%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT208^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n021.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 08:45:09 AM UTC 2026
% Result : Theorem 2.95s 1.73s
% Output : Refutation 2.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : DAT208^1 : TPTP v9.2.1. Released v7.0.0.
% 0.11/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.32 % Computer : n021.cluster.edu
% 0.15/0.32 % Model : x86_64 x86_64
% 0.15/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.32 % Memory : 8042.1875MB
% 0.15/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.32 % CPULimit : 300
% 0.15/0.32 % WCLimit : 300
% 0.15/0.32 % DateTime : Tue Feb 24 06:10:53 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.44 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : DAT208^1 : TPTP v9.2.1. Released v7.0.0.
% 0.29/0.45 % Domain : Data Structures
% 0.29/0.45 % Problem : Sorted list operations 293
% 0.29/0.45 % Version : [Bla16] axioms : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [Lam09] Lammich (2009), Collections Framework
% 0.29/0.45 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.29/0.45 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Bla16]
% 0.29/0.45 % Names : sorted_list_operations__293.p [Bla16]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.33 v9.1.0, 0.00 v7.5.0, 0.67 v7.3.0, 1.00 v7.1.0
% 0.29/0.45 % Syntax : Number of formulae : 329 ( 46 unt; 45 typ; 0 def)
% 0.29/0.45 % Number of atoms : 984 ( 188 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 13 ( 3 avg)
% 0.29/0.45 % Number of connectives : 3964 ( 87 ~; 28 |; 70 &;3248 @)
% 0.29/0.45 % ( 0 <=>; 531 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 21 ( 9 avg)
% 0.29/0.45 % Number of types : 2 ( 1 usr)
% 0.29/0.45 % Number of type conns : 149 ( 149 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 46 ( 44 usr; 4 con; 0-4 aty)
% 0.29/0.45 % Number of variables : 1070 ( 64 ^; 927 !; 41 ?;1070 :)
% 0.29/0.45 % ( 38 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.45 % SPC : TH1_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.29/0.45 % 2016-07-13 14:41:38.793
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 %----Could-be-implicit typings (4)
% 0.29/0.45 thf(ty_t_List_Olist,type,
% 0.29/0.45 list: $tType > $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_t_Set_Oset,type,
% 0.29/0.45 set: $tType > $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_t_itself,type,
% 0.29/0.45 itself: $tType > $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_tf_a,type,
% 0.29/0.45 a: $tType ).
% 0.29/0.45
% 0.29/0.45 %----Explicit typings (41)
% 0.29/0.45 thf(sy_cl_HOL_Otype,type,
% 0.29/0.45 type:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.45 ord:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.45 order:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Lattices_Olattice,type,
% 0.29/0.45 lattice:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Ono__bot,type,
% 0.29/0.45 no_bot:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Ono__top,type,
% 0.29/0.45 no_top:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.29/0.45 finite_finite:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.45 linorder:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.45 preorder:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.29/0.45 wellorder:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Odense__order,type,
% 0.29/0.45 dense_order:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/0.45 linordered_idom:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Fields_Olinordered__field,type,
% 0.29/0.45 linordered_field:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.29/0.45 semilattice_sup:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.29/0.45 dense_linorder:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.29/0.45 condit1656338222tinuum:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.29/0.45 condit1037483654norder:
% 0.29/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite,type,
% 0.29/0.45 finite_finite2:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.29/0.45 sup_sup:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_List_Ocoset,type,
% 0.29/0.45 coset:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_List_Odistinct,type,
% 0.29/0.45 distinct:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_List_Olinorder__class_Osorted,type,
% 0.29/0.45 linorder_sorted:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
% 0.29/0.45 linord467138063of_set:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_List_Olist_Oset,type,
% 0.29/0.45 set2:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_List_Omember,type,
% 0.29/0.45 member:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_List_Ounion,type,
% 0.29/0.45 union:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Misc_Omerge,type,
% 0.29/0.45 merge:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Misc_Omergesort__by__rel__merge,type,
% 0.29/0.45 merges2133357844_merge:
% 0.29/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Misc_Omergesort__remdups,type,
% 0.29/0.45 mergesort_remdups:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Misc_Osorted__by__rel,type,
% 0.29/0.45 sorted_by_rel:
% 0.29/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.45 ord_less:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.45 ord_less_eq:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Pure_Otype,type,
% 0.29/0.45 type2:
% 0.29/0.45 !>[A: $tType] : ( itself @ A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_OCollect,type,
% 0.29/0.45 collect:
% 0.29/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Odelete__sorted,type,
% 0.29/0.45 sorted763967275sorted:
% 0.29/0.45 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Omemb__sorted,type,
% 0.29/0.45 sorted873716653sorted:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Osubset__sorted,type,
% 0.29/0.45 sorted1061247458sorted:
% 0.29/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member,type,
% 0.29/0.45 member2:
% 0.29/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_a,type,
% 0.29/0.45 a2: a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_x,type,
% 0.29/0.45 x: a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_xsa,type,
% 0.29/0.45 xsa: list @ a ).
% 0.29/0.45
% 0.29/0.45 %----Relevant facts (256)
% 0.29/0.45 thf(fact_0_set__eq__sorted__correct,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [L1: list @ A,L2: list @ A] :
% 0.29/0.45 ( ( ( distinct @ A @ L1 )
% 0.29/0.45 & ( linorder_sorted @ A @ L1 ) )
% 0.29/0.45 => ( ( ( distinct @ A @ L2 )
% 0.29/0.45 & ( linorder_sorted @ A @ L2 ) )
% 0.29/0.45 => ( ( L1 = L2 )
% 0.29/0.45 = ( ( set2 @ A @ L1 )
% 0.29/0.45 = ( set2 @ A @ L2 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_eq_sorted_correct
% 0.29/0.45 thf(fact_1_sorted__distinct__set__unique,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Xs: list @ A,Ys: list @ A] :
% 0.29/0.45 ( ( linorder_sorted @ A @ Xs )
% 0.29/0.45 => ( ( distinct @ A @ Xs )
% 0.29/0.45 => ( ( linorder_sorted @ A @ Ys )
% 0.29/0.45 => ( ( distinct @ A @ Ys )
% 0.29/0.45 => ( ( ( set2 @ A @ Xs )
% 0.29/0.45 = ( set2 @ A @ Ys ) )
% 0.29/0.45 => ( Xs = Ys ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sorted_distinct_set_unique
% 0.29/0.45 thf(fact_2_order__refl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_refl
% 0.29/0.45 thf(fact_3_minf_I8_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(8)
% 0.29/0.45 thf(fact_4_minf_I6_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.45 => ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(6)
% 0.29/0.45 thf(fact_5_pinf_I8_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.45 => ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(8)
% 0.29/0.45 thf(fact_6_pinf_I6_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(6)
% 0.29/0.45 thf(fact_7_leD,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % leD
% 0.29/0.45 thf(fact_8_leI,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % leI
% 0.29/0.45 thf(fact_9_le__less,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less_eq @ A )
% 0.29/0.45 = ( ^ [X3: A,Y2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.29/0.45 | ( X3 = Y2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_less
% 0.29/0.45 thf(fact_10_less__le,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less @ A )
% 0.29/0.45 = ( ^ [X3: A,Y2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.45 & ( X3 != Y2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_le
% 0.29/0.45 thf(fact_11_order__le__less__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.29/0.45 => ( ( ord_less @ B @ B2 @ C )
% 0.29/0.45 => ( ! [X4: B,Y3: B] :
% 0.29/0.45 ( ( ord_less @ B @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_le_less_subst1
% 0.29/0.45 thf(fact_12_order__le__less__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C2: $tType] :
% 0.29/0.45 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.29/0.45 => ( ! [X4: A,Y3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_le_less_subst2
% 0.29/0.45 thf(fact_13_subset__code_I1_J,axiom,
% 0.29/0.45 ! [A: $tType,Xs: list @ A,B3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 )
% 0.29/0.45 = ( ! [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.29/0.45 => ( member2 @ A @ X3 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_code(1)
% 0.29/0.45 thf(fact_14_dual__order_Oantisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.antisym
% 0.29/0.45 thf(fact_15_dual__order_Otrans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.29/0.45 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.trans
% 0.29/0.45 thf(fact_16_linorder__wlog,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.29/0.45 ( ! [A3: A,B4: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.29/0.45 => ( P @ A3 @ B4 ) )
% 0.29/0.45 => ( ! [A3: A,B4: A] :
% 0.29/0.45 ( ( P @ B4 @ A3 )
% 0.29/0.45 => ( P @ A3 @ B4 ) )
% 0.29/0.45 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_wlog
% 0.29/0.45 thf(fact_17_dual__order_Orefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.refl
% 0.29/0.45 thf(fact_18_order__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.29/0.45 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_trans
% 0.29/0.45 thf(fact_19_order__class_Oorder_Oantisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.45 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.antisym
% 0.29/0.45 thf(fact_20_ord__le__eq__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( B2 = C )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_trans
% 0.29/0.45 thf(fact_21_ord__eq__le__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A] :
% 0.29/0.45 ( ( A2 = B2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_trans
% 0.29/0.45 thf(fact_22_antisym__conv,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 = ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv
% 0.29/0.45 thf(fact_23_le__cases3,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.45 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.29/0.45 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.29/0.45 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.29/0.45 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.29/0.45 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.29/0.45 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases3
% 0.29/0.45 thf(fact_24_order_Otrans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.trans
% 0.29/0.45 thf(fact_25_le__cases,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases
% 0.29/0.45 thf(fact_26_eq__refl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X = Y )
% 0.29/0.45 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_refl
% 0.29/0.45 thf(fact_27_linear,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linear
% 0.29/0.45 thf(fact_28_antisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym
% 0.29/0.45 thf(fact_29_eq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.29/0.45 = ( ^ [X3: A,Y2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.45 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_iff
% 0.29/0.45 thf(fact_30_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ( F @ B2 )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X4: A,Y3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_31_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 = ( F @ B2 ) )
% 0.29/0.45 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.29/0.45 => ( ! [X4: B,Y3: B] :
% 0.29/0.45 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_32_order__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C2: $tType] :
% 0.29/0.45 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.29/0.45 => ( ! [X4: A,Y3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_33_order__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.29/0.45 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.29/0.45 => ( ! [X4: B,Y3: B] :
% 0.29/0.45 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_34_le__fun__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.45 = ( ^ [F2: A > B,G: A > B] :
% 0.29/0.45 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_fun_def
% 0.29/0.45 thf(fact_35_le__funI,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ! [F: A > B,G2: A > B] :
% 0.29/0.45 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.45 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funI
% 0.29/0.45 thf(fact_36_le__funE,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ! [F: A > B,G2: A > B,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funE
% 0.29/0.45 thf(fact_37_le__funD,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ! [F: A > B,G2: A > B,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funD
% 0.29/0.45 thf(fact_38_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.45 => ( A2 != B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_implies_not_eq
% 0.29/0.45 thf(fact_39_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( A2 != B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_implies_not_eq
% 0.29/0.45 thf(fact_40_not__less__iff__gr__or__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.45 = ( ( ord_less @ A @ Y @ X )
% 0.29/0.45 | ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_less_iff_gr_or_eq
% 0.29/0.45 thf(fact_41_dual__order_Ostrict__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.45 => ( ( ord_less @ A @ C @ B2 )
% 0.29/0.45 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_trans
% 0.29/0.45 thf(fact_42_less__imp__not__less,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_not_less
% 0.29/0.45 thf(fact_43_order_Ostrict__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less @ A @ B2 @ C )
% 0.29/0.45 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_trans
% 0.29/0.45 thf(fact_44_dual__order_Oirrefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.irrefl
% 0.29/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.29/0.45 ( ( member2 @ A @ A2 @ ( collect @ A @ P ) )
% 0.29/0.45 = ( P @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] :
% 0.29/0.45 ( ( collect @ A
% 0.29/0.45 @ ^ [X3: A] : ( member2 @ A @ X3 @ A4 ) )
% 0.29/0.45 = A4 ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_47_Collect__cong,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.45 ( ! [X4: A] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 = ( Q @ X4 ) )
% 0.29/0.45 => ( ( collect @ A @ P )
% 0.29/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_cong
% 0.29/0.45 thf(fact_48_ext,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F: A > B,G2: A > B] :
% 0.29/0.45 ( ! [X4: A] :
% 0.29/0.45 ( ( F @ X4 )
% 0.29/0.45 = ( G2 @ X4 ) )
% 0.29/0.45 => ( F = G2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % ext
% 0.29/0.45 thf(fact_49_linorder__cases,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ( X != Y )
% 0.29/0.45 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_cases
% 0.29/0.45 thf(fact_50_less__imp__triv,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,P: $o] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less @ A @ Y @ X )
% 0.29/0.45 => P ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_triv
% 0.29/0.45 thf(fact_51_less__imp__not__eq2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( Y != X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_not_eq2
% 0.29/0.45 thf(fact_52_antisym__conv3,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,X: A] :
% 0.29/0.45 ( ~ ( ord_less @ A @ Y @ X )
% 0.29/0.45 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.45 = ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv3
% 0.29/0.45 thf(fact_53_less__induct,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [P: A > $o,A2: A] :
% 0.29/0.45 ( ! [X4: A] :
% 0.29/0.45 ( ! [Y5: A] :
% 0.29/0.45 ( ( ord_less @ A @ Y5 @ X4 )
% 0.29/0.45 => ( P @ Y5 ) )
% 0.29/0.45 => ( P @ X4 ) )
% 0.29/0.45 => ( P @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_induct
% 0.29/0.45 thf(fact_54_less__not__sym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_not_sym
% 0.29/0.45 thf(fact_55_less__imp__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( X != Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_not_eq
% 0.29/0.45 thf(fact_56_dual__order_Oasym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.45 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.asym
% 0.29/0.45 thf(fact_57_ord__less__eq__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( B2 = C )
% 0.29/0.45 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_less_eq_trans
% 0.29/0.45 thf(fact_58_ord__eq__less__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A] :
% 0.29/0.45 ( ( A2 = B2 )
% 0.29/0.45 => ( ( ord_less @ A @ B2 @ C )
% 0.29/0.45 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_less_trans
% 0.29/0.45 thf(fact_59_less__irrefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_irrefl
% 0.29/0.45 thf(fact_60_less__linear,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 | ( X = Y )
% 0.29/0.45 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_linear
% 0.29/0.45 thf(fact_61_less__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less @ A @ Y @ Z2 )
% 0.29/0.45 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_trans
% 0.29/0.45 thf(fact_62_less__asym_H,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_asym'
% 0.29/0.45 thf(fact_63_less__asym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_asym
% 0.29/0.45 thf(fact_64_less__imp__neq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( X != Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_neq
% 0.29/0.45 thf(fact_65_dense,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ? [Z: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Z )
% 0.29/0.45 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense
% 0.29/0.45 thf(fact_66_order_Oasym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.asym
% 0.29/0.45 thf(fact_67_neq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 = ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % neq_iff
% 0.29/0.45 thf(fact_68_neqE,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 => ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % neqE
% 0.29/0.45 thf(fact_69_gt__ex,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( no_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.29/0.45
% 0.29/0.45 % gt_ex
% 0.29/0.45 thf(fact_70_lt__ex,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % lt_ex
% 0.29/0.45 thf(fact_71_order__less__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C2: $tType] :
% 0.29/0.45 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.29/0.45 => ( ! [X4: A,Y3: A] :
% 0.29/0.45 ( ( ord_less @ A @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_less_subst2
% 0.29/0.45 thf(fact_72_order__less__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.29/0.45 => ( ( ord_less @ B @ B2 @ C )
% 0.29/0.45 => ( ! [X4: B,Y3: B] :
% 0.29/0.45 ( ( ord_less @ B @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_less_subst1
% 0.29/0.45 thf(fact_73_ord__less__eq__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ( F @ B2 )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X4: A,Y3: A] :
% 0.29/0.45 ( ( ord_less @ A @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_less_eq_subst
% 0.29/0.45 thf(fact_74_ord__eq__less__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 = ( F @ B2 ) )
% 0.29/0.45 => ( ( ord_less @ B @ B2 @ C )
% 0.29/0.45 => ( ! [X4: B,Y3: B] :
% 0.29/0.45 ( ( ord_less @ B @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_less_subst
% 0.29/0.45 thf(fact_75_pinf_I1_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.45 ( ? [Z4: A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z4 @ X4 )
% 0.29/0.45 => ( ( P @ X4 )
% 0.29/0.45 = ( P2 @ X4 ) ) )
% 0.29/0.45 => ( ? [Z4: A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z4 @ X4 )
% 0.29/0.45 => ( ( Q @ X4 )
% 0.29/0.45 = ( Q2 @ X4 ) ) )
% 0.29/0.45 => ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.45 => ( ( ( P @ X2 )
% 0.29/0.45 & ( Q @ X2 ) )
% 0.29/0.45 = ( ( P2 @ X2 )
% 0.29/0.45 & ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(1)
% 0.29/0.45 thf(fact_76_pinf_I2_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.45 ( ? [Z4: A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z4 @ X4 )
% 0.29/0.45 => ( ( P @ X4 )
% 0.29/0.45 = ( P2 @ X4 ) ) )
% 0.29/0.45 => ( ? [Z4: A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z4 @ X4 )
% 0.29/0.45 => ( ( Q @ X4 )
% 0.29/0.45 = ( Q2 @ X4 ) ) )
% 0.29/0.45 => ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.45 => ( ( ( P @ X2 )
% 0.29/0.45 | ( Q @ X2 ) )
% 0.29/0.45 = ( ( P2 @ X2 )
% 0.29/0.45 | ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(2)
% 0.29/0.45 thf(fact_77_pinf_I3_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.45 => ( X2 != T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(3)
% 0.29/0.45 thf(fact_78_pinf_I4_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.45 => ( X2 != T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(4)
% 0.29/0.45 thf(fact_79_pinf_I5_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.45 => ~ ( ord_less @ A @ X2 @ T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(5)
% 0.29/0.45 thf(fact_80_pinf_I7_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.45 => ( ord_less @ A @ T @ X2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(7)
% 0.29/0.45 thf(fact_81_pinf_I11_J,axiom,
% 0.29/0.45 ! [C2: $tType,D: $tType] :
% 0.29/0.45 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.29/0.45 => ! [F3: D] :
% 0.29/0.45 ? [Z: C2] :
% 0.29/0.45 ! [X2: C2] :
% 0.29/0.45 ( ( ord_less @ C2 @ Z @ X2 )
% 0.29/0.45 => ( F3 = F3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % pinf(11)
% 0.29/0.45 thf(fact_82_minf_I1_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.45 ( ? [Z4: A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ X4 @ Z4 )
% 0.29/0.45 => ( ( P @ X4 )
% 0.29/0.45 = ( P2 @ X4 ) ) )
% 0.29/0.45 => ( ? [Z4: A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ X4 @ Z4 )
% 0.29/0.45 => ( ( Q @ X4 )
% 0.29/0.45 = ( Q2 @ X4 ) ) )
% 0.29/0.45 => ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.45 => ( ( ( P @ X2 )
% 0.29/0.45 & ( Q @ X2 ) )
% 0.29/0.45 = ( ( P2 @ X2 )
% 0.29/0.45 & ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(1)
% 0.29/0.45 thf(fact_83_minf_I2_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.45 ( ? [Z4: A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ X4 @ Z4 )
% 0.29/0.45 => ( ( P @ X4 )
% 0.29/0.45 = ( P2 @ X4 ) ) )
% 0.29/0.45 => ( ? [Z4: A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ X4 @ Z4 )
% 0.29/0.45 => ( ( Q @ X4 )
% 0.29/0.45 = ( Q2 @ X4 ) ) )
% 0.29/0.45 => ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.45 => ( ( ( P @ X2 )
% 0.29/0.45 | ( Q @ X2 ) )
% 0.29/0.45 = ( ( P2 @ X2 )
% 0.29/0.45 | ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(2)
% 0.29/0.45 thf(fact_84_minf_I3_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.45 => ( X2 != T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(3)
% 0.29/0.45 thf(fact_85_minf_I4_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.45 => ( X2 != T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(4)
% 0.29/0.45 thf(fact_86_minf_I5_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.45 => ( ord_less @ A @ X2 @ T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(5)
% 0.29/0.45 thf(fact_87_minf_I7_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [T: A] :
% 0.29/0.45 ? [Z: A] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.45 => ~ ( ord_less @ A @ T @ X2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(7)
% 0.29/0.45 thf(fact_88_minf_I11_J,axiom,
% 0.29/0.45 ! [C2: $tType,D: $tType] :
% 0.29/0.45 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.29/0.45 => ! [F3: D] :
% 0.29/0.45 ? [Z: C2] :
% 0.29/0.45 ! [X2: C2] :
% 0.29/0.45 ( ( ord_less @ C2 @ X2 @ Z )
% 0.29/0.45 => ( F3 = F3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % minf(11)
% 0.29/0.45 thf(fact_89_order_Onot__eq__order__implies__strict,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( A2 != B2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.not_eq_order_implies_strict
% 0.29/0.45 thf(fact_90_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.45 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_implies_order
% 0.29/0.45 thf(fact_91_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less @ A )
% 0.29/0.45 = ( ^ [B5: A,A5: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.29/0.45 & ( A5 != B5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_iff_order
% 0.29/0.45 thf(fact_92_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less_eq @ A )
% 0.29/0.45 = ( ^ [B5: A,A5: A] :
% 0.29/0.45 ( ( ord_less @ A @ B5 @ A5 )
% 0.29/0.45 | ( A5 = B5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.order_iff_strict
% 0.29/0.45 thf(fact_93_order_Ostrict__implies__order,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_implies_order
% 0.29/0.45 thf(fact_94_dense__le__bounded,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ! [W: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ W )
% 0.29/0.45 => ( ( ord_less @ A @ W @ Y )
% 0.29/0.45 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense_le_bounded
% 0.29/0.45 thf(fact_95_dense__ge__bounded,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Z2: A,X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z2 @ X )
% 0.29/0.45 => ( ! [W: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z2 @ W )
% 0.29/0.45 => ( ( ord_less @ A @ W @ X )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense_ge_bounded
% 0.29/0.45 thf(fact_96_dual__order_Ostrict__trans2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.29/0.45 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_trans2
% 0.29/0.45 thf(fact_97_dual__order_Ostrict__trans1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.45 => ( ( ord_less @ A @ C @ B2 )
% 0.29/0.45 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_trans1
% 0.29/0.45 thf(fact_98_order_Ostrict__iff__order,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less @ A )
% 0.29/0.45 = ( ^ [A5: A,B5: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.29/0.45 & ( A5 != B5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_iff_order
% 0.29/0.45 thf(fact_99_order_Oorder__iff__strict,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less_eq @ A )
% 0.29/0.45 = ( ^ [A5: A,B5: A] :
% 0.29/0.45 ( ( ord_less @ A @ A5 @ B5 )
% 0.29/0.45 | ( A5 = B5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.order_iff_strict
% 0.29/0.45 thf(fact_100_order_Ostrict__trans2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.45 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_trans2
% 0.29/0.45 thf(fact_101_order_Ostrict__trans1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less @ A @ B2 @ C )
% 0.29/0.45 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_trans1
% 0.29/0.45 thf(fact_102_not__le__imp__less,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,X: A] :
% 0.29/0.45 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_le_imp_less
% 0.29/0.45 thf(fact_103_less__le__not__le,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less @ A )
% 0.29/0.45 = ( ^ [X3: A,Y2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.45 & ~ ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_le_not_le
% 0.29/0.45 thf(fact_104_le__imp__less__or__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 | ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_imp_less_or_eq
% 0.29/0.45 thf(fact_105_le__less__linear,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_less_linear
% 0.29/0.45 thf(fact_106_dense__le,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,Z2: A] :
% 0.29/0.45 ( ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ X4 @ Y )
% 0.29/0.45 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense_le
% 0.29/0.45 thf(fact_107_dense__ge,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Z2: A,Y: A] :
% 0.29/0.45 ( ! [X4: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z2 @ X4 )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ X4 ) )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense_ge
% 0.29/0.45 thf(fact_108_less__le__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.29/0.45 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_le_trans
% 0.29/0.45 thf(fact_109_le__less__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less @ A @ Y @ Z2 )
% 0.29/0.45 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_less_trans
% 0.29/0.45 thf(fact_110_antisym__conv2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.45 = ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv2
% 0.29/0.45 thf(fact_111_antisym__conv1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 = ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv1
% 0.29/0.45 thf(fact_112_less__imp__le,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_le
% 0.29/0.45 thf(fact_113_le__neq__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( A2 != B2 )
% 0.29/0.45 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_neq_trans
% 0.29/0.45 thf(fact_114_not__less,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.45 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_less
% 0.29/0.45 thf(fact_115_not__le,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.29/0.45 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_le
% 0.29/0.45 thf(fact_116_order__less__le__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C2: $tType] :
% 0.29/0.45 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.29/0.45 => ( ! [X4: A,Y3: A] :
% 0.29/0.45 ( ( ord_less @ A @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_less_le_subst2
% 0.29/0.45 thf(fact_117_order__less__le__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.29/0.45 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.29/0.45 => ( ! [X4: B,Y3: B] :
% 0.29/0.45 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_less_le_subst1
% 0.29/0.45 thf(fact_118_distinct__union,axiom,
% 0.29/0.45 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.45 ( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
% 0.29/0.45 = ( distinct @ A @ Ys ) ) ).
