54 ( ( top @ A8 @ ( type2 @ A8 ) )
% 2.97/1.54 => ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_fun___Orderings_Oord,axiom,
% 2.97/1.54 ! [A7: $tType,A8: $tType] :
% 2.97/1.54 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.97/1.54 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
% 2.97/1.54 ! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 2.97/1.54 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 2.97/1.54 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_Set_Oset___Orderings_Otop_4,axiom,
% 2.97/1.54 ! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
% 2.97/1.54 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
% 2.97/1.54 order_top @ $o @ ( type2 @ $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.97/1.54 preorder @ $o @ ( type2 @ $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.97/1.54 linorder @ $o @ ( type2 @ $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 2.97/1.54 order @ $o @ ( type2 @ $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_HOL_Obool___Orderings_Otop_9,axiom,
% 2.97/1.54 top @ $o @ ( type2 @ $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 2.97/1.54 ord @ $o @ ( type2 @ $o ) ).
% 2.97/1.54
% 2.97/1.54 thf(help_If_3_1_T,axiom,
% 2.97/1.54 ! [P: $o] :
% 2.97/1.54 ( ( P = $true )
% 2.97/1.54 | ( P = $false ) ) ).
% 2.97/1.54
% 2.97/1.54 thf(help_If_2_1_T,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,Y: A] :
% 2.97/1.54 ( ( if @ A @ $false @ X3 @ Y )
% 2.97/1.54 = Y ) ).
% 2.97/1.54
% 2.97/1.54 thf(help_If_1_1_T,axiom,
% 2.97/1.54 ! [A: $tType,X3: A,Y: A] :
% 2.97/1.54 ( ( if @ A @ $true @ X3 @ Y )
% 2.97/1.54 = X3 ) ).
% 2.97/1.54
% 2.97/1.54 thf(tfree_0,hypothesis,
% 2.97/1.54 linorder @ a @ ( type2 @ a ) ).
% 2.97/1.54
% 2.97/1.54 thf(conj_0,conjecture,
% 2.97/1.54 ord_less @ a @ x2 @ x1 ).
% 2.97/1.54 ------------------------
% 2.97/1.57 % (19343)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_19011 for (2999ds/2Mi)
% 2.97/1.57 % (19345)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19011 for (2999ds/18Mi)
% 2.97/1.57 % (19343)Instruction limit reached!
% 2.97/1.57 % (19343)------------------------------
% 2.97/1.57 % (19343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.57 % (19343)Termination reason: Unknown
% 2.97/1.57 % (19343)Termination phase: shuffling
% 2.97/1.57
% 2.97/1.57 % (19343)Memory used [KB]: 1407
% 2.97/1.57 % (19343)Time elapsed: 0.003 s
% 2.97/1.57 % (19343)Instructions burned: 5 (million)
% 2.97/1.57 % (19343)------------------------------
% 2.97/1.57 % (19343)------------------------------
% 2.97/1.57 % (19345)Instruction limit reached!
% 2.97/1.57 % (19345)------------------------------
% 2.97/1.57 % (19345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.57 % (19345)Termination reason: Unknown
% 2.97/1.57 % (19345)Termination phase: shuffling
% 2.97/1.57
% 2.97/1.57 % (19345)Memory used [KB]: 1791
% 2.97/1.57 % (19345)Time elapsed: 0.008 s
% 2.97/1.57 % (19345)Instructions burned: 19 (million)
% 2.97/1.57 % (19345)------------------------------
% 2.97/1.57 % (19345)------------------------------
% 2.97/1.57 % (19339)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19011 for (2999ds/183Mi)
% 2.97/1.57 % (19340)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_19011 for (2999ds/4Mi)
% 2.97/1.57 % (19341)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_19011 for (2999ds/27Mi)
% 2.97/1.57 % (19342)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19011 for (2999ds/2Mi)
% 2.97/1.57 % (19344)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_19011 for (2999ds/275Mi)
% 2.97/1.57 % (19342)Instruction limit reached!
% 2.97/1.57 % (19342)------------------------------
% 2.97/1.57 % (19342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.57 % (19342)Termination reason: Unknown
% 2.97/1.57 % (19342)Termination phase: shuffling
% 2.97/1.57
% 2.97/1.57 % (19342)Memory used [KB]: 1407
% 2.97/1.57 % (19342)Time elapsed: 0.004 s
% 2.97/1.57 % (19342)Instructions burned: 3 (million)
% 2.97/1.57 % (19342)------------------------------
% 2.97/1.57 % (19342)------------------------------
% 2.97/1.58 % (19340)Instruction limit reached!