% 0.29/0.45
% 0.29/0.45 % distinct_union
% 0.29/0.45 thf(fact_119_complete__interval,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,P: A > $o] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.45 => ( ( P @ A2 )
% 0.29/0.45 => ( ~ ( P @ B2 )
% 0.29/0.45 => ? [C3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.29/0.45 & ( ord_less_eq @ A @ C3 @ B2 )
% 0.29/0.45 & ! [X2: A] :
% 0.29/0.45 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.29/0.45 & ( ord_less @ A @ X2 @ C3 ) )
% 0.29/0.45 => ( P @ X2 ) )
% 0.29/0.45 & ! [D2: A] :
% 0.29/0.45 ( ! [X4: A] :
% 0.29/0.45 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 0.29/0.45 & ( ord_less @ A @ X4 @ D2 ) )
% 0.29/0.45 => ( P @ X4 ) )
% 0.29/0.45 => ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % complete_interval
% 0.29/0.45 thf(fact_120_subset__sorted__correct,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [L1: list @ A,L2: list @ A] :
% 0.29/0.45 ( ( ( distinct @ A @ L1 )
% 0.29/0.45 & ( linorder_sorted @ A @ L1 ) )
% 0.29/0.45 => ( ( ( distinct @ A @ L2 )
% 0.29/0.45 & ( linorder_sorted @ A @ L2 ) )
% 0.29/0.45 => ( ( sorted1061247458sorted @ A @ L1 @ L2 )
% 0.29/0.45 = ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_sorted_correct
% 0.29/0.45 thf(fact_121_mergesort__remdups__correct,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [L: list @ A] :
% 0.29/0.45 ( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
% 0.29/0.45 & ( linorder_sorted @ A @ ( mergesort_remdups @ A @ L ) )
% 0.29/0.45 & ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
% 0.29/0.45 = ( set2 @ A @ L ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mergesort_remdups_correct
% 0.29/0.45 thf(fact_122_memb__sorted__correct,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Xs: list @ A,X: A] :
% 0.29/0.45 ( ( linorder_sorted @ A @ Xs )
% 0.29/0.45 => ( ( sorted873716653sorted @ A @ Xs @ X )
% 0.29/0.45 = ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % memb_sorted_correct
% 0.29/0.45 thf(fact_123_in__set__member,axiom,
% 0.29/0.45 ! [A: $tType,X: A,Xs: list @ A] :
% 0.29/0.45 ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
% 0.29/0.45 = ( member @ A @ Xs @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % in_set_member
% 0.29/0.45 thf(fact_124_finite__sorted__distinct__unique,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A4: set @ A] :
% 0.29/0.45 ( ( finite_finite2 @ A @ A4 )
% 0.29/0.45 => ? [X4: list @ A] :
% 0.29/0.45 ( ( ( set2 @ A @ X4 )
% 0.29/0.45 = A4 )
% 0.29/0.45 & ( linorder_sorted @ A @ X4 )
% 0.29/0.45 & ( distinct @ A @ X4 )
% 0.29/0.45 & ! [Y5: list @ A] :
% 0.29/0.45 ( ( ( ( set2 @ A @ Y5 )
% 0.29/0.45 = A4 )
% 0.29/0.45 & ( linorder_sorted @ A @ Y5 )
% 0.29/0.45 & ( distinct @ A @ Y5 ) )
% 0.29/0.45 => ( Y5 = X4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_sorted_distinct_unique
% 0.29/0.45 thf(fact_125_ex__gt__or__lt,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ? [B4: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B4 )
% 0.29/0.45 | ( ord_less @ A @ B4 @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ex_gt_or_lt
% 0.29/0.45 thf(fact_126_linorder__neqE__linordered__idom,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 => ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_neqE_linordered_idom
% 0.29/0.45 thf(fact_127_linordered__field__no__ub,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X2: A] :
% 0.29/0.45 ? [X1: A] : ( ord_less @ A @ X2 @ X1 ) ) ).
% 0.29/0.45
% 0.29/0.45 % linordered_field_no_ub
% 0.29/0.45 thf(fact_128_linordered__field__no__lb,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X2: A] :
% 0.29/0.45 ? [Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % linordered_field_no_lb
% 0.29/0.45 thf(fact_129_List_Ofinite__set,axiom,
% 0.29/0.45 ! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% 0.29/0.45
% 0.29/0.45 % List.finite_set
% 0.29/0.45 thf(fact_130_less__fun__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ( ( ord_less @ ( A > B ) )
% 0.29/0.45 = ( ^ [F2: A > B,G: A > B] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 0.29/0.45 & ~ ( ord_less_eq @ ( A > B ) @ G @ F2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_fun_def
% 0.29/0.45 thf(fact_131_subset__Collect__conv,axiom,
% 0.29/0.45 ! [A: $tType,S: set @ A,P: A > $o] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 0.29/0.45 = ( ! [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ S )
% 0.29/0.45 => ( P @ X3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_Collect_conv
% 0.29/0.45 thf(fact_132_finite__list,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] :
% 0.29/0.45 ( ( finite_finite2 @ A @ A4 )
% 0.29/0.45 => ? [Xs2: list @ A] :
% 0.29/0.45 ( ( set2 @ A @ Xs2 )
% 0.29/0.45 = A4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_list
% 0.29/0.45 thf(fact_133_finite__distinct__list,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] :
% 0.29/0.45 ( ( finite_finite2 @ A @ A4 )
% 0.29/0.45 => ? [Xs2: list @ A] :
% 0.29/0.45 ( ( ( set2 @ A @ Xs2 )
% 0.29/0.45 = A4 )
% 0.29/0.45 & ( distinct @ A @ Xs2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_distinct_list
% 0.29/0.45 thf(fact_134_ord__eq__le__eq__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C: A,D3: A] :
% 0.29/0.45 ( ( A2 = B2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.45 => ( ( C = D3 )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ D3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_eq_trans
% 0.29/0.45 thf(fact_135_subsetI,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ! [X4: A] :
% 0.29/0.45 ( ( member2 @ A @ X4 @ A4 )
% 0.29/0.45 => ( member2 @ A @ X4 @ B3 ) )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetI
% 0.29/0.45 thf(fact_136_psubsetI,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( A4 != B3 )
% 0.29/0.45 => ( ord_less @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % psubsetI
% 0.29/0.45 thf(fact_137_subset__antisym,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
% 0.29/0.45 => ( A4 = B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_antisym
% 0.29/0.45 thf(fact_138_finite__code,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( finite_finite2 @ A )
% 0.29/0.45 = ( ^ [A6: set @ A] : $true ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_code
% 0.29/0.45 thf(fact_139_psubset__trans,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( ord_less @ ( set @ A ) @ B3 @ C4 )
% 0.29/0.45 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % psubset_trans
% 0.29/0.45 thf(fact_140_psubsetD,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( member2 @ A @ C @ A4 )
% 0.29/0.45 => ( member2 @ A @ C @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % psubsetD
% 0.29/0.45 thf(fact_141_finite__set__choice,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
% 0.29/0.45 ( ( finite_finite2 @ A @ A4 )
% 0.29/0.45 => ( ! [X4: A] :
% 0.29/0.45 ( ( member2 @ A @ X4 @ A4 )
% 0.29/0.45 => ? [X12: B] : ( P @ X4 @ X12 ) )
% 0.29/0.45 => ? [F4: A > B] :
% 0.29/0.45 ! [X2: A] :
% 0.29/0.45 ( ( member2 @ A @ X2 @ A4 )
% 0.29/0.45 => ( P @ X2 @ ( F4 @ X2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_set_choice
% 0.29/0.45 thf(fact_142_finite,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite
% 0.29/0.45 thf(fact_143_finite__psubset__induct,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
% 0.29/0.45 ( ( finite_finite2 @ A @ A4 )
% 0.29/0.45 => ( ! [A7: set @ A] :
% 0.29/0.45 ( ( finite_finite2 @ A @ A7 )
% 0.29/0.45 => ( ! [B6: set @ A] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) @ B6 @ A7 )
% 0.29/0.45 => ( P @ B6 ) )
% 0.29/0.45 => ( P @ A7 ) ) )
% 0.29/0.45 => ( P @ A4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_psubset_induct
% 0.29/0.45 thf(fact_144_subset__iff__psubset__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.29/0.45 | ( A6 = B7 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff_psubset_eq
% 0.29/0.45 thf(fact_145_subset__psubset__trans,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( ord_less @ ( set @ A ) @ B3 @ C4 )
% 0.29/0.45 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_psubset_trans
% 0.29/0.45 thf(fact_146_subset__not__subset__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.29/0.45 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_not_subset_eq
% 0.29/0.45 thf(fact_147_psubset__subset__trans,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
% 0.29/0.45 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % psubset_subset_trans
% 0.29/0.45 thf(fact_148_psubset__imp__subset,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % psubset_imp_subset
% 0.29/0.45 thf(fact_149_Collect__mono__iff,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.29/0.45 = ( ! [X3: A] :
% 0.29/0.45 ( ( P @ X3 )
% 0.29/0.45 => ( Q @ X3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono_iff
% 0.29/0.45 thf(fact_150_contra__subsetD,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ~ ( member2 @ A @ C @ B3 )
% 0.29/0.45 => ~ ( member2 @ A @ C @ A4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % contra_subsetD
% 0.29/0.45 thf(fact_151_set__eq__subset,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ^ [Y4: set @ A,Z3: set @ A] : ( Y4 = Z3 ) )
% 0.29/0.45 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.29/0.45 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_eq_subset
% 0.29/0.45 thf(fact_152_subset__trans,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_trans
% 0.29/0.45 thf(fact_153_Collect__mono,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.45 ( ! [X4: A] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 => ( Q @ X4 ) )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono
% 0.29/0.45 thf(fact_154_subset__refl,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.29/0.45
% 0.29/0.45 % subset_refl
% 0.29/0.45 thf(fact_155_rev__subsetD,axiom,
% 0.29/0.45 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( member2 @ A @ C @ A4 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( member2 @ A @ C @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rev_subsetD
% 0.29/0.45 thf(fact_156_subset__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.29/0.45 ! [T2: A] :
% 0.29/0.45 ( ( member2 @ A @ T2 @ A6 )
% 0.29/0.45 => ( member2 @ A @ T2 @ B7 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_157_set__rev__mp,axiom,
% 0.29/0.45 ! [A: $tType,X: A,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( member2 @ A @ X @ A4 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( member2 @ A @ X @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_rev_mp
% 0.29/0.45 thf(fact_158_psubset__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.29/0.45 & ( A6 != B7 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % psubset_eq
% 0.29/0.45 thf(fact_159_equalityD2,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( A4 = B3 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD2
% 0.29/0.45 thf(fact_160_equalityD1,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( A4 = B3 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD1
% 0.29/0.45 thf(fact_161_subset__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.29/0.45 ! [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ A6 )
% 0.29/0.45 => ( member2 @ A @ X3 @ B7 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_162_equalityE,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( A4 = B3 )
% 0.29/0.45 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityE
% 0.29/0.45 thf(fact_163_subsetCE,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( member2 @ A @ C @ A4 )
% 0.29/0.45 => ( member2 @ A @ C @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetCE
% 0.29/0.45 thf(fact_164_psubsetE,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % psubsetE
% 0.29/0.45 thf(fact_165_subsetD,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( member2 @ A @ C @ A4 )
% 0.29/0.45 => ( member2 @ A @ C @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_166_in__mono,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( member2 @ A @ X @ A4 )
% 0.29/0.45 => ( member2 @ A @ X @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % in_mono
% 0.29/0.45 thf(fact_167_set__mp,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( member2 @ A @ X @ A4 )
% 0.29/0.45 => ( member2 @ A @ X @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_mp
% 0.29/0.45 thf(fact_168_rev__finite__subset,axiom,
% 0.29/0.45 ! [A: $tType,B3: set @ A,A4: set @ A] :
% 0.29/0.45 ( ( finite_finite2 @ A @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( finite_finite2 @ A @ A4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rev_finite_subset
% 0.29/0.45 thf(fact_169_infinite__super,axiom,
% 0.29/0.45 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
% 0.29/0.45 => ( ~ ( finite_finite2 @ A @ S )
% 0.29/0.45 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % infinite_super
% 0.29/0.45 thf(fact_170_finite__subset,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( finite_finite2 @ A @ B3 )
% 0.29/0.45 => ( finite_finite2 @ A @ A4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_subset
% 0.29/0.45 thf(fact_171_sorted__list__of__set,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A4: set @ A] :
% 0.29/0.45 ( ( finite_finite2 @ A @ A4 )
% 0.29/0.45 => ( ( ( set2 @ A @ ( linord467138063of_set @ A @ A4 ) )
% 0.29/0.45 = A4 )
% 0.29/0.45 & ( linorder_sorted @ A @ ( linord467138063of_set @ A @ A4 ) )
% 0.29/0.45 & ( distinct @ A @ ( linord467138063of_set @ A @ A4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sorted_list_of_set
% 0.29/0.45 thf(fact_172_set__union,axiom,
% 0.29/0.45 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.45 ( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
% 0.29/0.45 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_union
% 0.29/0.45 thf(fact_173_subset__code_I2_J,axiom,
% 0.29/0.45 ! [B: $tType,A4: set @ B,Ys: list @ B] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ B ) @ A4 @ ( coset @ B @ Ys ) )
% 0.29/0.45 = ( ! [X3: B] :
% 0.29/0.45 ( ( member2 @ B @ X3 @ ( set2 @ B @ Ys ) )
% 0.29/0.45 => ~ ( member2 @ B @ X3 @ A4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_code(2)
% 0.29/0.45 thf(fact_174_sorted__by__rel__linord,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [L: list @ A] :
% 0.29/0.45 ( ( sorted_by_rel @ A @ ( ord_less_eq @ A ) @ L )
% 0.29/0.45 = ( linorder_sorted @ A @ L ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sorted_by_rel_linord
% 0.29/0.45 thf(fact_175_UnCI,axiom,
% 0.29/0.45 ! [A: $tType,C: A,B3: set @ A,A4: set @ A] :
% 0.29/0.45 ( ( ~ ( member2 @ A @ C @ B3 )
% 0.29/0.45 => ( member2 @ A @ C @ A4 ) )
% 0.29/0.45 => ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnCI
% 0.29/0.45 thf(fact_176_Un__iff,axiom,
% 0.29/0.45 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.29/0.45 = ( ( member2 @ A @ C @ A4 )
% 0.29/0.45 | ( member2 @ A @ C @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_iff
% 0.29/0.45 thf(fact_177_finite__Un,axiom,
% 0.29/0.45 ! [A: $tType,F3: set @ A,G3: set @ A] :
% 0.29/0.45 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F3 @ G3 ) )
% 0.29/0.45 = ( ( finite_finite2 @ A @ F3 )
% 0.29/0.45 & ( finite_finite2 @ A @ G3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_Un
% 0.29/0.45 thf(fact_178_Un__subset__iff,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 )
% 0.29/0.45 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
% 0.29/0.45 & ( ord_less_eq @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_subset_iff
% 0.29/0.45 thf(fact_179_Un__mono,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,C4: set @ A,B3: set @ A,D4: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D4 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_mono
% 0.29/0.45 thf(fact_180_Un__least,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,C4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_least
% 0.29/0.45 thf(fact_181_Un__upper1,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_upper1
% 0.29/0.45 thf(fact_182_Un__upper2,axiom,
% 0.29/0.45 ! [A: $tType,B3: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_upper2
% 0.29/0.45 thf(fact_183_Un__absorb1,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 => ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 = B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_absorb1
% 0.29/0.45 thf(fact_184_Un__absorb2,axiom,
% 0.29/0.45 ! [A: $tType,B3: set @ A,A4: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
% 0.29/0.45 => ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
% 0.29/0.45 = A4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_absorb2
% 0.29/0.45 thf(fact_185_subset__Un__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ A6 @ B7 )
% 0.29/0.45 = B7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_Un_eq
% 0.29/0.45 thf(fact_186_infinite__Un,axiom,
% 0.29/0.45 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.29/0.45 ( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) )
% 0.29/0.45 = ( ~ ( finite_finite2 @ A @ S )
% 0.29/0.45 | ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % infinite_Un
% 0.29/0.45 thf(fact_187_Un__infinite,axiom,
% 0.29/0.45 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.29/0.45 ( ~ ( finite_finite2 @ A @ S )
% 0.29/0.45 => ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_infinite
% 0.29/0.45 thf(fact_188_finite__UnI,axiom,
% 0.29/0.45 ! [A: $tType,F3: set @ A,G3: set @ A] :
% 0.29/0.45 ( ( finite_finite2 @ A @ F3 )
% 0.29/0.45 => ( ( finite_finite2 @ A @ G3 )
% 0.29/0.45 => ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F3 @ G3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % finite_UnI
% 0.29/0.45 thf(fact_189_UnE,axiom,
% 0.29/0.45 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.29/0.45 => ( ~ ( member2 @ A @ C @ A4 )
% 0.29/0.45 => ( member2 @ A @ C @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnE
% 0.29/0.45 thf(fact_190_UnI1,axiom,
% 0.29/0.45 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 0.29/0.45 ( ( member2 @ A @ C @ A4 )
% 0.29/0.45 => ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnI1
% 0.29/0.45 thf(fact_191_UnI2,axiom,
% 0.29/0.45 ! [A: $tType,C: A,B3: set @ A,A4: set @ A] :
% 0.29/0.45 ( ( member2 @ A @ C @ B3 )
% 0.29/0.45 => ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnI2
% 0.29/0.45 thf(fact_192_bex__Un,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
% 0.29/0.45 ( ( ? [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.29/0.45 & ( P @ X3 ) ) )
% 0.29/0.45 = ( ? [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ A4 )
% 0.29/0.45 & ( P @ X3 ) )
% 0.29/0.45 | ? [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ B3 )
% 0.29/0.45 & ( P @ X3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bex_Un
% 0.29/0.45 thf(fact_193_ball__Un,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
% 0.29/0.45 ( ( ! [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.29/0.45 => ( P @ X3 ) ) )
% 0.29/0.45 = ( ! [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ A4 )
% 0.29/0.45 => ( P @ X3 ) )
% 0.29/0.45 & ! [X3: A] :
% 0.29/0.45 ( ( member2 @ A @ X3 @ B3 )
% 0.29/0.45 => ( P @ X3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ball_Un
% 0.29/0.45 thf(fact_194_Un__assoc,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 0.29/0.46 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 )
% 0.29/0.46 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_assoc
% 0.29/0.46 thf(fact_195_Un__absorb,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A] :
% 0.29/0.46 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 0.29/0.46 = A4 ) ).
% 0.29/0.46
% 0.29/0.46 % Un_absorb
% 0.29/0.46 thf(fact_196_Un__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( sup_sup @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_commute
% 0.29/0.46 thf(fact_197_Un__left__absorb,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.29/0.46 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.29/0.46 = ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_left_absorb
% 0.29/0.46 thf(fact_198_Un__left__commute,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 0.29/0.46 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) )
% 0.29/0.46 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_left_commute
% 0.29/0.46 thf(fact_199_sorted__by__rel__weaken,axiom,
% 0.29/0.46 ! [A: $tType,L0: list @ A,R: A > A > $o,R2: A > A > $o] :
% 0.29/0.46 ( ! [X4: A,Y3: A] :
% 0.29/0.46 ( ( member2 @ A @ X4 @ ( set2 @ A @ L0 ) )
% 0.29/0.46 => ( ( member2 @ A @ Y3 @ ( set2 @ A @ L0 ) )
% 0.29/0.46 => ( ( R @ X4 @ Y3 )
% 0.29/0.46 => ( R2 @ X4 @ Y3 ) ) ) )
% 0.29/0.46 => ( ( sorted_by_rel @ A @ R @ L0 )
% 0.29/0.46 => ( sorted_by_rel @ A @ R2 @ L0 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sorted_by_rel_weaken
% 0.29/0.46 thf(fact_200_distinct__sorted__list__of__set,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A4: set @ A] : ( distinct @ A @ ( linord467138063of_set @ A @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % distinct_sorted_list_of_set
% 0.29/0.46 thf(fact_201_sup_Obounded__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,C: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
% 0.29/0.46 = ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 & ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.bounded_iff
% 0.29/0.46 thf(fact_202_le__sup__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
% 0.29/0.46 = ( ( ord_less_eq @ A @ X @ Z2 )
% 0.29/0.46 & ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_sup_iff
% 0.29/0.46 thf(fact_203_sup_Oright__idem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 0.29/0.46 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.right_idem
% 0.29/0.46 thf(fact_204_sup__apply,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.29/0.46 => ( ( sup_sup @ ( A > B ) )
% 0.29/0.46 = ( ^ [F2: A > B,G: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_apply
% 0.29/0.46 thf(fact_205_sup_Oidem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( sup_sup @ A @ A2 @ A2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.idem
% 0.29/0.46 thf(fact_206_sup__idem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ( ( sup_sup @ A @ X @ X )
% 0.29/0.46 = X ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_idem
% 0.29/0.46 thf(fact_207_sup_Oleft__idem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 0.29/0.46 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_idem
% 0.29/0.46 thf(fact_208_sup__left__idem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.29/0.46 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_idem
% 0.29/0.46 thf(fact_209_inf__sup__aci_I8_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.29/0.46 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(8)
% 0.29/0.46 thf(fact_210_inf__sup__aci_I7_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
% 0.29/0.46 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(7)
% 0.29/0.46 thf(fact_211_inf__sup__aci_I6_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
% 0.29/0.46 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(6)
% 0.29/0.46 thf(fact_212_inf__sup__aci_I5_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( sup_sup @ A )
% 0.29/0.46 = ( ^ [X3: A,Y2: A] : ( sup_sup @ A @ Y2 @ X3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(5)
% 0.29/0.46 thf(fact_213_sup__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.29/0.46 => ( ( sup_sup @ ( A > B ) )
% 0.29/0.46 = ( ^ [F2: A > B,G: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_fun_def
% 0.29/0.46 thf(fact_214_sup_Oassoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C )
% 0.29/0.46 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.assoc
% 0.29/0.46 thf(fact_215_sup__assoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
% 0.29/0.46 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_assoc
% 0.29/0.46 thf(fact_216_sup_Ocommute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( sup_sup @ A )
% 0.29/0.46 = ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.commute
% 0.29/0.46 thf(fact_217_sup__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( sup_sup @ A )
% 0.29/0.46 = ( ^ [X3: A,Y2: A] : ( sup_sup @ A @ Y2 @ X3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_commute
% 0.29/0.46 thf(fact_218_sup_Oleft__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C ) )
% 0.29/0.46 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_commute
% 0.29/0.46 thf(fact_219_sup__left__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
% 0.29/0.46 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_commute
% 0.29/0.46 thf(fact_220_inf__sup__ord_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(4)
% 0.29/0.46 thf(fact_221_inf__sup__ord_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(3)
% 0.29/0.46 thf(fact_222_le__supE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B2: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ A2 @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_supE
% 0.29/0.46 thf(fact_223_le__supI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,X: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ X )
% 0.29/0.46 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_supI
% 0.29/0.46 thf(fact_224_sup__ge1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_ge1
% 0.29/0.46 thf(fact_225_sup__ge2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_ge2
% 0.29/0.46 thf(fact_226_le__supI1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ A2 )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_supI1
% 0.29/0.46 thf(fact_227_le__supI2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_supI2
% 0.29/0.46 thf(fact_228_sup_Omono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [C: A,A2: A,D3: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ C @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ D3 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( sup_sup @ A @ C @ D3 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.mono
% 0.29/0.46 thf(fact_229_sup__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,C: A,B2: A,D3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ C )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ D3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C @ D3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_mono
% 0.29/0.46 thf(fact_230_sup__least,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A,Z2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Z2 @ X )
% 0.29/0.46 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_least
% 0.29/0.46 thf(fact_231_le__iff__sup,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [X3: A,Y2: A] :
% 0.29/0.46 ( ( sup_sup @ A @ X3 @ Y2 )
% 0.29/0.46 = Y2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_sup
% 0.29/0.46 thf(fact_232_sup_OorderE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 => ( A2
% 0.29/0.46 = ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.orderE
% 0.29/0.46 thf(fact_233_sup_OorderI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( sup_sup @ A @ A2 @ B2 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.orderI
% 0.29/0.46 thf(fact_234_sup__unique,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [F: A > A > A,X: A,Y: A] :
% 0.29/0.46 ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ X4 @ ( F @ X4 @ Y3 ) )
% 0.29/0.46 => ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X4 @ Y3 ) )
% 0.29/0.46 => ( ! [X4: A,Y3: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Z @ X4 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ Y3 @ Z ) @ X4 ) ) )
% 0.29/0.46 => ( ( sup_sup @ A @ X @ Y )
% 0.29/0.46 = ( F @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_unique
% 0.29/0.46 thf(fact_235_sup_Oabsorb1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 => ( ( sup_sup @ A @ A2 @ B2 )
% 0.29/0.46 = A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.absorb1
% 0.29/0.46 thf(fact_236_sup_Oabsorb2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( sup_sup @ A @ A2 @ B2 )
% 0.29/0.46 = B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.absorb2
% 0.29/0.46 thf(fact_237_sup__absorb1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( sup_sup @ A @ X @ Y )
% 0.29/0.46 = X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_absorb1
% 0.29/0.46 thf(fact_238_sup__absorb2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( sup_sup @ A @ X @ Y )
% 0.29/0.46 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_absorb2
% 0.29/0.46 thf(fact_239_sup_OboundedE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,C: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.boundedE
% 0.29/0.46 thf(fact_240_sup_OboundedI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ A2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.boundedI
% 0.29/0.46 thf(fact_241_sup_Oorder__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [B5: A,A5: A] :
% 0.29/0.46 ( A5
% 0.29/0.46 = ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.order_iff
% 0.29/0.46 thf(fact_242_sup_Ocobounded1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.cobounded1
% 0.29/0.46 thf(fact_243_sup_Ocobounded2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.cobounded2
% 0.29/0.46 thf(fact_244_sup_Oabsorb__iff1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [B5: A,A5: A] :
% 0.29/0.46 ( ( sup_sup @ A @ A5 @ B5 )
% 0.29/0.46 = A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.absorb_iff1
% 0.29/0.46 thf(fact_245_sup_Oabsorb__iff2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [A5: A,B5: A] :
% 0.29/0.46 ( ( sup_sup @ A @ A5 @ B5 )
% 0.29/0.46 = B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.absorb_iff2
% 0.29/0.46 thf(fact_246_sup_OcoboundedI1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ C @ A2 )
% 0.29/0.46 => ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.coboundedI1
% 0.29/0.46 thf(fact_247_sup_OcoboundedI2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [C: A,B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ C @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.coboundedI2
% 0.29/0.46 thf(fact_248_sup_Ostrict__coboundedI2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [C: A,B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ C @ B2 )
% 0.29/0.46 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_coboundedI2
% 0.29/0.46 thf(fact_249_sup_Ostrict__coboundedI1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ C @ A2 )
% 0.29/0.46 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_coboundedI1
% 0.29/0.46 thf(fact_250_sup_Ostrict__order__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [B5: A,A5: A] :
% 0.29/0.46 ( ( A5
% 0.29/0.46 = ( sup_sup @ A @ A5 @ B5 ) )
% 0.29/0.46 & ( A5 != B5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_order_iff
% 0.29/0.46 thf(fact_251_sup_Ostrict__boundedE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,C: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
% 0.29/0.46 => ~ ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.46 => ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.strict_boundedE
% 0.29/0.46 thf(fact_252_less__supI2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ B2 )
% 0.29/0.46 => ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_supI2
% 0.29/0.46 thf(fact_253_less__supI1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ A2 )
% 0.29/0.46 => ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_supI1
% 0.29/0.46 thf(fact_254_merge__correct,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [L1: list @ A,L2: list @ A] :
% 0.29/0.46 ( ( ( distinct @ A @ L1 )
% 0.29/0.46 & ( linorder_sorted @ A @ L1 ) )
% 0.29/0.46 => ( ( ( distinct @ A @ L2 )
% 0.29/0.46 & ( linorder_sorted @ A @ L2 ) )
% 0.29/0.46 => ( ( distinct @ A @ ( merge @ A @ L1 @ L2 ) )
% 0.29/0.46 & ( linorder_sorted @ A @ ( merge @ A @ L1 @ L2 ) )
% 0.29/0.46 & ( ( set2 @ A @ ( merge @ A @ L1 @ L2 ) )
% 0.29/0.46 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % merge_correct
% 0.29/0.46 thf(fact_255_set__mergesort__by__rel__merge,axiom,
% 0.29/0.46 ! [A: $tType,R: A > A > $o,Xs: list @ A,Ys: list @ A] :
% 0.29/0.46 ( ( set2 @ A @ ( merges2133357844_merge @ A @ R @ Xs @ Ys ) )
% 0.29/0.46 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_mergesort_by_rel_merge
% 0.29/0.46
% 0.29/0.46 %----Subclasses (4)
% 0.29/0.46 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( type @ A @ ( type2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ord @ A @ ( type2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( order @ A @ ( type2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( preorder @ A @ ( type2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 %----Type constructors (19)
% 0.29/0.46 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( semilattice_sup @ A9 @ ( type2 @ A9 ) )
% 0.29/0.46 => ( semilattice_sup @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 0.29/0.46 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 0.29/0.46 => ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( lattice @ A9 @ ( type2 @ A9 ) )
% 0.29/0.46 => ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( order @ A9 @ ( type2 @ A9 ) )
% 0.29/0.46 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 0.29/0.46 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 0.29/0.46 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 0.29/0.46 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
% 0.29/0.46 ! [A8: $tType] :
% 0.29/0.46 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
% 0.29/0.46 ! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.29/0.46 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.29/0.46 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
% 0.29/0.46 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 0.29/0.46 preorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.29/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_9,axiom,
% 0.29/0.46 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Olattice_10,axiom,
% 0.29/0.46 lattice @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 0.29/0.46 order @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 0.29/0.46 ord @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 %----Free types (1)
% 0.29/0.46 thf(tfree_0,hypothesis,
% 0.29/0.46 linorder @ a @ ( type2 @ a ) ).
% 0.29/0.46
% 0.29/0.46 %----Conjectures (4)
% 0.29/0.46 thf(conj_0,hypothesis,
% 0.29/0.46 ( ( distinct @ a @ ( sorted763967275sorted @ a @ x @ xsa ) )
% 0.29/0.46 & ( linorder_sorted @ a @ ( sorted763967275sorted @ a @ x @ xsa ) )
% 0.29/0.46 & ! [X2: a] :
% 0.29/0.46 ( ( member2 @ a @ X2 @ ( set2 @ a @ ( sorted763967275sorted @ a @ x @ xsa ) ) )
% 0.29/0.46 = ( ( member2 @ a @ X2 @ ( set2 @ a @ xsa ) )
% 0.29/0.46 & ( X2 != x ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(conj_1,hypothesis,
% 0.29/0.46 ( ( linorder_sorted @ a @ xsa )
% 0.29/0.46 & ! [X2: a] :
% 0.29/0.46 ( ( member2 @ a @ X2 @ ( set2 @ a @ xsa ) )
% 0.29/0.46 => ( ord_less_eq @ a @ a2 @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(conj_2,hypothesis,
% 0.29/0.46 ( ~ ( member2 @ a @ a2 @ ( set2 @ a @ xsa ) )
% 0.29/0.46 & ( distinct @ a @ xsa ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(conj_3,conjecture,
% 0.29/0.46 ( ( ( x != a2 )
% 0.29/0.46 | ! [X4: a] :
% 0.29/0.46 ( ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) )
% 0.29/0.46 = ( ( ( X4 = a2 )
% 0.29/0.46 | ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
% 0.29/0.46 & ( X4 != a2 ) ) ) )
% 0.29/0.46 & ( ( x = a2 )
% 0.29/0.46 | ( ( ~ ( ord_less @ a @ a2 @ x )
% 0.29/0.46 | ! [X4: a] :
% 0.29/0.46 ( ( ( X4 = a2 )
% 0.29/0.46 | ( ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) )
% 0.29/0.46 & ( X4 != x ) ) )
% 0.29/0.46 = ( ( ( X4 = a2 )
% 0.29/0.46 | ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
% 0.29/0.46 & ( X4 != x ) ) ) )
% 0.29/0.46 & ( ( ord_less @ a @ a2 @ x )
% 0.29/0.46 | ! [X4: a] :
% 0.29/0.46 ( ( ( X4 = a2 )
% 0.29/0.46 | ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
% 0.29/0.46 = ( ( ( X4 = a2 )
% 0.29/0.46 | ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
% 0.29/0.46 & ( X4 != x ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.65/1.48 ---- Embedded in THF ---
% 2.65/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.65/1.48 %%% Generated on Tue Feb 24 06:10:54 EST 2026
% 2.65/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.65/1.48 %%% Logic specification used:
% 2.65/1.48 %%% thf(spec, logic, $$dhol).
% 2.65/1.48
% 2.65/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.brU5k5c5ot/DTF2DTF27099.p
% 2.65/1.48 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.65/1.48 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.65/1.48 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.65/1.48 thf(ty_tf_a, type, a: $tType).
% 2.65/1.48 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.65/1.48 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.65/1.48 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.65/1.48 thf(sy_c_List_Ocoset, type, coset: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.65/1.48 thf(sy_c_List_Odistinct, type, distinct: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.65/1.48 thf(sy_c_List_Olinorder__class_Osorted, type, linorder_sorted: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.65/1.48 thf(sy_c_List_Olinorder__class_Osorted__list__of__set, type, linord467138063of_set: (!> [A:$tType]: (((set @ A) > (list @ A))))).
% 2.65/1.48 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.65/1.48 thf(sy_c_List_Omember, type, member: (!> [A:$tType]: (((list @ A) > (A > $o))))).
% 2.65/1.48 thf(sy_c_List_Ounion, type, union: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.65/1.48 thf(sy_c_Misc_Omerge, type, merge: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.65/1.48 thf(sy_c_Misc_Omergesort__by__rel__merge, type, merges2133357844_merge: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > (list @ A))))))).
% 2.65/1.48 thf(sy_c_Misc_Omergesort__remdups, type, mergesort_remdups: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.65/1.48 thf(sy_c_Misc_Osorted__by__rel, type, sorted_by_rel: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > $o))))).
% 2.65/1.48 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.65/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.65/1.48 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.65/1.48 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.65/1.48 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Odelete__sorted, type, sorted763967275sorted: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 2.65/1.48 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Omemb__sorted, type, sorted873716653sorted: (!> [A:$tType]: (((list @ A) > (A > $o))))).
% 2.65/1.48 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Osubset__sorted, type, sorted1061247458sorted: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 2.65/1.48 thf(sy_c_member, type, member2: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.65/1.48 thf(sy_v_a, type, a2: a).