% 2.97/1.58 % (19340)------------------------------
% 2.97/1.58 % (19340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.58 % (19340)Termination reason: Unknown
% 2.97/1.58 % (19340)Termination phase: shuffling
% 2.97/1.58
% 2.97/1.58 % (19340)Memory used [KB]: 1407
% 2.97/1.58 % (19340)Time elapsed: 0.005 s
% 2.97/1.58 % (19340)Instructions burned: 5 (million)
% 2.97/1.58 % (19340)------------------------------
% 2.97/1.58 % (19340)------------------------------
% 2.97/1.58 % (19346)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19011 for (2999ds/3Mi)
% 2.97/1.58 % (19346)Instruction limit reached!
% 2.97/1.58 % (19346)------------------------------
% 2.97/1.58 % (19346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.58 % (19346)Termination reason: Unknown
% 2.97/1.58 % (19346)Termination phase: shuffling
% 2.97/1.58
% 2.97/1.58 % (19346)Memory used [KB]: 1407
% 2.97/1.58 % (19346)Time elapsed: 0.002 s
% 2.97/1.58 % (19346)Instructions burned: 3 (million)
% 2.97/1.58 % (19346)------------------------------
% 2.97/1.58 % (19346)------------------------------
% 2.97/1.58 % (19347)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_19011 for (2999ds/37Mi)
% 2.97/1.59 % (19350)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_19011 for (2999ds/1041Mi)
% 2.97/1.59 % (19341)Instruction limit reached!
% 2.97/1.59 % (19341)------------------------------
% 2.97/1.59 % (19341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.59 % (19341)Termination reason: Unknown
% 2.97/1.59 % (19341)Termination phase: Property scanning
% 2.97/1.59
% 2.97/1.59 % (19341)Memory used [KB]: 1918
% 2.97/1.59 % (19341)Time elapsed: 0.019 s
% 2.97/1.59 % (19341)Instructions burned: 27 (million)
% 2.97/1.59 % (19341)------------------------------
% 2.97/1.59 % (19341)------------------------------
% 2.97/1.59 % (19347)Instruction limit reached!
% 2.97/1.59 % (19347)------------------------------
% 2.97/1.59 % (19347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.59 % (19347)Termination reason: Unknown
% 2.97/1.59 % (19347)Termination phase: shuffling
% 2.97/1.59
% 2.97/1.59 % (19347)Memory used [KB]: 2174
% 2.97/1.59 % (19347)Time elapsed: 0.012 s
% 2.97/1.59 % (19347)Instructions burned: 38 (million)
% 2.97/1.59 % (19347)------------------------------
% 2.97/1.59 % (19347)------------------------------
% 2.97/1.59 % (19348)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_19011 for (2999ds/15Mi)
% 2.97/1.59 % (19349)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_19011 for (2999ds/3Mi)
% 2.97/1.60 % (19349)Instruction limit reached!
% 2.97/1.60 % (19349)------------------------------
% 2.97/1.60 % (19349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.60 % (19349)Termination reason: Unknown
% 2.97/1.60 % (19349)Termination phase: shuffling
% 2.97/1.60
% 2.97/1.60 % (19349)Memory used [KB]: 1407
% 2.97/1.60 % (19349)Time elapsed: 0.004 s
% 2.97/1.60 % (19349)Instructions burned: 3 (million)
% 2.97/1.60 % (19349)------------------------------
% 2.97/1.60 % (19349)------------------------------
% 3.02/1.60 % (19352)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_19011 for (2999ds/16Mi)
% 3.02/1.60 % (19348)Instruction limit reached!
% 3.02/1.60 % (19348)------------------------------
% 3.02/1.60 % (19348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.60 % (19348)Termination reason: Unknown
% 3.02/1.60 % (19348)Termination phase: shuffling
% 3.02/1.60
% 3.02/1.60 % (19348)Memory used [KB]: 1663
% 3.02/1.60 % (19348)Time elapsed: 0.012 s
% 3.02/1.60 % (19348)Instructions burned: 16 (million)
% 3.02/1.60 % (19348)------------------------------
% 3.02/1.60 % (19348)------------------------------
% 3.02/1.60 % (19351)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_19011 for (2999ds/7Mi)
% 3.02/1.61 % (19351)Instruction limit reached!