% 2.65/1.48 thf(sy_v_x, type, x: a).
% 2.65/1.48 thf(sy_v_xsa, type, xsa: (list @ a)).
% 2.65/1.48 thf(fact_0_set__eq__sorted__correct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [L1:(list @ A),L2:(list @ A)]: (((((distinct @ A) @ L1) & ((linorder_sorted @ A) @ L1)) => ((((distinct @ A) @ L2) & ((linorder_sorted @ A) @ L2)) => ((L1 = L2) = (((set2 @ A) @ L1) = ((set2 @ A) @ L2))))))))))).
% 2.65/1.48 thf(fact_1_sorted__distinct__set__unique, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A),Ys:(list @ A)]: ((((linorder_sorted @ A) @ Xs) => (((distinct @ A) @ Xs) => (((linorder_sorted @ A) @ Ys) => (((distinct @ A) @ Ys) => ((((set2 @ A) @ Xs) = ((set2 @ A) @ Ys)) => (Xs = Ys)))))))))))).
% 2.65/1.48 thf(fact_2_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.65/1.48 thf(fact_3_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (~ ((((ord_less_eq @ A) @ T) @ X2)))))))))))))).
% 2.65/1.48 thf(fact_4_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (((ord_less_eq @ A) @ X2) @ T)))))))))))).
% 2.65/1.48 thf(fact_5_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((ord_less_eq @ A) @ T) @ X2)))))))))))).
% 2.65/1.48 thf(fact_6_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ T)))))))))))))).
% 2.65/1.48 thf(fact_7_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 2.65/1.48 thf(fact_8_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.65/1.48 thf(fact_9_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) | (X3 = Y2))))))))).
% 2.65/1.48 thf(fact_10_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (X3 != Y2))))))))).
% 2.65/1.48 thf(fact_11_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X4:B,Y3:B]: (((((ord_less @ B) @ X4) @ Y3) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.65/1.48 thf(fact_12_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ C2) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.65/1.48 thf(fact_13_subset__code_I1_J, axiom, (! [A:$tType,Xs:(list @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ B3) = (! [X3:A]: (((((member2 @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((member2 @ A) @ X3) @ B3)))))))).
% 2.65/1.48 thf(fact_14_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.65/1.48 thf(fact_15_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.65/1.48 thf(fact_16_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A3:A,B4:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((P @ A3) @ B4)))) => ((! [A3:A,B4:A]: ((((P @ B4) @ A3) => ((P @ A3) @ B4)))) => ((P @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_17_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.65/1.48 thf(fact_18_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.65/1.48 thf(fact_19_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.65/1.48 thf(fact_20_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.65/1.48 thf(fact_21_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.65/1.48 thf(fact_22_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.65/1.48 thf(fact_23_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.65/1.48 thf(fact_24_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.65/1.48 thf(fact_25_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.65/1.48 thf(fact_26_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_27_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.65/1.48 thf(fact_28_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.65/1.48 thf(fact_29_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (((ord_less_eq @ A) @ Y2) @ X3))))))))).
% 2.65/1.48 thf(fact_30_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 2.65/1.48 thf(fact_31_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.65/1.48 thf(fact_32_order__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ C2) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.65/1.48 thf(fact_33_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.65/1.48 thf(fact_34_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G @ X3))))))))))).
% 2.65/1.48 thf(fact_35_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G2:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G2 @ X4)))) => (((ord_less_eq @ (A > B)) @ F) @ G2)))))))).
% 2.65/1.48 thf(fact_36_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X)) @ (G2 @ X))))))))).
% 2.65/1.48 thf(fact_37_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X)) @ (G2 @ X))))))))).
% 2.65/1.48 thf(fact_38_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.65/1.48 thf(fact_39_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.65/1.48 thf(fact_40_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 2.65/1.48 thf(fact_41_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.65/1.48 thf(fact_42_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.65/1.48 thf(fact_43_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.65/1.48 thf(fact_44_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.65/1.48 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member2 @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.65/1.48 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member2 @ A) @ X3) @ A4)))) = A4)))).
% 2.65/1.48 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.65/1.48 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G2:(A > B)]: (((! [X4:A]: (((F @ X4) = (G2 @ X4)))) => (F = G2))))).
% 2.65/1.48 thf(fact_49_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.65/1.48 thf(fact_50_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 2.65/1.48 thf(fact_51_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 2.65/1.48 thf(fact_52_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.65/1.48 thf(fact_53_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X4) => (P @ Y5)))) => (P @ X4)))) => (P @ A2)))))))).
% 2.65/1.48 thf(fact_54_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.65/1.48 thf(fact_55_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.65/1.48 thf(fact_56_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 2.65/1.48 thf(fact_57_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.65/1.48 thf(fact_58_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.65/1.48 thf(fact_59_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.65/1.48 thf(fact_60_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.65/1.48 thf(fact_61_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.65/1.48 thf(fact_62_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.65/1.48 thf(fact_63_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.65/1.48 thf(fact_64_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.65/1.48 thf(fact_65_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z:A]: (((((ord_less @ A) @ X) @ Z) & (((ord_less @ A) @ Z) @ Y))))))))))).
% 2.65/1.48 thf(fact_66_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.65/1.48 thf(fact_67_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.65/1.48 thf(fact_68_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.65/1.48 thf(fact_69_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.65/1.48 thf(fact_70_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X))))))))).
% 2.65/1.48 thf(fact_71_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (((ord_less @ C2) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.65/1.48 thf(fact_72_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X4:B,Y3:B]: (((((ord_less @ B) @ X4) @ Y3) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.65/1.48 thf(fact_73_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (((ord_less @ B) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 2.65/1.48 thf(fact_74_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X4:B,Y3:B]: (((((ord_less @ B) @ X4) @ Y3) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.65/1.48 thf(fact_75_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((P @ X4) = (P2 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((P @ X2) & (Q @ X2)) = ((P2 @ X2) & (Q2 @ X2)))))))))))))))).
% 2.65/1.48 thf(fact_76_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((P @ X4) = (P2 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((P @ X2) | (Q @ X2)) = ((P2 @ X2) | (Q2 @ X2)))))))))))))))).
% 2.65/1.48 thf(fact_77_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (X2 != T)))))))))))).
% 2.65/1.48 thf(fact_78_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (X2 != T)))))))))))).
% 2.65/1.48 thf(fact_79_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (~ ((((ord_less @ A) @ X2) @ T)))))))))))))).
% 2.65/1.48 thf(fact_80_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((ord_less @ A) @ T) @ X2)))))))))))).
% 2.65/1.48 thf(fact_81_pinf_I11_J, axiom, (! [C2:$tType,D:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D]: ((? [Z:C2]: ((! [X2:C2]: (((((ord_less @ C2) @ Z) @ X2) => (F3 = F3)))))))))))).
% 2.65/1.48 thf(fact_82_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((P @ X4) = (P2 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (((P @ X2) & (Q @ X2)) = ((P2 @ X2) & (Q2 @ X2)))))))))))))))).
% 2.65/1.48 thf(fact_83_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((P @ X4) = (P2 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((Q @ X4) = (Q2 @ X4))))))) => (? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (((P @ X2) | (Q @ X2)) = ((P2 @ X2) | (Q2 @ X2)))))))))))))))).
% 2.65/1.48 thf(fact_84_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (X2 != T)))))))))))).
% 2.65/1.48 thf(fact_85_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (X2 != T)))))))))))).
% 2.65/1.48 thf(fact_86_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (((ord_less @ A) @ X2) @ T)))))))))))).
% 2.65/1.48 thf(fact_87_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (~ ((((ord_less @ A) @ T) @ X2)))))))))))))).
% 2.65/1.48 thf(fact_88_minf_I11_J, axiom, (! [C2:$tType,D:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D]: ((? [Z:C2]: ((! [X2:C2]: (((((ord_less @ C2) @ X2) @ Z) => (F3 = F3)))))))))))).
% 2.65/1.48 thf(fact_89_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_90_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.65/1.48 thf(fact_91_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B5:A,A5:A]: (((((ord_less_eq @ A) @ B5) @ A5) & (A5 != B5))))))))).
% 2.65/1.48 thf(fact_92_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: (((((ord_less @ A) @ B5) @ A5) | (A5 = B5))))))))).
% 2.65/1.48 thf(fact_93_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.65/1.48 thf(fact_94_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 2.65/1.48 thf(fact_95_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,X:A,Y:A]: (((((ord_less @ A) @ Z2) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 2.65/1.48 thf(fact_96_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.65/1.48 thf(fact_97_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.65/1.48 thf(fact_98_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) & (A5 != B5))))))))).
% 2.65/1.48 thf(fact_99_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: (((((ord_less @ A) @ A5) @ B5) | (A5 = B5))))))))).
% 2.65/1.48 thf(fact_100_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.65/1.48 thf(fact_101_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.65/1.48 thf(fact_102_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_103_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (~ ((((ord_less_eq @ A) @ Y2) @ X3))))))))))).
% 2.65/1.48 thf(fact_104_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 2.65/1.48 thf(fact_105_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 2.65/1.48 thf(fact_106_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y:A,Z2:A]: (((! [X4:A]: (((((ord_less @ A) @ X4) @ Y) => (((ord_less_eq @ A) @ X4) @ Z2)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 2.65/1.48 thf(fact_107_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,Y:A]: (((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((ord_less_eq @ A) @ Y) @ X4)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 2.65/1.48 thf(fact_108_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.65/1.48 thf(fact_109_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.65/1.48 thf(fact_110_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.65/1.48 thf(fact_111_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.65/1.48 thf(fact_112_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_113_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_114_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.65/1.48 thf(fact_115_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 2.65/1.48 thf(fact_116_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (((ord_less @ C2) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.65/1.48 thf(fact_117_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.65/1.48 thf(fact_118_distinct__union, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((distinct @ A) @ (((union @ A) @ Xs) @ Ys)) = ((distinct @ A) @ Ys))))).
% 2.65/1.48 thf(fact_119_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((P @ A2) => ((~ ((P @ B2))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B2) & ((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ C3)) => (P @ X2)))) & (! [D2:A]: (((! [X4:A]: ((((((ord_less_eq @ A) @ A2) @ X4) & (((ord_less @ A) @ X4) @ D2)) => (P @ X4)))) => (((ord_less_eq @ A) @ D2) @ C3)))))))))))))))))).
% 2.65/1.48 thf(fact_120_subset__sorted__correct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [L1:(list @ A),L2:(list @ A)]: (((((distinct @ A) @ L1) & ((linorder_sorted @ A) @ L1)) => ((((distinct @ A) @ L2) & ((linorder_sorted @ A) @ L2)) => ((((sorted1061247458sorted @ A) @ L1) @ L2) = (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ L1)) @ ((set2 @ A) @ L2))))))))))).
% 2.65/1.48 thf(fact_121_mergesort__remdups__correct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [L:(list @ A)]: ((((distinct @ A) @ ((mergesort_remdups @ A) @ L)) & (((linorder_sorted @ A) @ ((mergesort_remdups @ A) @ L)) & (((set2 @ A) @ ((mergesort_remdups @ A) @ L)) = ((set2 @ A) @ L)))))))))).
% 2.65/1.48 thf(fact_122_memb__sorted__correct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(list @ A),X:A]: ((((linorder_sorted @ A) @ Xs) => ((((sorted873716653sorted @ A) @ Xs) @ X) = (((member2 @ A) @ X) @ ((set2 @ A) @ Xs)))))))))).
% 2.65/1.48 thf(fact_123_in__set__member, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member2 @ A) @ X) @ ((set2 @ A) @ Xs)) = (((member @ A) @ Xs) @ X))))).
% 2.65/1.48 thf(fact_124_finite__sorted__distinct__unique, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (? [X4:(list @ A)]: (((((set2 @ A) @ X4) = A4) & (((linorder_sorted @ A) @ X4) & (((distinct @ A) @ X4) & (! [Y5:(list @ A)]: ((((((set2 @ A) @ Y5) = A4) & (((linorder_sorted @ A) @ Y5) & ((distinct @ A) @ Y5))) => (Y5 = X4)))))))))))))))).
% 2.65/1.48 thf(fact_125_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B4:A]: (((((ord_less @ A) @ A2) @ B4) | (((ord_less @ A) @ B4) @ A2)))))))))).
% 2.65/1.48 thf(fact_126_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.65/1.48 thf(fact_127_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X2:A]: ((? [X1:A]: ((((ord_less @ A) @ X2) @ X1))))))))).
% 2.65/1.48 thf(fact_128_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X2:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X2))))))))).
% 2.65/1.48 thf(fact_129_List_Ofinite__set, axiom, (! [A:$tType,Xs:(list @ A)]: (((finite_finite2 @ A) @ ((set2 @ A) @ Xs))))).
% 2.65/1.48 thf(fact_130_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G) & (~ ((((ord_less_eq @ (A > B)) @ G) @ F2))))))))))).
% 2.65/1.48 thf(fact_131_subset__Collect__conv, axiom, (! [A:$tType,S:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ S) @ ((collect @ A) @ P)) = (! [X3:A]: (((((member2 @ A) @ X3) @ S) => (P @ X3)))))))).
% 2.65/1.48 thf(fact_132_finite__list, axiom, (! [A:$tType,A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (? [Xs2:(list @ A)]: ((((set2 @ A) @ Xs2) = A4))))))).
% 2.65/1.48 thf(fact_133_finite__distinct__list, axiom, (! [A:$tType,A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (? [Xs2:(list @ A)]: (((((set2 @ A) @ Xs2) = A4) & ((distinct @ A) @ Xs2)))))))).
% 2.65/1.48 thf(fact_134_ord__eq__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A,D3:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => ((C = D3) => (((ord_less_eq @ A) @ A2) @ D3)))))))))).
% 2.65/1.48 thf(fact_135_subsetI, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((! [X4:A]: (((((member2 @ A) @ X4) @ A4) => (((member2 @ A) @ X4) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B3))))).
% 2.65/1.48 thf(fact_136_psubsetI, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((A4 != B3) => (((ord_less @ (set @ A)) @ A4) @ B3)))))).
% 2.65/1.48 thf(fact_137_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A4) => (A4 = B3)))))).
% 2.65/1.48 thf(fact_138_finite__code, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) = (^ [A6:(set @ A)]: ($true))))))).
% 2.65/1.48 thf(fact_139_psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C4) => (((ord_less @ (set @ A)) @ A4) @ C4)))))).
% 2.65/1.48 thf(fact_140_psubsetD, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A4) @ B3) => ((((member2 @ A) @ C) @ A4) => (((member2 @ A) @ C) @ B3)))))).
% 2.65/1.48 thf(fact_141_finite__set__choice, axiom, (! [B:$tType,A:$tType,A4:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A4) => ((! [X4:A]: (((((member2 @ A) @ X4) @ A4) => (? [X12:B]: (((P @ X4) @ X12)))))) => (? [F4:(A > B)]: ((! [X2:A]: (((((member2 @ A) @ X2) @ A4) => ((P @ X2) @ (F4 @ X2)))))))))))).
% 2.65/1.48 thf(fact_142_finite, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => (! [A4:(set @ A)]: (((finite_finite2 @ A) @ A4))))))).
% 2.65/1.48 thf(fact_143_finite__psubset__induct, axiom, (! [A:$tType,A4:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A4) => ((! [A7:(set @ A)]: ((((finite_finite2 @ A) @ A7) => ((! [B6:(set @ A)]: (((((ord_less @ (set @ A)) @ B6) @ A7) => (P @ B6)))) => (P @ A7))))) => (P @ A4)))))).
% 2.65/1.48 thf(fact_144_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B7) | (A6 = B7)))))))).
% 2.65/1.48 thf(fact_145_subset__psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C4) => (((ord_less @ (set @ A)) @ A4) @ C4)))))).
% 2.65/1.48 thf(fact_146_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (~ ((((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))))).
% 2.65/1.48 thf(fact_147_psubset__subset__trans, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C4) => (((ord_less @ (set @ A)) @ A4) @ C4)))))).
% 2.65/1.48 thf(fact_148_psubset__imp__subset, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B3) => (((ord_less_eq @ (set @ A)) @ A4) @ B3))))).
% 2.65/1.48 thf(fact_149_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 2.65/1.48 thf(fact_150_contra__subsetD, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((~ ((((member2 @ A) @ C) @ B3))) => (~ ((((member2 @ A) @ C) @ A4)))))))).
% 2.65/1.48 thf(fact_151_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z3:(set @ A)]: ((Y4 = Z3))) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))).
% 2.65/1.48 thf(fact_152_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C4) => (((ord_less_eq @ (set @ A)) @ A4) @ C4)))))).
% 2.65/1.48 thf(fact_153_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.65/1.48 thf(fact_154_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 2.65/1.48 thf(fact_155_rev__subsetD, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member2 @ A) @ C) @ A4) => ((((ord_less_eq @ (set @ A)) @ A4) @ B3) => (((member2 @ A) @ C) @ B3)))))).
% 2.65/1.48 thf(fact_156_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [T2:A]: (((((member2 @ A) @ T2) @ A6) => (((member2 @ A) @ T2) @ B7)))))))))).
% 2.65/1.48 thf(fact_157_set__rev__mp, axiom, (! [A:$tType,X:A,A4:(set @ A),B3:(set @ A)]: (((((member2 @ A) @ X) @ A4) => ((((ord_less_eq @ (set @ A)) @ A4) @ B3) => (((member2 @ A) @ X) @ B3)))))).
% 2.65/1.48 thf(fact_158_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (A6 != B7)))))))).
% 2.65/1.48 thf(fact_159_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((A4 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A4))))).
% 2.65/1.48 thf(fact_160_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((A4 = B3) => (((ord_less_eq @ (set @ A)) @ A4) @ B3))))).
% 2.65/1.48 thf(fact_161_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [X3:A]: (((((member2 @ A) @ X3) @ A6) => (((member2 @ A) @ X3) @ B7)))))))))).
% 2.65/1.48 thf(fact_162_equalityE, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((A4 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A4)))))))))).
% 2.65/1.48 thf(fact_163_subsetCE, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((member2 @ A) @ C) @ A4) => (((member2 @ A) @ C) @ B3)))))).
% 2.65/1.48 thf(fact_164_psubsetE, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B3) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A4)))))))).
% 2.65/1.48 thf(fact_165_subsetD, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((member2 @ A) @ C) @ A4) => (((member2 @ A) @ C) @ B3)))))).
% 2.65/1.48 thf(fact_166_in__mono, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((member2 @ A) @ X) @ A4) => (((member2 @ A) @ X) @ B3)))))).
% 2.65/1.48 thf(fact_167_set__mp, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((member2 @ A) @ X) @ A4) => (((member2 @ A) @ X) @ B3)))))).
% 2.65/1.48 thf(fact_168_rev__finite__subset, axiom, (! [A:$tType,B3:(set @ A),A4:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((finite_finite2 @ A) @ A4)))))).
% 2.65/1.48 thf(fact_169_infinite__super, axiom, (! [A:$tType,S:(set @ A),T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S) @ T3) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ T3)))))))).
% 2.65/1.48 thf(fact_170_finite__subset, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => (((finite_finite2 @ A) @ B3) => ((finite_finite2 @ A) @ A4)))))).
% 2.65/1.48 thf(fact_171_sorted__list__of__set, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) => ((((set2 @ A) @ ((linord467138063of_set @ A) @ A4)) = A4) & (((linorder_sorted @ A) @ ((linord467138063of_set @ A) @ A4)) & ((distinct @ A) @ ((linord467138063of_set @ A) @ A4))))))))))).
% 2.65/1.48 thf(fact_172_set__union, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((set2 @ A) @ (((union @ A) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)))))).
% 2.65/1.48 thf(fact_173_subset__code_I2_J, axiom, (! [B:$tType,A4:(set @ B),Ys:(list @ B)]: (((((ord_less_eq @ (set @ B)) @ A4) @ ((coset @ B) @ Ys)) = (! [X3:B]: (((((member2 @ B) @ X3) @ ((set2 @ B) @ Ys)) => (~ ((((member2 @ B) @ X3) @ A4)))))))))).
% 2.65/1.48 thf(fact_174_sorted__by__rel__linord, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [L:(list @ A)]: (((((sorted_by_rel @ A) @ (ord_less_eq @ A)) @ L) = ((linorder_sorted @ A) @ L)))))))).
% 2.65/1.48 thf(fact_175_UnCI, axiom, (! [A:$tType,C:A,B3:(set @ A),A4:(set @ A)]: ((((~ ((((member2 @ A) @ C) @ B3))) => (((member2 @ A) @ C) @ A4)) => (((member2 @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B3)))))).
% 2.65/1.48 thf(fact_176_Un__iff, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member2 @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) = ((((member2 @ A) @ C) @ A4) | (((member2 @ A) @ C) @ B3)))))).
% 2.65/1.48 thf(fact_177_finite__Un, axiom, (! [A:$tType,F3:(set @ A),G3:(set @ A)]: ((((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ F3) @ G3)) = (((finite_finite2 @ A) @ F3) & ((finite_finite2 @ A) @ G3)))))).
% 2.65/1.48 thf(fact_178_Un__subset__iff, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) @ C4) = ((((ord_less_eq @ (set @ A)) @ A4) @ C4) & (((ord_less_eq @ (set @ A)) @ B3) @ C4)))))).
% 2.65/1.48 thf(fact_179_Un__mono, axiom, (! [A:$tType,A4:(set @ A),C4:(set @ A),B3:(set @ A),D4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ C4) => ((((ord_less_eq @ (set @ A)) @ B3) @ D4) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) @ (((sup_sup @ (set @ A)) @ C4) @ D4))))))).
% 2.65/1.48 thf(fact_180_Un__least, axiom, (! [A:$tType,A4:(set @ A),C4:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ C4) => ((((ord_less_eq @ (set @ A)) @ B3) @ C4) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) @ C4)))))).
% 2.65/1.48 thf(fact_181_Un__upper1, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ A4) @ B3))))).
% 2.65/1.48 thf(fact_182_Un__upper2, axiom, (! [A:$tType,B3:(set @ A),A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B3) @ (((sup_sup @ (set @ A)) @ A4) @ B3))))).
% 2.65/1.48 thf(fact_183_Un__absorb1, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((sup_sup @ (set @ A)) @ A4) @ B3) = B3))))).
% 2.65/1.48 thf(fact_184_Un__absorb2, axiom, (! [A:$tType,B3:(set @ A),A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A4) => ((((sup_sup @ (set @ A)) @ A4) @ B3) = A4))))).
% 2.65/1.48 thf(fact_185_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ B7) = B7))))))).
% 2.65/1.48 thf(fact_186_infinite__Un, axiom, (! [A:$tType,S:(set @ A),T3:(set @ A)]: (((~ (((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ S) @ T3)))) = ((~ (((finite_finite2 @ A) @ S))) | (~ (((finite_finite2 @ A) @ T3)))))))).
% 2.65/1.48 thf(fact_187_Un__infinite, axiom, (! [A:$tType,S:(set @ A),T3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ S) @ T3)))))))).
% 2.65/1.48 thf(fact_188_finite__UnI, axiom, (! [A:$tType,F3:(set @ A),G3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => (((finite_finite2 @ A) @ G3) => ((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ F3) @ G3))))))).
% 2.65/1.48 thf(fact_189_UnE, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member2 @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) => ((~ ((((member2 @ A) @ C) @ A4))) => (((member2 @ A) @ C) @ B3)))))).
% 2.65/1.48 thf(fact_190_UnI1, axiom, (! [A:$tType,C:A,A4:(set @ A),B3:(set @ A)]: (((((member2 @ A) @ C) @ A4) => (((member2 @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B3)))))).
% 2.65/1.48 thf(fact_191_UnI2, axiom, (! [A:$tType,C:A,B3:(set @ A),A4:(set @ A)]: (((((member2 @ A) @ C) @ B3) => (((member2 @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B3)))))).
% 2.65/1.48 thf(fact_192_bex__Un, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),P:(A > $o)]: (((? [X3:A]: (((((member2 @ A) @ X3) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) & (P @ X3)))) = ((? [X3:A]: (((((member2 @ A) @ X3) @ A4) & (P @ X3)))) | (? [X3:A]: (((((member2 @ A) @ X3) @ B3) & (P @ X3))))))))).
% 2.65/1.48 thf(fact_193_ball__Un, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),P:(A > $o)]: (((! [X3:A]: (((((member2 @ A) @ X3) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) => (P @ X3)))) = ((! [X3:A]: (((((member2 @ A) @ X3) @ A4) => (P @ X3)))) & (! [X3:A]: (((((member2 @ A) @ X3) @ B3) => (P @ X3))))))))).
% 2.65/1.48 thf(fact_194_Un__assoc, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) @ C4) = (((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B3) @ C4)))))).
% 2.65/1.48 thf(fact_195_Un__absorb, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ A4) = A4)))).
% 2.65/1.48 thf(fact_196_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((((sup_sup @ (set @ A)) @ B7) @ A6))))))).
% 2.65/1.48 thf(fact_197_Un__left__absorb, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) = (((sup_sup @ (set @ A)) @ A4) @ B3))))).
% 2.65/1.48 thf(fact_198_Un__left__commute, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B3) @ C4)) = (((sup_sup @ (set @ A)) @ B3) @ (((sup_sup @ (set @ A)) @ A4) @ C4)))))).
% 2.65/1.48 thf(fact_199_sorted__by__rel__weaken, axiom, (! [A:$tType,L0:(list @ A),R:(A > (A > $o)),R2:(A > (A > $o))]: (((! [X4:A,Y3:A]: (((((member2 @ A) @ X4) @ ((set2 @ A) @ L0)) => ((((member2 @ A) @ Y3) @ ((set2 @ A) @ L0)) => (((R @ X4) @ Y3) => ((R2 @ X4) @ Y3)))))) => ((((sorted_by_rel @ A) @ R) @ L0) => (((sorted_by_rel @ A) @ R2) @ L0)))))).
% 2.65/1.48 thf(fact_200_distinct__sorted__list__of__set, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:(set @ A)]: (((distinct @ A) @ ((linord467138063of_set @ A) @ A4)))))))).
% 2.65/1.48 thf(fact_201_sup_Obounded__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,C:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C)) @ A2) = ((((ord_less_eq @ A) @ B2) @ A2) & (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.65/1.48 thf(fact_202_le__sup__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z2) = ((((ord_less_eq @ A) @ X) @ Z2) & (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 2.65/1.48 thf(fact_203_sup_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ B2) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.65/1.48 thf(fact_204_sup__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G:(A > B),X3:A]: ((((sup_sup @ B) @ (F2 @ X3)) @ (G @ X3))))))))).
% 2.65/1.48 thf(fact_205_sup_Oidem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 2.65/1.48 thf(fact_206_sup__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 2.65/1.48 thf(fact_207_sup_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.65/1.48 thf(fact_208_sup__left__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_209_inf__sup__aci_I8_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_210_inf__sup__aci_I7_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z2)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z2))))))))).
% 2.65/1.48 thf(fact_211_inf__sup__aci_I6_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z2) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z2))))))))).
% 2.65/1.48 thf(fact_212_inf__sup__aci_I5_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X3:A,Y2:A]: ((((sup_sup @ A) @ Y2) @ X3)))))))).
% 2.65/1.48 thf(fact_213_sup__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G:(A > B),X3:A]: ((((sup_sup @ B) @ (F2 @ X3)) @ (G @ X3))))))))).
% 2.65/1.48 thf(fact_214_sup_Oassoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ C) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C))))))))).
% 2.65/1.48 thf(fact_215_sup__assoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z2) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z2))))))))).
% 2.65/1.48 thf(fact_216_sup_Ocommute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [A5:A,B5:A]: ((((sup_sup @ A) @ B5) @ A5)))))))).
% 2.65/1.48 thf(fact_217_sup__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X3:A,Y2:A]: ((((sup_sup @ A) @ Y2) @ X3)))))))).
% 2.65/1.48 thf(fact_218_sup_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((sup_sup @ A) @ B2) @ (((sup_sup @ A) @ A2) @ C)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C))))))))).
% 2.65/1.48 thf(fact_219_sup__left__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z2:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z2)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z2))))))))).
% 2.65/1.48 thf(fact_220_inf__sup__ord_I4_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_221_inf__sup__ord_I3_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_222_le__supE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A,X:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ X) => (~ (((((ord_less_eq @ A) @ A2) @ X) => (~ ((((ord_less_eq @ A) @ B2) @ X))))))))))))).
% 2.65/1.48 thf(fact_223_le__supI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,X:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ X) => ((((ord_less_eq @ A) @ B2) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ X))))))))).
% 2.65/1.48 thf(fact_224_sup__ge1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_225_sup__ge2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.65/1.48 thf(fact_226_le__supI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X) @ A2) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_227_le__supI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ X) @ B2) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_228_sup_Omono, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C:A,A2:A,D3:A,B2:A]: (((((ord_less_eq @ A) @ C) @ A2) => ((((ord_less_eq @ A) @ D3) @ B2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C) @ D3)) @ (((sup_sup @ A) @ A2) @ B2)))))))))).
% 2.65/1.48 thf(fact_229_sup__mono, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A,D3:A]: (((((ord_less_eq @ A) @ A2) @ C) => ((((ord_less_eq @ A) @ B2) @ D3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ (((sup_sup @ A) @ C) @ D3)))))))))).
% 2.65/1.48 thf(fact_230_sup__least, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A,Z2:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ Z2) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y) @ Z2)) @ X))))))))).
% 2.65/1.48 thf(fact_231_le__iff__sup, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X3:A,Y2:A]: (((((sup_sup @ A) @ X3) @ Y2) = Y2)))))))).
% 2.65/1.48 thf(fact_232_sup_OorderE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (A2 = (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_233_sup_OorderI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 = (((sup_sup @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.65/1.48 thf(fact_234_sup__unique, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [F:(A > (A > A)),X:A,Y:A]: (((! [X4:A,Y3:A]: ((((ord_less_eq @ A) @ X4) @ ((F @ X4) @ Y3)))) => ((! [X4:A,Y3:A]: ((((ord_less_eq @ A) @ Y3) @ ((F @ X4) @ Y3)))) => ((! [X4:A,Y3:A,Z:A]: (((((ord_less_eq @ A) @ Y3) @ X4) => ((((ord_less_eq @ A) @ Z) @ X4) => (((ord_less_eq @ A) @ ((F @ Y3) @ Z)) @ X4))))) => ((((sup_sup @ A) @ X) @ Y) = ((F @ X) @ Y))))))))))).
% 2.65/1.48 thf(fact_235_sup_Oabsorb1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((sup_sup @ A) @ A2) @ B2) = A2)))))))).
% 2.65/1.48 thf(fact_236_sup_Oabsorb2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((sup_sup @ A) @ A2) @ B2) = B2)))))))).
% 2.65/1.48 thf(fact_237_sup__absorb1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((sup_sup @ A) @ X) @ Y) = X)))))))).
% 2.65/1.48 thf(fact_238_sup__absorb2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((sup_sup @ A) @ X) @ Y) = Y)))))))).
% 2.65/1.48 thf(fact_239_sup_OboundedE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,C:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C)) @ A2) => (~ (((((ord_less_eq @ A) @ B2) @ A2) => (~ ((((ord_less_eq @ A) @ C) @ A2))))))))))))).
% 2.65/1.48 thf(fact_240_sup_OboundedI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ A2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C)) @ A2))))))))).
% 2.65/1.48 thf(fact_241_sup_Oorder__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: ((A5 = (((sup_sup @ A) @ A5) @ B5))))))))).
% 2.65/1.48 thf(fact_242_sup_Ocobounded1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.65/1.48 thf(fact_243_sup_Ocobounded2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: ((((ord_less_eq @ A) @ B2) @ (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.65/1.48 thf(fact_244_sup_Oabsorb__iff1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: (((((sup_sup @ A) @ A5) @ B5) = A5)))))))).
% 2.65/1.48 thf(fact_245_sup_Oabsorb__iff2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: (((((sup_sup @ A) @ A5) @ B5) = B5)))))))).
% 2.65/1.48 thf(fact_246_sup_OcoboundedI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ C) @ A2) => (((ord_less_eq @ A) @ C) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_247_sup_OcoboundedI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_248_sup_Ostrict__coboundedI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C:A,B2:A,A2:A]: (((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_249_sup_Ostrict__coboundedI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ C) @ A2) => (((ord_less @ A) @ C) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_250_sup_Ostrict__order__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B5:A,A5:A]: (((A5 = (((sup_sup @ A) @ A5) @ B5)) & (A5 != B5))))))))).
% 2.65/1.48 thf(fact_251_sup_Ostrict__boundedE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,C:A,A2:A]: (((((ord_less @ A) @ (((sup_sup @ A) @ B2) @ C)) @ A2) => (~ (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ C) @ A2))))))))))))).
% 2.65/1.48 thf(fact_252_less__supI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,B2:A,A2:A]: (((((ord_less @ A) @ X) @ B2) => (((ord_less @ A) @ X) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_253_less__supI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,A2:A,B2:A]: (((((ord_less @ A) @ X) @ A2) => (((ord_less @ A) @ X) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.65/1.48 thf(fact_254_merge__correct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [L1:(list @ A),L2:(list @ A)]: (((((distinct @ A) @ L1) & ((linorder_sorted @ A) @ L1)) => ((((distinct @ A) @ L2) & ((linorder_sorted @ A) @ L2)) => (((distinct @ A) @ (((merge @ A) @ L1) @ L2)) & (((linorder_sorted @ A) @ (((merge @ A) @ L1) @ L2)) & (((set2 @ A) @ (((merge @ A) @ L1) @ L2)) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ L1)) @ ((set2 @ A) @ L2))))))))))))).
% 2.65/1.48 thf(fact_255_set__mergesort__by__rel__merge, axiom, (! [A:$tType,R:(A > (A > $o)),Xs:(list @ A),Ys:(list @ A)]: ((((set2 @ A) @ ((((merges2133357844_merge @ A) @ R) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)))))).
% 2.65/1.48 thf(subcl_Orderings_Olinorder___HOL_Otype, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((type @ A) @ (type2 @ A)))))).
% 2.65/1.48 thf(subcl_Orderings_Olinorder___Orderings_Oord, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((ord @ A) @ (type2 @ A)))))).
% 2.65/1.48 thf(subcl_Orderings_Olinorder___Orderings_Oorder, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((order @ A) @ (type2 @ A)))))).
% 2.65/1.48 thf(subcl_Orderings_Olinorder___Orderings_Opreorder, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((preorder @ A) @ (type2 @ A)))))).
% 2.65/1.48 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A8:$tType,A9:$tType]: ((((semilattice_sup @ A9) @ (type2 @ A9)) => ((semilattice_sup @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.65/1.48 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: ((((preorder @ A9) @ (type2 @ A9)) => ((preorder @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.65/1.48 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: (((((finite_finite @ A8) @ (type2 @ A8)) & ((finite_finite @ A9) @ (type2 @ A9))) => ((finite_finite @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.65/1.48 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: ((((lattice @ A9) @ (type2 @ A9)) => ((lattice @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.65/1.48 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: ((((order @ A9) @ (type2 @ A9)) => ((order @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.65/1.48 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: ((((ord @ A9) @ (type2 @ A9)) => ((ord @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.65/1.48 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1, axiom, (! [A8:$tType]: (((semilattice_sup @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.65/1.48 thf(tcon_Set_Oset___Orderings_Opreorder_2, axiom, (! [A8:$tType]: (((preorder @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.65/1.48 thf(tcon_Set_Oset___Finite__Set_Ofinite_3, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((finite_finite @ (set @ A8)) @ (type2 @ (set @ A8))))))).
% 2.65/1.48 thf(tcon_Set_Oset___Lattices_Olattice_4, axiom, (! [A8:$tType]: (((lattice @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.65/1.48 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A8:$tType]: (((order @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.65/1.48 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A8:$tType]: (((ord @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.65/1.48 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 2.65/1.48 thf(tcon_HOL_Obool___Orderings_Opreorder_8, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.65/1.48 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.65/1.48 thf(tcon_HOL_Obool___Finite__Set_Ofinite_9, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 2.65/1.48 thf(tcon_HOL_Obool___Lattices_Olattice_10, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 2.65/1.48 thf(tcon_HOL_Obool___Orderings_Oorder_11, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.65/1.48 thf(tcon_HOL_Obool___Orderings_Oord_12, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.65/1.48 thf(tfree_0, hypothesis, ((linorder @ a) @ (type2 @ a))).
% 2.65/1.48 thf(conj_0, hypothesis, (((distinct @ a) @ (((sorted763967275sorted @ a) @ x) @ xsa)) & (((linorder_sorted @ a) @ (((sorted763967275sorted @ a) @ x) @ xsa)) & (! [X2:a]: (((((member2 @ a) @ X2) @ ((set2 @ a) @ (((sorted763967275sorted @ a) @ x) @ xsa))) = ((((member2 @ a) @ X2) @ ((set2 @ a) @ xsa)) & (X2 != x)))))))).
% 2.65/1.48 thf(conj_1, hypothesis, (((linorder_sorted @ a) @ xsa) & (! [X2:a]: (((((member2 @ a) @ X2) @ ((set2 @ a) @ xsa)) => (((ord_less_eq @ a) @ a2) @ X2)))))).
% 2.65/1.48 thf(conj_2, hypothesis, ((~ ((((member2 @ a) @ a2) @ ((set2 @ a) @ xsa)))) & ((distinct @ a) @ xsa))).
% 2.65/1.48 thf(conj_3, conjecture, (((x != a2) | (! [X4:a]: (((((member2 @ a) @ X4) @ ((set2 @ a) @ xsa)) = (((X4 = a2) | (((member2 @ a) @ X4) @ ((set2 @ a) @ xsa))) & (X4 != a2)))))) & ((x = a2) | (((~ ((((ord_less @ a) @ a2) @ x))) | (! [X4:a]: ((((X4 = a2) | ((((member2 @ a) @ X4) @ ((set2 @ a) @ xsa)) & (X4 != x))) = (((X4 = a2) | (((member2 @ a) @ X4) @ ((set2 @ a) @ xsa))) & (X4 != x)))))) & ((((ord_less @ a) @ a2) @ x) | (! [X4:a]: ((((X4 = a2) | (((member2 @ a) @ X4) @ ((set2 @ a) @ xsa))) = (((X4 = a2) | (((member2 @ a) @ X4) @ ((set2 @ a) @ xsa))) & (X4 != x)))))))))).
% 2.65/1.48 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.brU5k5c5ot/DTF2DTF27099.p
% 2.65/1.48 ------------------------
% 2.93/1.51 ---- Cleaned THF ---
% 2.93/1.51 thf(ty_t_List_Olist,type,
% 2.93/1.51 list: $tType > $tType ).
% 2.93/1.51
% 2.93/1.51 thf(ty_t_Set_Oset,type,
% 2.93/1.51 set: $tType > $tType ).
% 2.93/1.51
% 2.93/1.51 thf(ty_t_itself,type,
% 2.93/1.51 itself: $tType > $tType ).