% 3.02/1.61 % (19351)------------------------------
% 3.02/1.61 % (19351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.61 % (19351)Termination reason: Unknown
% 3.02/1.61 % (19351)Termination phase: shuffling
% 3.02/1.61
% 3.02/1.61 % (19351)Memory used [KB]: 1535
% 3.02/1.61 % (19351)Time elapsed: 0.004 s
% 3.02/1.61 % (19351)Instructions burned: 7 (million)
% 3.02/1.61 % (19351)------------------------------
% 3.02/1.61 % (19351)------------------------------
% 3.02/1.61 % (19352)Instruction limit reached!
% 3.02/1.61 % (19352)------------------------------
% 3.02/1.61 % (19352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.61 % (19352)Termination reason: Unknown
% 3.02/1.61 % (19352)Termination phase: shuffling
% 3.02/1.61
% 3.02/1.61 % (19352)Memory used [KB]: 1791
% 3.02/1.61 % (19352)Time elapsed: 0.007 s
% 3.02/1.61 % (19352)Instructions burned: 17 (million)
% 3.02/1.61 % (19352)------------------------------
% 3.02/1.61 % (19352)------------------------------
% 3.02/1.61 % (19353)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_19011 for (2999ds/3Mi)
% 3.02/1.61 % (19353)Instruction limit reached!
% 3.02/1.61 % (19353)------------------------------
% 3.02/1.61 % (19353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.61 % (19353)Termination reason: Unknown
% 3.02/1.61 % (19353)Termination phase: shuffling
% 3.02/1.61
% 3.02/1.61 % (19353)Memory used [KB]: 1407
% 3.02/1.61 % (19353)Time elapsed: 0.003 s
% 3.02/1.61 % (19353)Instructions burned: 3 (million)
% 3.02/1.61 % (19353)------------------------------
% 3.02/1.61 % (19353)------------------------------
% 3.02/1.61 % (19344)First to succeed.
% 3.02/1.62 % (19355)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_19011 for (2999ds/7Mi)
% 3.02/1.62 % (19344)Refutation found. Thanks to Tanya!
% 3.02/1.62 % SZS status Theorem for DTF2THF_19011
% 3.02/1.62 % SZS output start Proof for DTF2THF_19011
% 3.02/1.62 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.02/1.62 thf(type_def_7, type, list: ($tType) > $tType).
% 3.02/1.62 thf(type_def_8, type, set: ($tType) > $tType).
% 3.02/1.62 thf(type_def_9, type, a: $tType).
% 3.02/1.62 thf(func_def_0, type, list: ($tType) > $tType).
% 3.02/1.62 thf(func_def_1, type, set: ($tType) > $tType).
% 3.02/1.62 thf(func_def_2, type, itself: ($tType) > $tType).
% 3.02/1.62 thf(func_def_3, type, a: $tType).
% 3.02/1.62 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_5, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_6, type, top: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_7, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_8, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_9, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_10, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_11, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_12, type, order_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_13, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_14, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_15, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_16, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_17, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_18, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_19, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.02/1.62 thf(func_def_20, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.02/1.62 thf(func_def_21, type, append: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 3.02/1.62 thf(func_def_22, type, coset: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 3.02/1.62 thf(func_def_23, type, distinct: !>[X0: $tType]:(list @ X0 > $o)).
% 3.02/1.62 thf(func_def_24, type, insert: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 3.02/1.62 thf(func_def_25, type, linorder_sorted: !>[X0: $tType]:(list @ X0 > $o)).
% 3.02/1.62 thf(func_def_26, type, cons: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 3.02/1.62 thf(func_def_27, type, nil: !>[X0: $tType]:(list @ X0)).
% 3.02/1.62 thf(func_def_28, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 3.02/1.62 thf(func_def_29, type, product_lists: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 3.02/1.62 thf(func_def_30, type, sublists: !>[X0: $tType]:(list @ X0 > list @ list @ X0)).