% 2.93/1.51
% 2.93/1.51 thf(ty_tf_a,type,
% 2.93/1.51 a: $tType ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_HOL_Otype,type,
% 2.93/1.51 type:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Oord,type,
% 2.93/1.51 ord:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Oorder,type,
% 2.93/1.51 order:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Lattices_Olattice,type,
% 2.93/1.51 lattice:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Ono__bot,type,
% 2.93/1.51 no_bot:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Ono__top,type,
% 2.93/1.51 no_top:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.93/1.51 finite_finite:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Olinorder,type,
% 2.93/1.51 linorder:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Opreorder,type,
% 2.93/1.51 preorder:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Owellorder,type,
% 2.93/1.51 wellorder:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Odense__order,type,
% 2.93/1.51 dense_order:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.93/1.51 linordered_idom:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Fields_Olinordered__field,type,
% 2.93/1.51 linordered_field:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.93/1.51 semilattice_sup:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.93/1.51 dense_linorder:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.93/1.51 condit1656338222tinuum:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.93/1.51 condit1037483654norder:
% 2.93/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Finite__Set_Ofinite,type,
% 2.93/1.51 finite_finite2:
% 2.93/1.51 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.93/1.51 sup_sup:
% 2.93/1.51 !>[A: $tType] : ( A > A > A ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_List_Ocoset,type,
% 2.93/1.51 coset:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_List_Odistinct,type,
% 2.93/1.51 distinct:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_List_Olinorder__class_Osorted,type,
% 2.93/1.51 linorder_sorted:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
% 2.93/1.51 linord467138063of_set:
% 2.93/1.51 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_List_Olist_Oset,type,
% 2.93/1.51 set2:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_List_Omember,type,
% 2.93/1.51 member:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > A > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_List_Ounion,type,
% 2.93/1.51 union:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Misc_Omerge,type,
% 2.93/1.51 merge:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Misc_Omergesort__by__rel__merge,type,
% 2.93/1.51 merges2133357844_merge:
% 2.93/1.51 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Misc_Omergesort__remdups,type,
% 2.93/1.51 mergesort_remdups:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Misc_Osorted__by__rel,type,
% 2.93/1.51 sorted_by_rel:
% 2.93/1.51 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.93/1.51 ord_less:
% 2.93/1.51 !>[A: $tType] : ( A > A > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.93/1.51 ord_less_eq:
% 2.93/1.51 !>[A: $tType] : ( A > A > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Pure_Otype,type,
% 2.93/1.51 type2:
% 2.93/1.51 !>[A: $tType] : ( itself @ A ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Set_OCollect,type,
% 2.93/1.51 collect:
% 2.93/1.51 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Odelete__sorted,type,
% 2.93/1.51 sorted763967275sorted:
% 2.93/1.51 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Omemb__sorted,type,
% 2.93/1.51 sorted873716653sorted:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > A > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Osubset__sorted,type,
% 2.93/1.51 sorted1061247458sorted:
% 2.93/1.51 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_c_member,type,
% 2.93/1.51 member2:
% 2.93/1.51 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(sy_v_a,type,
% 2.93/1.51 a2: a ).
% 2.93/1.51
% 2.93/1.51 thf(sy_v_x,type,
% 2.93/1.51 x: a ).
% 2.93/1.51
% 2.93/1.51 thf(sy_v_xsa,type,
% 2.93/1.51 xsa: list @ a ).
% 2.93/1.51
% 2.93/1.51 thf(fact_0_set__eq__sorted__correct,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [L1: list @ A,L2: list @ A] :
% 2.93/1.51 ( ( ( distinct @ A @ L1 )
% 2.93/1.51 & ( linorder_sorted @ A @ L1 ) )
% 2.93/1.51 => ( ( ( distinct @ A @ L2 )
% 2.93/1.51 & ( linorder_sorted @ A @ L2 ) )
% 2.93/1.51 => ( ( L1 = L2 )
% 2.93/1.51 = ( ( set2 @ A @ L1 )
% 2.93/1.51 = ( set2 @ A @ L2 ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_1_sorted__distinct__set__unique,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Xs: list @ A,Ys: list @ A] :
% 2.93/1.51 ( ( linorder_sorted @ A @ Xs )
% 2.93/1.51 => ( ( distinct @ A @ Xs )
% 2.93/1.51 => ( ( linorder_sorted @ A @ Ys )
% 2.93/1.51 => ( ( distinct @ A @ Ys )
% 2.93/1.51 => ( ( ( set2 @ A @ Xs )
% 2.93/1.51 = ( set2 @ A @ Ys ) )
% 2.93/1.51 => ( Xs = Ys ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_2_order__refl,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_3_minf_I8_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X2 @ Z )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_4_minf_I6_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X2 @ Z )
% 2.93/1.51 => ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_5_pinf_I8_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z @ X2 )
% 2.93/1.51 => ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_6_pinf_I6_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z @ X2 )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_7_leD,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,X: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.51 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_8_leI,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ~ ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_9_le__less,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less_eq @ A )
% 2.93/1.51 = ( ^ [X3: A,Y2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X3 @ Y2 )
% 2.93/1.51 | ( X3 = Y2 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_10_less__le,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less @ A )
% 2.93/1.51 = ( ^ [X3: A,Y2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.93/1.51 & ( X3 != Y2 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_11_order__le__less__subst1,axiom,
% 2.93/1.51 ! [A: $tType,B: $tType] :
% 2.93/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.93/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.93/1.51 => ( ( ord_less @ B @ B2 @ C )
% 2.93/1.51 => ( ! [X4: B,Y3: B] :
% 2.93/1.51 ( ( ord_less @ B @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_12_order__le__less__subst2,axiom,
% 2.93/1.51 ! [A: $tType,C2: $tType] :
% 2.93/1.51 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.93/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 2.93/1.51 => ( ! [X4: A,Y3: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_13_subset__code_I1_J,axiom,
% 2.93/1.51 ! [A: $tType,Xs: list @ A,B3: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 )
% 2.93/1.51 = ( ! [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.93/1.51 => ( member2 @ A @ X3 @ B3 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_14_dual__order_Oantisym,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( A2 = B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_15_dual__order_Otrans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.93/1.51 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_16_linorder__wlog,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.93/1.51 ( ! [A3: A,B4: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.93/1.51 => ( P @ A3 @ B4 ) )
% 2.93/1.51 => ( ! [A3: A,B4: A] :
% 2.93/1.51 ( ( P @ B4 @ A3 )
% 2.93/1.51 => ( P @ A3 @ B4 ) )
% 2.93/1.51 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_17_dual__order_Orefl,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_18_order__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.93/1.51 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_19_order__class_Oorder_Oantisym,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 => ( A2 = B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_20_ord__le__eq__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( B2 = C )
% 2.93/1.51 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_21_ord__eq__le__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( A2 = B2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.93/1.51 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_22_antisym__conv,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,X: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.51 => ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 = ( X = Y ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_23_le__cases3,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 2.93/1.51 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.93/1.51 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 2.93/1.51 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.93/1.51 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.93/1.51 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_24_order_Otrans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.93/1.51 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_25_le__cases,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_26_eq__refl,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( X = Y )
% 2.93/1.51 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_27_linear,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_28_antisym,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 => ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.51 => ( X = Y ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_29_eq__iff,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.93/1.51 = ( ^ [X3: A,Y2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.93/1.51 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_30_ord__le__eq__subst,axiom,
% 2.93/1.51 ! [A: $tType,B: $tType] :
% 2.93/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,B2: A,F: A > B,C: B] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ( F @ B2 )
% 2.93/1.51 = C )
% 2.93/1.51 => ( ! [X4: A,Y3: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_31_ord__eq__le__subst,axiom,
% 2.93/1.51 ! [A: $tType,B: $tType] :
% 2.93/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.93/1.51 ( ( A2
% 2.93/1.51 = ( F @ B2 ) )
% 2.93/1.51 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.93/1.51 => ( ! [X4: B,Y3: B] :
% 2.93/1.51 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_32_order__subst2,axiom,
% 2.93/1.51 ! [A: $tType,C2: $tType] :
% 2.93/1.51 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.93/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 2.93/1.51 => ( ! [X4: A,Y3: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_33_order__subst1,axiom,
% 2.93/1.51 ! [A: $tType,B: $tType] :
% 2.93/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.93/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.93/1.51 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.93/1.51 => ( ! [X4: B,Y3: B] :
% 2.93/1.51 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_34_le__fun__def,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType] :
% 2.93/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 => ( ( ord_less_eq @ ( A > B ) )
% 2.93/1.51 = ( ^ [F2: A > B,G: A > B] :
% 2.93/1.51 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_35_le__funI,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType] :
% 2.93/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 => ! [F: A > B,G2: A > B] :
% 2.93/1.51 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) )
% 2.93/1.51 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_36_le__funE,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType] :
% 2.93/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 => ! [F: A > B,G2: A > B,X: A] :
% 2.93/1.51 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.93/1.51 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_37_le__funD,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType] :
% 2.93/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 => ! [F: A > B,G2: A > B,X: A] :
% 2.93/1.51 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.93/1.51 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_38_dual__order_Ostrict__implies__not__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less @ A @ B2 @ A2 )
% 2.93/1.51 => ( A2 != B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_39_order_Ostrict__implies__not__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( A2 != B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_40_not__less__iff__gr__or__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.93/1.51 = ( ( ord_less @ A @ Y @ X )
% 2.93/1.51 | ( X = Y ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_41_dual__order_Ostrict__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A,C: A] :
% 2.93/1.51 ( ( ord_less @ A @ B2 @ A2 )
% 2.93/1.51 => ( ( ord_less @ A @ C @ B2 )
% 2.93/1.51 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_42_less__imp__not__less,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_43_order_Ostrict__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less @ A @ B2 @ C )
% 2.93/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_44_dual__order_Oirrefl,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A] :
% 2.93/1.51 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_45_mem__Collect__eq,axiom,
% 2.93/1.51 ! [A: $tType,A2: A,P: A > $o] :
% 2.93/1.51 ( ( member2 @ A @ A2 @ ( collect @ A @ P ) )
% 2.93/1.51 = ( P @ A2 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_46_Collect__mem__eq,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A] :
% 2.93/1.51 ( ( collect @ A
% 2.93/1.51 @ ^ [X3: A] : ( member2 @ A @ X3 @ A4 ) )
% 2.93/1.51 = A4 ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_47_Collect__cong,axiom,
% 2.93/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.93/1.51 ( ! [X4: A] :
% 2.93/1.51 ( ( P @ X4 )
% 2.93/1.51 = ( Q @ X4 ) )
% 2.93/1.51 => ( ( collect @ A @ P )
% 2.93/1.51 = ( collect @ A @ Q ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_48_ext,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType,F: A > B,G2: A > B] :
% 2.93/1.51 ( ! [X4: A] :
% 2.93/1.51 ( ( F @ X4 )
% 2.93/1.51 = ( G2 @ X4 ) )
% 2.93/1.51 => ( F = G2 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_49_linorder__cases,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ~ ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ( X != Y )
% 2.93/1.51 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_50_less__imp__triv,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,P: $o] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ( ord_less @ A @ Y @ X )
% 2.93/1.51 => P ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_51_less__imp__not__eq2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( Y != X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_52_antisym__conv3,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,X: A] :
% 2.93/1.51 ( ~ ( ord_less @ A @ Y @ X )
% 2.93/1.51 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.93/1.51 = ( X = Y ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_53_less__induct,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [P: A > $o,A2: A] :
% 2.93/1.51 ( ! [X4: A] :
% 2.93/1.51 ( ! [Y5: A] :
% 2.93/1.51 ( ( ord_less @ A @ Y5 @ X4 )
% 2.93/1.51 => ( P @ Y5 ) )
% 2.93/1.51 => ( P @ X4 ) )
% 2.93/1.51 => ( P @ A2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_54_less__not__sym,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_55_less__imp__not__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( X != Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_56_dual__order_Oasym,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less @ A @ B2 @ A2 )
% 2.93/1.51 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_57_ord__less__eq__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( B2 = C )
% 2.93/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_58_ord__eq__less__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( A2 = B2 )
% 2.93/1.51 => ( ( ord_less @ A @ B2 @ C )
% 2.93/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_59_less__irrefl,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A] :
% 2.93/1.51 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_60_less__linear,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 | ( X = Y )
% 2.93/1.51 | ( ord_less @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_61_less__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ( ord_less @ A @ Y @ Z2 )
% 2.93/1.51 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_62_less__asym_H,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_63_less__asym,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_64_less__imp__neq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( X != Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_65_dense,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ? [Z: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Z )
% 2.93/1.51 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_66_order_Oasym,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_67_neq__iff,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( X != Y )
% 2.93/1.51 = ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_68_neqE,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( X != Y )
% 2.93/1.51 => ( ~ ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_69_gt__ex,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( no_top @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A] :
% 2.93/1.51 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_70_lt__ex,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A] :
% 2.93/1.51 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_71_order__less__subst2,axiom,
% 2.93/1.51 ! [A: $tType,C2: $tType] :
% 2.93/1.51 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.93/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 2.93/1.51 => ( ! [X4: A,Y3: A] :
% 2.93/1.51 ( ( ord_less @ A @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_72_order__less__subst1,axiom,
% 2.93/1.51 ! [A: $tType,B: $tType] :
% 2.93/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.93/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.93/1.51 => ( ( ord_less @ B @ B2 @ C )
% 2.93/1.51 => ( ! [X4: B,Y3: B] :
% 2.93/1.51 ( ( ord_less @ B @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_73_ord__less__eq__subst,axiom,
% 2.93/1.51 ! [A: $tType,B: $tType] :
% 2.93/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,B2: A,F: A > B,C: B] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ( F @ B2 )
% 2.93/1.51 = C )
% 2.93/1.51 => ( ! [X4: A,Y3: A] :
% 2.93/1.51 ( ( ord_less @ A @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_74_ord__eq__less__subst,axiom,
% 2.93/1.51 ! [A: $tType,B: $tType] :
% 2.93/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.93/1.51 ( ( A2
% 2.93/1.51 = ( F @ B2 ) )
% 2.93/1.51 => ( ( ord_less @ B @ B2 @ C )
% 2.93/1.51 => ( ! [X4: B,Y3: B] :
% 2.93/1.51 ( ( ord_less @ B @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_75_pinf_I1_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.93/1.51 ( ? [Z4: A] :
% 2.93/1.51 ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z4 @ X4 )
% 2.93/1.51 => ( ( P @ X4 )
% 2.93/1.51 = ( P2 @ X4 ) ) )
% 2.93/1.51 => ( ? [Z4: A] :
% 2.93/1.51 ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z4 @ X4 )
% 2.93/1.51 => ( ( Q @ X4 )
% 2.93/1.51 = ( Q2 @ X4 ) ) )
% 2.93/1.51 => ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z @ X2 )
% 2.93/1.51 => ( ( ( P @ X2 )
% 2.93/1.51 & ( Q @ X2 ) )
% 2.93/1.51 = ( ( P2 @ X2 )
% 2.93/1.51 & ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_76_pinf_I2_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.93/1.51 ( ? [Z4: A] :
% 2.93/1.51 ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z4 @ X4 )
% 2.93/1.51 => ( ( P @ X4 )
% 2.93/1.51 = ( P2 @ X4 ) ) )
% 2.93/1.51 => ( ? [Z4: A] :
% 2.93/1.51 ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z4 @ X4 )
% 2.93/1.51 => ( ( Q @ X4 )
% 2.93/1.51 = ( Q2 @ X4 ) ) )
% 2.93/1.51 => ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z @ X2 )
% 2.93/1.51 => ( ( ( P @ X2 )
% 2.93/1.51 | ( Q @ X2 ) )
% 2.93/1.51 = ( ( P2 @ X2 )
% 2.93/1.51 | ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_77_pinf_I3_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z @ X2 )
% 2.93/1.51 => ( X2 != T ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_78_pinf_I4_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z @ X2 )
% 2.93/1.51 => ( X2 != T ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_79_pinf_I5_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z @ X2 )
% 2.93/1.51 => ~ ( ord_less @ A @ X2 @ T ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_80_pinf_I7_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z @ X2 )
% 2.93/1.51 => ( ord_less @ A @ T @ X2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_81_pinf_I11_J,axiom,
% 2.93/1.51 ! [C2: $tType,D: $tType] :
% 2.93/1.51 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.93/1.51 => ! [F3: D] :
% 2.93/1.51 ? [Z: C2] :
% 2.93/1.51 ! [X2: C2] :
% 2.93/1.51 ( ( ord_less @ C2 @ Z @ X2 )
% 2.93/1.51 => ( F3 = F3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_82_minf_I1_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.93/1.51 ( ? [Z4: A] :
% 2.93/1.51 ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ X4 @ Z4 )
% 2.93/1.51 => ( ( P @ X4 )
% 2.93/1.51 = ( P2 @ X4 ) ) )
% 2.93/1.51 => ( ? [Z4: A] :
% 2.93/1.51 ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ X4 @ Z4 )
% 2.93/1.51 => ( ( Q @ X4 )
% 2.93/1.51 = ( Q2 @ X4 ) ) )
% 2.93/1.51 => ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X2 @ Z )
% 2.93/1.51 => ( ( ( P @ X2 )
% 2.93/1.51 & ( Q @ X2 ) )
% 2.93/1.51 = ( ( P2 @ X2 )
% 2.93/1.51 & ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_83_minf_I2_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.93/1.51 ( ? [Z4: A] :
% 2.93/1.51 ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ X4 @ Z4 )
% 2.93/1.51 => ( ( P @ X4 )
% 2.93/1.51 = ( P2 @ X4 ) ) )
% 2.93/1.51 => ( ? [Z4: A] :
% 2.93/1.51 ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ X4 @ Z4 )
% 2.93/1.51 => ( ( Q @ X4 )
% 2.93/1.51 = ( Q2 @ X4 ) ) )
% 2.93/1.51 => ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X2 @ Z )
% 2.93/1.51 => ( ( ( P @ X2 )
% 2.93/1.51 | ( Q @ X2 ) )
% 2.93/1.51 = ( ( P2 @ X2 )
% 2.93/1.51 | ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_84_minf_I3_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X2 @ Z )
% 2.93/1.51 => ( X2 != T ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_85_minf_I4_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X2 @ Z )
% 2.93/1.51 => ( X2 != T ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_86_minf_I5_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X2 @ Z )
% 2.93/1.51 => ( ord_less @ A @ X2 @ T ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_87_minf_I7_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [T: A] :
% 2.93/1.51 ? [Z: A] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X2 @ Z )
% 2.93/1.51 => ~ ( ord_less @ A @ T @ X2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_88_minf_I11_J,axiom,
% 2.93/1.51 ! [C2: $tType,D: $tType] :
% 2.93/1.51 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.93/1.51 => ! [F3: D] :
% 2.93/1.51 ? [Z: C2] :
% 2.93/1.51 ! [X2: C2] :
% 2.93/1.51 ( ( ord_less @ C2 @ X2 @ Z )
% 2.93/1.51 => ( F3 = F3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_89_order_Onot__eq__order__implies__strict,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( A2 != B2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_90_dual__order_Ostrict__implies__order,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less @ A @ B2 @ A2 )
% 2.93/1.51 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_91_dual__order_Ostrict__iff__order,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less @ A )
% 2.93/1.51 = ( ^ [B5: A,A5: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ B5 @ A5 )
% 2.93/1.51 & ( A5 != B5 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_92_dual__order_Oorder__iff__strict,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less_eq @ A )
% 2.93/1.51 = ( ^ [B5: A,A5: A] :
% 2.93/1.51 ( ( ord_less @ A @ B5 @ A5 )
% 2.93/1.51 | ( A5 = B5 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_93_order_Ostrict__implies__order,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_94_dense__le__bounded,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ! [W: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ W )
% 2.93/1.51 => ( ( ord_less @ A @ W @ Y )
% 2.93/1.51 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.93/1.51 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_95_dense__ge__bounded,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Z2: A,X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z2 @ X )
% 2.93/1.51 => ( ! [W: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z2 @ W )
% 2.93/1.51 => ( ( ord_less @ A @ W @ X )
% 2.93/1.51 => ( ord_less_eq @ A @ Y @ W ) ) )
% 2.93/1.51 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_96_dual__order_Ostrict__trans2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A,C: A] :
% 2.93/1.51 ( ( ord_less @ A @ B2 @ A2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.93/1.51 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_97_dual__order_Ostrict__trans1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 => ( ( ord_less @ A @ C @ B2 )
% 2.93/1.51 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_98_order_Ostrict__iff__order,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less @ A )
% 2.93/1.51 = ( ^ [A5: A,B5: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.93/1.51 & ( A5 != B5 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_99_order_Oorder__iff__strict,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less_eq @ A )
% 2.93/1.51 = ( ^ [A5: A,B5: A] :
% 2.93/1.51 ( ( ord_less @ A @ A5 @ B5 )
% 2.93/1.51 | ( A5 = B5 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_100_order_Ostrict__trans2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.93/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_101_order_Ostrict__trans1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less @ A @ B2 @ C )
% 2.93/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_102_not__le__imp__less,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,X: A] :
% 2.93/1.51 ( ~ ( ord_less_eq @ A @ Y @ X )
% 2.93/1.51 => ( ord_less @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_103_less__le__not__le,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less @ A )
% 2.93/1.51 = ( ^ [X3: A,Y2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.93/1.51 & ~ ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_104_le__imp__less__or__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 => ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 | ( X = Y ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_105_le__less__linear,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 | ( ord_less @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_106_dense__le,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,Z2: A] :
% 2.93/1.51 ( ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ X4 @ Y )
% 2.93/1.51 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.93/1.51 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_107_dense__ge,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Z2: A,Y: A] :
% 2.93/1.51 ( ! [X4: A] :
% 2.93/1.51 ( ( ord_less @ A @ Z2 @ X4 )
% 2.93/1.51 => ( ord_less_eq @ A @ Y @ X4 ) )
% 2.93/1.51 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_108_less__le__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.93/1.51 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_109_le__less__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 => ( ( ord_less @ A @ Y @ Z2 )
% 2.93/1.51 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_110_antisym__conv2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.93/1.51 = ( X = Y ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_111_antisym__conv1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ~ ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 = ( X = Y ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_112_less__imp__le,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_113_le__neq__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( A2 != B2 )
% 2.93/1.51 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_114_not__less,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.93/1.51 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_115_not__le,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 2.93/1.51 = ( ord_less @ A @ Y @ X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_116_order__less__le__subst2,axiom,
% 2.93/1.51 ! [A: $tType,C2: $tType] :
% 2.93/1.51 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.93/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 2.93/1.51 => ( ! [X4: A,Y3: A] :
% 2.93/1.51 ( ( ord_less @ A @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_117_order__less__le__subst1,axiom,
% 2.93/1.51 ! [A: $tType,B: $tType] :
% 2.93/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.93/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.51 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.93/1.51 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.93/1.51 => ( ! [X4: B,Y3: B] :
% 2.93/1.51 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.93/1.51 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.93/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_118_distinct__union,axiom,
% 2.93/1.51 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.93/1.51 ( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
% 2.93/1.51 = ( distinct @ A @ Ys ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_119_complete__interval,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,P: A > $o] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( P @ A2 )
% 2.93/1.51 => ( ~ ( P @ B2 )
% 2.93/1.51 => ? [C3: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.93/1.51 & ( ord_less_eq @ A @ C3 @ B2 )
% 2.93/1.51 & ! [X2: A] :
% 2.93/1.51 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 2.93/1.51 & ( ord_less @ A @ X2 @ C3 ) )
% 2.93/1.51 => ( P @ X2 ) )
% 2.93/1.51 & ! [D2: A] :
% 2.93/1.51 ( ! [X4: A] :
% 2.93/1.51 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 2.93/1.51 & ( ord_less @ A @ X4 @ D2 ) )
% 2.93/1.51 => ( P @ X4 ) )
% 2.93/1.51 => ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_120_subset__sorted__correct,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [L1: list @ A,L2: list @ A] :
% 2.93/1.51 ( ( ( distinct @ A @ L1 )
% 2.93/1.51 & ( linorder_sorted @ A @ L1 ) )
% 2.93/1.51 => ( ( ( distinct @ A @ L2 )
% 2.93/1.51 & ( linorder_sorted @ A @ L2 ) )
% 2.93/1.51 => ( ( sorted1061247458sorted @ A @ L1 @ L2 )
% 2.93/1.51 = ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_121_mergesort__remdups__correct,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [L: list @ A] :
% 2.93/1.51 ( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
% 2.93/1.51 & ( linorder_sorted @ A @ ( mergesort_remdups @ A @ L ) )
% 2.93/1.51 & ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
% 2.93/1.51 = ( set2 @ A @ L ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_122_memb__sorted__correct,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Xs: list @ A,X: A] :
% 2.93/1.51 ( ( linorder_sorted @ A @ Xs )
% 2.93/1.51 => ( ( sorted873716653sorted @ A @ Xs @ X )
% 2.93/1.51 = ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_123_in__set__member,axiom,
% 2.93/1.51 ! [A: $tType,X: A,Xs: list @ A] :
% 2.93/1.51 ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
% 2.93/1.51 = ( member @ A @ Xs @ X ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_124_finite__sorted__distinct__unique,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A4: set @ A] :
% 2.93/1.51 ( ( finite_finite2 @ A @ A4 )
% 2.93/1.51 => ? [X4: list @ A] :
% 2.93/1.51 ( ( ( set2 @ A @ X4 )
% 2.93/1.51 = A4 )
% 2.93/1.51 & ( linorder_sorted @ A @ X4 )
% 2.93/1.51 & ( distinct @ A @ X4 )
% 2.93/1.51 & ! [Y5: list @ A] :
% 2.93/1.51 ( ( ( ( set2 @ A @ Y5 )
% 2.93/1.51 = A4 )
% 2.93/1.51 & ( linorder_sorted @ A @ Y5 )
% 2.93/1.51 & ( distinct @ A @ Y5 ) )
% 2.93/1.51 => ( Y5 = X4 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_125_ex__gt__or__lt,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A] :
% 2.93/1.51 ? [B4: A] :
% 2.93/1.51 ( ( ord_less @ A @ A2 @ B4 )
% 2.93/1.51 | ( ord_less @ A @ B4 @ A2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_126_linorder__neqE__linordered__idom,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( X != Y )
% 2.93/1.51 => ( ~ ( ord_less @ A @ X @ Y )
% 2.93/1.51 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_127_linordered__field__no__ub,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X2: A] :
% 2.93/1.51 ? [X1: A] : ( ord_less @ A @ X2 @ X1 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_128_linordered__field__no__lb,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X2: A] :
% 2.93/1.51 ? [Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_129_List_Ofinite__set,axiom,
% 2.93/1.51 ! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_130_less__fun__def,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType] :
% 2.93/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.51 => ( ( ord_less @ ( A > B ) )
% 2.93/1.51 = ( ^ [F2: A > B,G: A > B] :
% 2.93/1.51 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 2.93/1.51 & ~ ( ord_less_eq @ ( A > B ) @ G @ F2 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_131_subset__Collect__conv,axiom,
% 2.93/1.51 ! [A: $tType,S: set @ A,P: A > $o] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 2.93/1.51 = ( ! [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ S )
% 2.93/1.51 => ( P @ X3 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_132_finite__list,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A] :
% 2.93/1.51 ( ( finite_finite2 @ A @ A4 )
% 2.93/1.51 => ? [Xs2: list @ A] :
% 2.93/1.51 ( ( set2 @ A @ Xs2 )
% 2.93/1.51 = A4 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_133_finite__distinct__list,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A] :
% 2.93/1.51 ( ( finite_finite2 @ A @ A4 )
% 2.93/1.51 => ? [Xs2: list @ A] :
% 2.93/1.51 ( ( ( set2 @ A @ Xs2 )
% 2.93/1.51 = A4 )
% 2.93/1.51 & ( distinct @ A @ Xs2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_134_ord__eq__le__eq__trans,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A,D3: A] :
% 2.93/1.51 ( ( A2 = B2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.93/1.51 => ( ( C = D3 )
% 2.93/1.51 => ( ord_less_eq @ A @ A2 @ D3 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_135_subsetI,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ! [X4: A] :
% 2.93/1.51 ( ( member2 @ A @ X4 @ A4 )
% 2.93/1.51 => ( member2 @ A @ X4 @ B3 ) )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_136_psubsetI,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( A4 != B3 )
% 2.93/1.51 => ( ord_less @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_137_subset__antisym,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
% 2.93/1.51 => ( A4 = B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_138_finite__code,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( finite_finite2 @ A )
% 2.93/1.51 = ( ^ [A6: set @ A] : $true ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_139_psubset__trans,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( ord_less @ ( set @ A ) @ B3 @ C4 )
% 2.93/1.51 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_140_psubsetD,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( member2 @ A @ C @ A4 )
% 2.93/1.51 => ( member2 @ A @ C @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_141_finite__set__choice,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
% 2.93/1.51 ( ( finite_finite2 @ A @ A4 )
% 2.93/1.51 => ( ! [X4: A] :
% 2.93/1.51 ( ( member2 @ A @ X4 @ A4 )
% 2.93/1.51 => ? [X12: B] : ( P @ X4 @ X12 ) )
% 2.93/1.51 => ? [F4: A > B] :
% 2.93/1.51 ! [X2: A] :
% 2.93/1.51 ( ( member2 @ A @ X2 @ A4 )
% 2.93/1.51 => ( P @ X2 @ ( F4 @ X2 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_142_finite,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_143_finite__psubset__induct,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
% 2.93/1.51 ( ( finite_finite2 @ A @ A4 )
% 2.93/1.51 => ( ! [A7: set @ A] :
% 2.93/1.51 ( ( finite_finite2 @ A @ A7 )
% 2.93/1.51 => ( ! [B6: set @ A] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) @ B6 @ A7 )
% 2.93/1.51 => ( P @ B6 ) )
% 2.93/1.51 => ( P @ A7 ) ) )
% 2.93/1.51 => ( P @ A4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_144_subset__iff__psubset__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 2.93/1.51 | ( A6 = B7 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_145_subset__psubset__trans,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( ord_less @ ( set @ A ) @ B3 @ C4 )
% 2.93/1.51 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_146_subset__not__subset__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) )
% 2.93/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 2.93/1.51 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_147_psubset__subset__trans,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
% 2.93/1.51 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_148_psubset__imp__subset,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_149_Collect__mono__iff,axiom,
% 2.93/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.93/1.51 = ( ! [X3: A] :
% 2.93/1.51 ( ( P @ X3 )
% 2.93/1.51 => ( Q @ X3 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_150_contra__subsetD,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ~ ( member2 @ A @ C @ B3 )
% 2.93/1.51 => ~ ( member2 @ A @ C @ A4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_151_set__eq__subset,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ^ [Y4: set @ A,Z3: set @ A] : ( Y4 = Z3 ) )
% 2.93/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 2.93/1.51 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_152_subset__trans,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_153_Collect__mono,axiom,
% 2.93/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.93/1.51 ( ! [X4: A] :
% 2.93/1.51 ( ( P @ X4 )
% 2.93/1.51 => ( Q @ X4 ) )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_154_subset__refl,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_155_rev__subsetD,axiom,
% 2.93/1.51 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( member2 @ A @ C @ A4 )
% 2.93/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( member2 @ A @ C @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_156_subset__iff,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.93/1.51 ! [T2: A] :
% 2.93/1.51 ( ( member2 @ A @ T2 @ A6 )
% 2.93/1.51 => ( member2 @ A @ T2 @ B7 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_157_set__rev__mp,axiom,
% 2.93/1.51 ! [A: $tType,X: A,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( member2 @ A @ X @ A4 )
% 2.93/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( member2 @ A @ X @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_158_psubset__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) )
% 2.93/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 2.93/1.51 & ( A6 != B7 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_159_equalityD2,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( A4 = B3 )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_160_equalityD1,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( A4 = B3 )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_161_subset__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.93/1.51 ! [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ A6 )
% 2.93/1.51 => ( member2 @ A @ X3 @ B7 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_162_equalityE,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( A4 = B3 )
% 2.93/1.51 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_163_subsetCE,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( member2 @ A @ C @ A4 )
% 2.93/1.51 => ( member2 @ A @ C @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_164_psubsetE,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_165_subsetD,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( member2 @ A @ C @ A4 )
% 2.93/1.51 => ( member2 @ A @ C @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_166_in__mono,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,X: A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( member2 @ A @ X @ A4 )
% 2.93/1.51 => ( member2 @ A @ X @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_167_set__mp,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,X: A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( member2 @ A @ X @ A4 )
% 2.93/1.51 => ( member2 @ A @ X @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_168_rev__finite__subset,axiom,
% 2.93/1.51 ! [A: $tType,B3: set @ A,A4: set @ A] :
% 2.93/1.51 ( ( finite_finite2 @ A @ B3 )
% 2.93/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( finite_finite2 @ A @ A4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_169_infinite__super,axiom,
% 2.93/1.51 ! [A: $tType,S: set @ A,T3: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
% 2.93/1.51 => ( ~ ( finite_finite2 @ A @ S )
% 2.93/1.51 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_170_finite__subset,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( finite_finite2 @ A @ B3 )
% 2.93/1.51 => ( finite_finite2 @ A @ A4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_171_sorted__list__of__set,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A4: set @ A] :
% 2.93/1.51 ( ( finite_finite2 @ A @ A4 )
% 2.93/1.51 => ( ( ( set2 @ A @ ( linord467138063of_set @ A @ A4 ) )
% 2.93/1.51 = A4 )
% 2.93/1.51 & ( linorder_sorted @ A @ ( linord467138063of_set @ A @ A4 ) )
% 2.93/1.51 & ( distinct @ A @ ( linord467138063of_set @ A @ A4 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_172_set__union,axiom,
% 2.93/1.51 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.93/1.51 ( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
% 2.93/1.51 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_173_subset__code_I2_J,axiom,
% 2.93/1.51 ! [B: $tType,A4: set @ B,Ys: list @ B] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ B ) @ A4 @ ( coset @ B @ Ys ) )
% 2.93/1.51 = ( ! [X3: B] :
% 2.93/1.51 ( ( member2 @ B @ X3 @ ( set2 @ B @ Ys ) )
% 2.93/1.51 => ~ ( member2 @ B @ X3 @ A4 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_174_sorted__by__rel__linord,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [L: list @ A] :
% 2.93/1.51 ( ( sorted_by_rel @ A @ ( ord_less_eq @ A ) @ L )
% 2.93/1.51 = ( linorder_sorted @ A @ L ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_175_UnCI,axiom,
% 2.93/1.51 ! [A: $tType,C: A,B3: set @ A,A4: set @ A] :
% 2.93/1.51 ( ( ~ ( member2 @ A @ C @ B3 )
% 2.93/1.51 => ( member2 @ A @ C @ A4 ) )
% 2.93/1.51 => ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_176_Un__iff,axiom,
% 2.93/1.51 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.93/1.51 = ( ( member2 @ A @ C @ A4 )
% 2.93/1.51 | ( member2 @ A @ C @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_177_finite__Un,axiom,
% 2.93/1.51 ! [A: $tType,F3: set @ A,G3: set @ A] :
% 2.93/1.51 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F3 @ G3 ) )
% 2.93/1.51 = ( ( finite_finite2 @ A @ F3 )
% 2.93/1.51 & ( finite_finite2 @ A @ G3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_178_Un__subset__iff,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 )
% 2.93/1.51 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
% 2.93/1.51 & ( ord_less_eq @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_179_Un__mono,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,C4: set @ A,B3: set @ A,D4: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
% 2.93/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D4 )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_180_Un__least,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,C4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
% 2.93/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
% 2.93/1.51 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_181_Un__upper1,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_182_Un__upper2,axiom,
% 2.93/1.51 ! [A: $tType,B3: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_183_Un__absorb1,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 => ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 = B3 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_184_Un__absorb2,axiom,
% 2.93/1.51 ! [A: $tType,B3: set @ A,A4: set @ A] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
% 2.93/1.51 => ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
% 2.93/1.51 = A4 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_185_subset__Un__eq,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.93/1.51 ( ( sup_sup @ ( set @ A ) @ A6 @ B7 )
% 2.93/1.51 = B7 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_186_infinite__Un,axiom,
% 2.93/1.51 ! [A: $tType,S: set @ A,T3: set @ A] :
% 2.93/1.51 ( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) )
% 2.93/1.51 = ( ~ ( finite_finite2 @ A @ S )
% 2.93/1.51 | ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_187_Un__infinite,axiom,
% 2.93/1.51 ! [A: $tType,S: set @ A,T3: set @ A] :
% 2.93/1.51 ( ~ ( finite_finite2 @ A @ S )
% 2.93/1.51 => ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_188_finite__UnI,axiom,
% 2.93/1.51 ! [A: $tType,F3: set @ A,G3: set @ A] :
% 2.93/1.51 ( ( finite_finite2 @ A @ F3 )
% 2.93/1.51 => ( ( finite_finite2 @ A @ G3 )
% 2.93/1.51 => ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F3 @ G3 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_189_UnE,axiom,
% 2.93/1.51 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.93/1.51 => ( ~ ( member2 @ A @ C @ A4 )
% 2.93/1.51 => ( member2 @ A @ C @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_190_UnI1,axiom,
% 2.93/1.51 ! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( member2 @ A @ C @ A4 )
% 2.93/1.51 => ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_191_UnI2,axiom,
% 2.93/1.51 ! [A: $tType,C: A,B3: set @ A,A4: set @ A] :
% 2.93/1.51 ( ( member2 @ A @ C @ B3 )
% 2.93/1.51 => ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_192_bex__Un,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
% 2.93/1.51 ( ( ? [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.93/1.51 & ( P @ X3 ) ) )
% 2.93/1.51 = ( ? [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ A4 )
% 2.93/1.51 & ( P @ X3 ) )
% 2.93/1.51 | ? [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ B3 )
% 2.93/1.51 & ( P @ X3 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_193_ball__Un,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
% 2.93/1.51 ( ( ! [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.93/1.51 => ( P @ X3 ) ) )
% 2.93/1.51 = ( ! [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ A4 )
% 2.93/1.51 => ( P @ X3 ) )
% 2.93/1.51 & ! [X3: A] :
% 2.93/1.51 ( ( member2 @ A @ X3 @ B3 )
% 2.93/1.51 => ( P @ X3 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_194_Un__assoc,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 2.93/1.51 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 )
% 2.93/1.51 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_195_Un__absorb,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A] :
% 2.93/1.51 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 2.93/1.51 = A4 ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_196_Un__commute,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( sup_sup @ ( set @ A ) )
% 2.93/1.51 = ( ^ [A6: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_197_Un__left__absorb,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.93/1.51 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.93/1.51 = ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_198_Un__left__commute,axiom,
% 2.93/1.51 ! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
% 2.93/1.51 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) )
% 2.93/1.51 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_199_sorted__by__rel__weaken,axiom,
% 2.93/1.51 ! [A: $tType,L0: list @ A,R: A > A > $o,R2: A > A > $o] :
% 2.93/1.51 ( ! [X4: A,Y3: A] :
% 2.93/1.51 ( ( member2 @ A @ X4 @ ( set2 @ A @ L0 ) )