% 3.02/1.62 thf(func_def_31, type, union: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 3.02/1.62 thf(func_def_32, type, list_c659805718roduct: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 3.02/1.62 thf(func_def_33, type, mergesort_remdups: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 3.02/1.62 thf(func_def_34, type, max: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 3.02/1.62 thf(func_def_35, type, min: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 3.02/1.62 thf(func_def_36, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.02/1.62 thf(func_def_37, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.02/1.62 thf(func_def_38, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 3.02/1.62 thf(func_def_39, type, ordering_top: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0 > $o)).
% 3.02/1.62 thf(func_def_40, type, top_top: !>[X0: $tType]:(X0)).
% 3.02/1.62 thf(func_def_41, type, type2: !>[X0: $tType]:(itself @ X0)).
% 3.02/1.62 thf(func_def_42, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.02/1.62 thf(func_def_43, type, remove: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.02/1.62 thf(func_def_44, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 3.02/1.62 thf(func_def_45, type, sorted1061247458sorted: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 3.02/1.62 thf(func_def_46, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.02/1.62 thf(func_def_47, type, l1: list @ a).
% 3.02/1.62 thf(func_def_48, type, l1a: list @ a).
% 3.02/1.62 thf(func_def_49, type, l2: list @ a).
% 3.02/1.62 thf(func_def_50, type, l2b: list @ a).
% 3.02/1.62 thf(func_def_51, type, x1: a).
% 3.02/1.62 thf(func_def_52, type, x2: a).
% 3.02/1.62 thf(func_def_67, type, sP0: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.02/1.62 thf(func_def_68, type, sP1: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.02/1.62 thf(func_def_69, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.02/1.62 thf(func_def_70, type, sP3: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 3.02/1.62 thf(func_def_71, type, sP4: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 3.02/1.62 thf(func_def_72, type, sP5: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 3.02/1.62 thf(func_def_73, type, sP6: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 3.02/1.62 thf(func_def_74, type, sK7: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_75, type, sK8: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_76, type, sK9: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_77, type, sK10: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_78, type, sK11: !>[X0: $tType]:(set @ X0 > list @ X0 > X0)).
% 3.02/1.62 thf(func_def_79, type, sK12: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_80, type, sK13: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_81, type, sK14: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_82, type, sK15: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.02/1.62 thf(func_def_83, type, sK16: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_84, type, sK17: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_85, type, sK18: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_86, type, sK19: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.02/1.62 thf(func_def_87, type, sK20: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_88, type, sK21: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_89, type, sK22: !>[X0: $tType]:(X0 > list @ X0 > X0)).
% 3.02/1.62 thf(func_def_90, type, sK23: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_91, type, sK24: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_92, type, sK25: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_93, type, sK26: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_94, type, sK27: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_95, type, sK28: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_96, type, sK29: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_97, type, sK30: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_98, type, sK31: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 3.02/1.62 thf(func_def_99, type, sK32: !>[X0: $tType]:(list @ X0 > X0 > X0)).
% 3.02/1.62 thf(func_def_100, type, sK33: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 3.02/1.62 thf(func_def_101, type, sK34: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_102, type, sK35: !>[X0: $tType]:(X0 > X0)).
% 3.02/1.62 thf(func_def_103, type, sK36: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_104, type, sK37: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_105, type, sK38: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_106, type, sK39: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 3.02/1.62 thf(func_def_107, type, sK40: !>[X0: $tType]:(list @ X0 > X0)).
% 3.02/1.62 thf(func_def_108, type, sK41: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 3.02/1.62 thf(func_def_109, type, sK42: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_110, type, sK43: !>[X0: $tType]:(list @ X0 > X0 > X0)).
% 3.02/1.62 thf(func_def_111, type, sK44: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_112, type, sK45: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_113, type, sK46: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 3.02/1.62 thf(func_def_115, type, ph48: !>[X0: $tType]:(X0)).
% 3.02/1.62 thf(f1257,plain,(
% 3.02/1.62 $false),
% 3.02/1.62 inference(subsumption_resolution,[status(thm)],[f1256,f1114])).
% 3.02/1.62 thf(f1114,plain,(
% 3.02/1.62 (x1 != x2)),
% 3.02/1.62 inference(cnf_transformation,[status(thm)],[f1])).
% 3.02/1.62 thf(f1,axiom,(
% 3.02/1.62 (x1 != x2)),
% 3.02/1.62 file('/export/starexec/sandbox/tmp/tmp.TYv64KhOhS/DTF2THF_19011.p',fact_0_x1__neq__x2)).
% 3.02/1.62 thf(f1256,plain,(
% 3.02/1.62 (x1 = x2)),
% 3.02/1.62 inference(subsumption_resolution,[status(thm)],[f1255,f1086])).
% 3.02/1.62 thf(f1086,plain,(
% 3.02/1.62 ((ord_less @ a @ x1 @ x2) != $true)),
% 3.02/1.62 inference(cnf_transformation,[status(thm)],[f822])).
% 3.02/1.62 thf(f822,plain,(
% 3.02/1.62 ((ord_less @ a @ x1 @ x2) != $true)),
% 3.02/1.62 inference(flattening,[status(thm)],[f500])).
% 3.02/1.62 thf(f500,plain,(
% 3.02/1.62 ~((ord_less @ a @ x1 @ x2) = $true)),
% 3.02/1.62 inference(fool_elimination,[status(thm)],[f499])).
% 3.02/1.62 thf(f499,plain,(
% 3.02/1.62 ~(ord_less @ a @ x1 @ x2)),
% 3.02/1.62 inference(rectify,[status(thm)],[f2])).
% 3.02/1.62 thf(f2,axiom,(
% 3.02/1.62 ~(ord_less @ a @ x1 @ x2)),
% 3.02/1.62 file('/export/starexec/sandbox/tmp/tmp.TYv64KhOhS/DTF2THF_19011.p',fact_1_x2__le__x1)).
% 3.02/1.62 thf(f1255,plain,(
% 3.02/1.62 ((ord_less @ a @ x1 @ x2) = $true) | (x1 = x2)),
% 3.02/1.62 inference(subsumption_resolution,[status(thm)],[f1252,f1051])).
% 3.02/1.62 thf(f1051,plain,(
% 3.02/1.62 ((linorder @ a @ type2 @ a) = $true)),
% 3.02/1.62 inference(cnf_transformation,[status(thm)],[f606])).
% 3.02/1.62 thf(f606,plain,(
% 3.02/1.62 ((linorder @ a @ type2 @ a) = $true)),
% 3.02/1.62 inference(fool_elimination,[status(thm)],[f605])).
% 3.02/1.62 thf(f605,plain,(
% 3.02/1.62 (linorder @ a @ type2 @ a)),
% 3.02/1.62 inference(rectify,[status(thm)],[f279])).
% 3.02/1.62 thf(f279,axiom,(
% 3.02/1.62 (linorder @ a @ type2 @ a)),
% 3.02/1.62 file('/export/starexec/sandbox/tmp/tmp.TYv64KhOhS/DTF2THF_19011.p',tfree_0)).
% 3.02/1.62 thf(f1252,plain,(
% 3.02/1.62 ((linorder @ a @ type2 @ a) != $true) | ((ord_less @ a @ x1 @ x2) = $true) | (x1 = x2)),
% 3.02/1.62 inference(trivial_inequality_removal,[status(thm)],[f1238])).
% 3.02/1.62 thf(f1238,plain,(
% 3.02/1.62 (x1 = x2) | ((linorder @ a @ type2 @ a) != $true) | ($true != $true) | ((ord_less @ a @ x1 @ x2) = $true)),
% 3.02/1.62 inference(superposition,[status(thm)],[f1099,f1103])).
% 3.02/1.62 thf(f1103,plain,(
% 3.02/1.62 ( ! [X0 : $tType,X2 : X0,X1 : X0] : (($true = (ord_less @ X0 @ X1 @ X2)) | ((ord_less @ X0 @ X2 @ X1) = $true) | (X1 = X2) | ((linorder @ X0 @ type2 @ X0) != $true)) )),
% 3.02/1.62 inference(cnf_transformation,[status(thm)],[f967])).
% 3.02/1.62 thf(f967,plain,(
% 3.02/1.62 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) != $true) | ! [X1 : X0,X2 : X0] : (((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true) | ($true = (ord_less @ X0 @ X1 @ X2))) & (($true != (ord_less @ X0 @ X1 @ X2)) | ((X1 != X2) & ((ord_less @ X0 @ X2 @ X1) != $true)))))),
% 3.02/1.62 inference(flattening,[status(thm)],[f966])).
% 3.02/1.62 thf(f966,plain,(
% 3.02/1.62 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) != $true) | ! [X1 : X0,X2 : X0] : ((((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true)) | ($true = (ord_less @ X0 @ X1 @ X2))) & (($true != (ord_less @ X0 @ X1 @ X2)) | ((X1 != X2) & ((ord_less @ X0 @ X2 @ X1) != $true)))))),
% 3.02/1.62 inference(nnf_transformation,[status(thm)],[f833])).
% 3.02/1.62 thf(f833,plain,(
% 3.02/1.62 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) != $true) | ! [X1 : X0,X2 : X0] : (((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true)) <=> ($true != (ord_less @ X0 @ X1 @ X2))))),
% 3.02/1.62 inference(ennf_transformation,[status(thm)],[f812])).
% 3.02/1.62 thf(f812,plain,(
% 3.02/1.62 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) = $true) => ! [X1 : X0,X2 : X0] : (((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true)) <=> ($true != (ord_less @ X0 @ X1 @ X2))))),
% 3.02/1.62 inference(flattening,[status(thm)],[f486])).
% 3.02/1.62 thf(f486,plain,(
% 3.02/1.62 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) = $true) => ! [X1 : X0,X2 : X0] : (~($true = (ord_less @ X0 @ X1 @ X2)) <=> ((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true))))),
% 3.02/1.62 inference(fool_elimination,[status(thm)],[f485])).
% 3.02/1.62 thf(f485,plain,(
% 3.02/1.62 ! [X0 : $tType] : ((linorder @ X0 @ type2 @ X0) => ! [X1 : X0,X2 : X0] : (~(ord_less @ X0 @ X1 @ X2) = (ord_less @ X0 @ X2 @ X1) | (X1 = X2)))),
% 3.02/1.62 inference(rectify,[status(thm)],[f24])).
% 3.02/1.62 thf(f24,axiom,(
% 3.02/1.62 ! [X0 : $tType] : ((linorder @ X0 @ type2 @ X0) => ! [X16 : X0,X17 : X0] : (~(ord_less @ X0 @ X16 @ X17) = (ord_less @ X0 @ X17 @ X16) | (X16 = X17)))),
% 3.02/1.62 file('/export/starexec/sandbox/tmp/tmp.TYv64KhOhS/DTF2THF_19011.p',fact_23_not__less__iff__gr__or__eq)).
% 3.02/1.62 thf(f1099,plain,(
% 3.02/1.62 ((ord_less @ a @ x2 @ x1) != $true)),
% 3.02/1.62 inference(cnf_transformation,[status(thm)],[f801])).
% 3.02/1.62 thf(f801,plain,(
% 3.02/1.62 ((ord_less @ a @ x2 @ x1) != $true)),
% 3.02/1.62 inference(flattening,[status(thm)],[f780])).
% 3.02/1.62 thf(f780,plain,(
% 3.02/1.62 ~((ord_less @ a @ x2 @ x1) = $true)),
% 3.02/1.62 inference(fool_elimination,[status(thm)],[f779])).
% 3.02/1.62 thf(f779,plain,(
% 3.02/1.62 ~(ord_less @ a @ x2 @ x1)),
% 3.02/1.62 inference(rectify,[status(thm)],[f281])).
% 3.02/1.62 thf(f281,negated_conjecture,(
% 3.02/1.62 ~(ord_less @ a @ x2 @ x1)),
% 3.02/1.62 inference(negated_conjecture,[status(cth)],[f280])).
% 3.02/1.62 thf(f280,conjecture,(
% 3.02/1.62 (ord_less @ a @ x2 @ x1)),
% 3.02/1.62 file('/export/starexec/sandbox/tmp/tmp.TYv64KhOhS/DTF2THF_19011.p',conj_0)).
% 3.02/1.62 % SZS output end Proof for DTF2THF_19011
% 3.02/1.62 % (19344)------------------------------
% 3.02/1.62 % (19344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.62 % (19344)Termination reason: Refutation
% 3.02/1.62
% 3.02/1.62 % (19344)Memory used [KB]: 6780
% 3.02/1.62 % (19344)Time elapsed: 0.044 s
% 3.02/1.62 % (19344)Instructions burned: 78 (million)
% 3.02/1.62 % (19344)------------------------------
% 3.02/1.62 % (19344)------------------------------
% 3.02/1.62 % (19338)Success in time 0.065 s
%------------------------------------------------------------------------------