% 2.93/1.51 => ( ( member2 @ A @ Y3 @ ( set2 @ A @ L0 ) )
% 2.93/1.51 => ( ( R @ X4 @ Y3 )
% 2.93/1.51 => ( R2 @ X4 @ Y3 ) ) ) )
% 2.93/1.51 => ( ( sorted_by_rel @ A @ R @ L0 )
% 2.93/1.51 => ( sorted_by_rel @ A @ R2 @ L0 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_200_distinct__sorted__list__of__set,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A4: set @ A] : ( distinct @ A @ ( linord467138063of_set @ A @ A4 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_201_sup_Obounded__iff,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,C: A,A2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
% 2.93/1.51 = ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 & ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_202_le__sup__iff,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
% 2.93/1.51 = ( ( ord_less_eq @ A @ X @ Z2 )
% 2.93/1.51 & ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_203_sup_Oright__idem,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 2.93/1.51 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_204_sup__apply,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.93/1.51 => ( ( sup_sup @ ( A > B ) )
% 2.93/1.51 = ( ^ [F2: A > B,G: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_205_sup_Oidem,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A] :
% 2.93/1.51 ( ( sup_sup @ A @ A2 @ A2 )
% 2.93/1.51 = A2 ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_206_sup__idem,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A] :
% 2.93/1.51 ( ( sup_sup @ A @ X @ X )
% 2.93/1.51 = X ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_207_sup_Oleft__idem,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 2.93/1.51 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_208_sup__left__idem,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.93/1.51 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_209_inf__sup__aci_I8_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( lattice @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.93/1.51 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_210_inf__sup__aci_I7_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( lattice @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
% 2.93/1.51 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_211_inf__sup__aci_I6_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( lattice @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
% 2.93/1.51 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_212_inf__sup__aci_I5_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( lattice @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( sup_sup @ A )
% 2.93/1.51 = ( ^ [X3: A,Y2: A] : ( sup_sup @ A @ Y2 @ X3 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_213_sup__fun__def,axiom,
% 2.93/1.51 ! [B: $tType,A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.93/1.51 => ( ( sup_sup @ ( A > B ) )
% 2.93/1.51 = ( ^ [F2: A > B,G: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_214_sup_Oassoc,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,C: A] :
% 2.93/1.51 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C )
% 2.93/1.51 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_215_sup__assoc,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
% 2.93/1.51 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_216_sup_Ocommute,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( sup_sup @ A )
% 2.93/1.51 = ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_217_sup__commute,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( sup_sup @ A )
% 2.93/1.51 = ( ^ [X3: A,Y2: A] : ( sup_sup @ A @ Y2 @ X3 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_218_sup_Oleft__commute,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A,C: A] :
% 2.93/1.51 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C ) )
% 2.93/1.51 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_219_sup__left__commute,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A,Z2: A] :
% 2.93/1.51 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
% 2.93/1.51 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_220_inf__sup__ord_I4_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( lattice @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_221_inf__sup__ord_I3_J,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( lattice @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_222_le__supE,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A,X: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X )
% 2.93/1.51 => ~ ( ( ord_less_eq @ A @ A2 @ X )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_223_le__supI,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,X: A,B2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ X )
% 2.93/1.51 => ( ( ord_less_eq @ A @ B2 @ X )
% 2.93/1.51 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_224_sup__ge1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_225_sup__ge2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_226_le__supI1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ A2 )
% 2.93/1.51 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_227_le__supI2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ B2 )
% 2.93/1.51 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_228_sup_Omono,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [C: A,A2: A,D3: A,B2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ C @ A2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ D3 @ B2 )
% 2.93/1.51 => ( ord_less_eq @ A @ ( sup_sup @ A @ C @ D3 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_229_sup__mono,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,C: A,B2: A,D3: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ C )
% 2.93/1.51 => ( ( ord_less_eq @ A @ B2 @ D3 )
% 2.93/1.51 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C @ D3 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_230_sup__least,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,X: A,Z2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.51 => ( ( ord_less_eq @ A @ Z2 @ X )
% 2.93/1.51 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_231_le__iff__sup,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less_eq @ A )
% 2.93/1.51 = ( ^ [X3: A,Y2: A] :
% 2.93/1.51 ( ( sup_sup @ A @ X3 @ Y2 )
% 2.93/1.51 = Y2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_232_sup_OorderE,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 => ( A2
% 2.93/1.51 = ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_233_sup_OorderI,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( A2
% 2.93/1.51 = ( sup_sup @ A @ A2 @ B2 ) )
% 2.93/1.51 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_234_sup__unique,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [F: A > A > A,X: A,Y: A] :
% 2.93/1.51 ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ X4 @ ( F @ X4 @ Y3 ) )
% 2.93/1.51 => ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X4 @ Y3 ) )
% 2.93/1.51 => ( ! [X4: A,Y3: A,Z: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ Z @ X4 )
% 2.93/1.51 => ( ord_less_eq @ A @ ( F @ Y3 @ Z ) @ X4 ) ) )
% 2.93/1.51 => ( ( sup_sup @ A @ X @ Y )
% 2.93/1.51 = ( F @ X @ Y ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_235_sup_Oabsorb1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 => ( ( sup_sup @ A @ A2 @ B2 )
% 2.93/1.51 = A2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_236_sup_Oabsorb2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.51 => ( ( sup_sup @ A @ A2 @ B2 )
% 2.93/1.51 = B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_237_sup__absorb1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [Y: A,X: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.51 => ( ( sup_sup @ A @ X @ Y )
% 2.93/1.51 = X ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_238_sup__absorb2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,Y: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.51 => ( ( sup_sup @ A @ X @ Y )
% 2.93/1.51 = Y ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_239_sup_OboundedE,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,C: A,A2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
% 2.93/1.51 => ~ ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 => ~ ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_240_sup_OboundedI,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A,C: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.51 => ( ( ord_less_eq @ A @ C @ A2 )
% 2.93/1.51 => ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_241_sup_Oorder__iff,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less_eq @ A )
% 2.93/1.51 = ( ^ [B5: A,A5: A] :
% 2.93/1.51 ( A5
% 2.93/1.51 = ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_242_sup_Ocobounded1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_243_sup_Ocobounded2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_244_sup_Oabsorb__iff1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less_eq @ A )
% 2.93/1.51 = ( ^ [B5: A,A5: A] :
% 2.93/1.51 ( ( sup_sup @ A @ A5 @ B5 )
% 2.93/1.51 = A5 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_245_sup_Oabsorb__iff2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less_eq @ A )
% 2.93/1.51 = ( ^ [A5: A,B5: A] :
% 2.93/1.51 ( ( sup_sup @ A @ A5 @ B5 )
% 2.93/1.51 = B5 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_246_sup_OcoboundedI1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [C: A,A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ C @ A2 )
% 2.93/1.51 => ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_247_sup_OcoboundedI2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [C: A,B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less_eq @ A @ C @ B2 )
% 2.93/1.51 => ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_248_sup_Ostrict__coboundedI2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [C: A,B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less @ A @ C @ B2 )
% 2.93/1.51 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_249_sup_Ostrict__coboundedI1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [C: A,A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less @ A @ C @ A2 )
% 2.93/1.51 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_250_sup_Ostrict__order__iff,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ( ord_less @ A )
% 2.93/1.51 = ( ^ [B5: A,A5: A] :
% 2.93/1.51 ( ( A5
% 2.93/1.51 = ( sup_sup @ A @ A5 @ B5 ) )
% 2.93/1.51 & ( A5 != B5 ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_251_sup_Ostrict__boundedE,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [B2: A,C: A,A2: A] :
% 2.93/1.51 ( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
% 2.93/1.51 => ~ ( ( ord_less @ A @ B2 @ A2 )
% 2.93/1.51 => ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_252_less__supI2,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,B2: A,A2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ B2 )
% 2.93/1.51 => ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_253_less__supI1,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [X: A,A2: A,B2: A] :
% 2.93/1.51 ( ( ord_less @ A @ X @ A2 )
% 2.93/1.51 => ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_254_merge__correct,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ! [L1: list @ A,L2: list @ A] :
% 2.93/1.51 ( ( ( distinct @ A @ L1 )
% 2.93/1.51 & ( linorder_sorted @ A @ L1 ) )
% 2.93/1.51 => ( ( ( distinct @ A @ L2 )
% 2.93/1.51 & ( linorder_sorted @ A @ L2 ) )
% 2.93/1.51 => ( ( distinct @ A @ ( merge @ A @ L1 @ L2 ) )
% 2.93/1.51 & ( linorder_sorted @ A @ ( merge @ A @ L1 @ L2 ) )
% 2.93/1.51 & ( ( set2 @ A @ ( merge @ A @ L1 @ L2 ) )
% 2.93/1.51 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(fact_255_set__mergesort__by__rel__merge,axiom,
% 2.93/1.51 ! [A: $tType,R: A > A > $o,Xs: list @ A,Ys: list @ A] :
% 2.93/1.51 ( ( set2 @ A @ ( merges2133357844_merge @ A @ R @ Xs @ Ys ) )
% 2.93/1.51 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( type @ A @ ( type2 @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( ord @ A @ ( type2 @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( order @ A @ ( type2 @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
% 2.93/1.51 ! [A: $tType] :
% 2.93/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.51 => ( preorder @ A @ ( type2 @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.93/1.51 ! [A8: $tType,A9: $tType] :
% 2.93/1.51 ( ( semilattice_sup @ A9 @ ( type2 @ A9 ) )
% 2.93/1.51 => ( semilattice_sup @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.93/1.51 ! [A8: $tType,A9: $tType] :
% 2.93/1.51 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 2.93/1.51 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.93/1.51 ! [A8: $tType,A9: $tType] :
% 2.93/1.51 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 2.93/1.51 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 2.93/1.51 => ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.93/1.51 ! [A8: $tType,A9: $tType] :
% 2.93/1.51 ( ( lattice @ A9 @ ( type2 @ A9 ) )
% 2.93/1.51 => ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.93/1.51 ! [A8: $tType,A9: $tType] :
% 2.93/1.51 ( ( order @ A9 @ ( type2 @ A9 ) )
% 2.93/1.51 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_fun___Orderings_Oord,axiom,
% 2.93/1.51 ! [A8: $tType,A9: $tType] :
% 2.93/1.51 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 2.93/1.51 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 2.93/1.51 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 2.93/1.51 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
% 2.93/1.51 ! [A8: $tType] :
% 2.93/1.51 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 2.93/1.51 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
% 2.93/1.51 ! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.93/1.51 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 2.93/1.51 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
% 2.93/1.51 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 2.93/1.51 preorder @ $o @ ( type2 @ $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.93/1.51 linorder @ $o @ ( type2 @ $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_HOL_Obool___Finite__Set_Ofinite_9,axiom,
% 2.93/1.51 finite_finite @ $o @ ( type2 @ $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_HOL_Obool___Lattices_Olattice_10,axiom,
% 2.93/1.51 lattice @ $o @ ( type2 @ $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 2.93/1.51 order @ $o @ ( type2 @ $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 2.93/1.51 ord @ $o @ ( type2 @ $o ) ).
% 2.93/1.51
% 2.93/1.51 thf(tfree_0,hypothesis,
% 2.93/1.51 linorder @ a @ ( type2 @ a ) ).
% 2.93/1.51
% 2.93/1.51 thf(conj_0,hypothesis,
% 2.93/1.51 ( ( distinct @ a @ ( sorted763967275sorted @ a @ x @ xsa ) )
% 2.93/1.51 & ( linorder_sorted @ a @ ( sorted763967275sorted @ a @ x @ xsa ) )
% 2.93/1.51 & ! [X2: a] :
% 2.93/1.51 ( ( member2 @ a @ X2 @ ( set2 @ a @ ( sorted763967275sorted @ a @ x @ xsa ) ) )
% 2.93/1.51 = ( ( member2 @ a @ X2 @ ( set2 @ a @ xsa ) )
% 2.93/1.51 & ( X2 != x ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(conj_1,hypothesis,
% 2.93/1.51 ( ( linorder_sorted @ a @ xsa )
% 2.93/1.51 & ! [X2: a] :
% 2.93/1.51 ( ( member2 @ a @ X2 @ ( set2 @ a @ xsa ) )
% 2.93/1.51 => ( ord_less_eq @ a @ a2 @ X2 ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(conj_2,hypothesis,
% 2.93/1.51 ( ~ ( member2 @ a @ a2 @ ( set2 @ a @ xsa ) )
% 2.93/1.51 & ( distinct @ a @ xsa ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(conj_3,conjecture,
% 2.93/1.51 ( ( ( x != a2 )
% 2.93/1.51 | ! [X4: a] :
% 2.93/1.51 ( ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) )
% 2.93/1.51 = ( ( ( X4 = a2 )
% 2.93/1.51 | ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
% 2.93/1.51 & ( X4 != a2 ) ) ) )
% 2.93/1.51 & ( ( x = a2 )
% 2.93/1.51 | ( ( ~ ( ord_less @ a @ a2 @ x )
% 2.93/1.51 | ! [X4: a] :
% 2.93/1.51 ( ( ( X4 = a2 )
% 2.93/1.51 | ( ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) )
% 2.93/1.51 & ( X4 != x ) ) )
% 2.93/1.51 = ( ( ( X4 = a2 )
% 2.93/1.51 | ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
% 2.93/1.51 & ( X4 != x ) ) ) )
% 2.93/1.51 & ( ( ord_less @ a @ a2 @ x )
% 2.93/1.51 | ! [X4: a] :
% 2.93/1.51 ( ( ( X4 = a2 )
% 2.93/1.51 | ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
% 2.93/1.51 = ( ( ( X4 = a2 )
% 2.93/1.51 | ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
% 2.93/1.51 & ( X4 != x ) ) ) ) ) ) ) ).
% 2.93/1.51 ------------------------
% 2.93/1.53 % (27280)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/183Mi)
% 2.93/1.53 % (27281)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_27099 for (2999ds/4Mi)
% 2.93/1.53 % (27284)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_27099 for (2999ds/2Mi)
% 2.93/1.53 % (27286)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/18Mi)
% 2.93/1.54 % (27282)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_27099 for (2999ds/27Mi)
% 2.93/1.54 % (27284)Instruction limit reached!
% 2.93/1.54 % (27284)------------------------------
% 2.93/1.54 % (27284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.54 % (27284)Termination reason: Unknown
% 2.93/1.54 % (27284)Termination phase: shuffling
% 2.93/1.54
% 2.93/1.54 % (27284)Memory used [KB]: 1407
% 2.93/1.54 % (27284)Time elapsed: 0.003 s
% 2.93/1.54 % (27284)Instructions burned: 3 (million)
% 2.93/1.54 % (27284)------------------------------
% 2.93/1.54 % (27284)------------------------------
% 2.93/1.54 % (27283)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/2Mi)
% 2.93/1.54 % (27285)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_27099 for (2999ds/275Mi)
% 2.93/1.54 % (27281)Instruction limit reached!
% 2.93/1.54 % (27281)------------------------------
% 2.93/1.54 % (27281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.54 % (27281)Termination reason: Unknown
% 2.93/1.54 % (27281)Termination phase: shuffling
% 2.93/1.54
% 2.93/1.54 % (27281)Memory used [KB]: 1407
% 2.93/1.54 % (27281)Time elapsed: 0.004 s
% 2.93/1.54 % (27283)Instruction limit reached!
% 2.93/1.54 % (27283)------------------------------
% 2.93/1.54 % (27283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.54 % (27283)Termination reason: Unknown
% 2.93/1.54 % (27283)Termination phase: shuffling
% 2.93/1.54
% 2.93/1.54 % (27283)Memory used [KB]: 1279
% 2.93/1.54 % (27283)Time elapsed: 0.003 s
% 2.93/1.54 % (27283)Instructions burned: 2 (million)
% 2.93/1.54 % (27283)------------------------------
% 2.93/1.54 % (27283)------------------------------
% 2.93/1.54 % (27281)Instructions burned: 5 (million)
% 2.93/1.54 % (27281)------------------------------
% 2.93/1.54 % (27281)------------------------------
% 2.93/1.54 % (27286)Instruction limit reached!
% 2.93/1.54 % (27286)------------------------------
% 2.93/1.54 % (27286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.54 % (27286)Termination reason: Unknown
% 2.93/1.54 % (27286)Termination phase: shuffling
% 2.93/1.54
% 2.93/1.54 % (27286)Memory used [KB]: 1663
% 2.93/1.54 % (27286)Time elapsed: 0.011 s
% 2.93/1.54 % (27286)Instructions burned: 19 (million)
% 2.93/1.54 % (27286)------------------------------
% 2.93/1.54 % (27286)------------------------------
% 2.93/1.55 % (27282)Instruction limit reached!
% 2.93/1.55 % (27282)------------------------------
% 2.93/1.55 % (27282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.55 % (27282)Termination reason: Unknown
% 2.93/1.55 % (27282)Termination phase: Property scanning
% 2.93/1.55
% 2.93/1.55 % (27282)Memory used [KB]: 1791
% 2.93/1.55 % (27282)Time elapsed: 0.017 s
% 2.93/1.55 % (27282)Instructions burned: 28 (million)
% 2.93/1.55 % (27282)------------------------------
% 2.93/1.55 % (27282)------------------------------
% 2.93/1.55 % (27290)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_27099 for (2999ds/15Mi)
% 2.93/1.55 % (27289)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_27099 for (2999ds/37Mi)
% 2.93/1.55 % (27287)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/3Mi)
% 2.93/1.56 % (27287)Instruction limit reached!
% 2.93/1.56 % (27287)------------------------------
% 2.93/1.56 % (27287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.56 % (27287)Termination reason: Unknown
% 2.93/1.56 % (27287)Termination phase: shuffling
% 2.93/1.56
% 2.93/1.56 % (27287)Memory used [KB]: 1407
% 2.93/1.56 % (27287)Time elapsed: 0.004 s
% 2.93/1.56 % (27287)Instructions burned: 4 (million)
% 2.93/1.56 % (27287)------------------------------
% 2.93/1.56 % (27287)------------------------------
% 2.93/1.56 % (27291)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_27099 for (2999ds/3Mi)
% 2.93/1.56 % (27290)Instruction limit reached!
% 2.93/1.56 % (27290)------------------------------
% 2.93/1.56 % (27290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.56 % (27290)Termination reason: Unknown
% 2.93/1.56 % (27290)Termination phase: shuffling
% 2.93/1.56
% 2.93/1.56 % (27290)Memory used [KB]: 1663
% 2.93/1.56 % (27290)Time elapsed: 0.009 s
% 2.93/1.56 % (27290)Instructions burned: 15 (million)
% 2.93/1.56 % (27290)------------------------------
% 2.93/1.56 % (27290)------------------------------
% 2.93/1.56 % (27291)Instruction limit reached!
% 2.93/1.56 % (27291)------------------------------
% 2.93/1.56 % (27291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.56 % (27291)Termination reason: Unknown
% 2.93/1.56 % (27291)Termination phase: shuffling
% 2.93/1.56
% 2.93/1.56 % (27291)Memory used [KB]: 1279
% 2.93/1.56 % (27291)Time elapsed: 0.003 s
% 2.93/1.56 % (27291)Instructions burned: 3 (million)
% 2.93/1.56 % (27291)------------------------------
% 2.93/1.56 % (27291)------------------------------
% 2.93/1.57 % (27289)Instruction limit reached!
% 2.93/1.57 % (27289)------------------------------
% 2.93/1.57 % (27289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.57 % (27289)Termination reason: Unknown
% 2.93/1.57 % (27289)Termination phase: Property scanning
% 2.93/1.57
% 2.93/1.57 % (27289)Memory used [KB]: 2046
% 2.93/1.57 % (27289)Time elapsed: 0.016 s
% 2.93/1.57 % (27289)Instructions burned: 37 (million)
% 2.93/1.57 % (27289)------------------------------
% 2.93/1.57 % (27289)------------------------------
% 2.93/1.57 % (27292)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_27099 for (2999ds/1041Mi)
% 2.93/1.58 % (27293)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/7Mi)
% 2.93/1.58 % (27297)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/3Mi)
% 2.93/1.58 % (27297)Instruction limit reached!
% 2.93/1.58 % (27297)------------------------------
% 2.93/1.58 % (27297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (27297)Termination reason: Unknown
% 2.93/1.58 % (27297)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (27297)Memory used [KB]: 1279
% 2.93/1.58 % (27297)Time elapsed: 0.002 s
% 2.93/1.58 % (27297)Instructions burned: 3 (million)
% 2.93/1.58 % (27297)------------------------------
% 2.93/1.58 % (27297)------------------------------
% 2.93/1.58 % (27294)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/16Mi)
% 2.93/1.58 % (27295)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_27099 for (2999ds/3Mi)
% 2.93/1.58 % (27293)Instruction limit reached!
% 2.93/1.58 % (27293)------------------------------
% 2.93/1.58 % (27293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (27293)Termination reason: Unknown
% 2.93/1.58 % (27293)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (27293)Memory used [KB]: 1407
% 2.93/1.58 % (27293)Time elapsed: 0.006 s
% 2.93/1.58 % (27293)Instructions burned: 8 (million)
% 2.93/1.58 % (27293)------------------------------
% 2.93/1.58 % (27293)------------------------------
% 2.93/1.58 % (27295)Instruction limit reached!
% 2.93/1.58 % (27295)------------------------------
% 2.93/1.58 % (27295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (27295)Termination reason: Unknown
% 2.93/1.58 % (27295)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (27295)Memory used [KB]: 1279
% 2.93/1.58 % (27295)Time elapsed: 0.003 s
% 2.93/1.58 % (27295)Instructions burned: 3 (million)
% 2.93/1.58 % (27295)------------------------------
% 2.93/1.58 % (27295)------------------------------
% 2.93/1.59 % (27294)Instruction limit reached!
% 2.93/1.59 % (27294)------------------------------
% 2.93/1.59 % (27294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (27294)Termination reason: Unknown
% 2.93/1.59 % (27294)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (27294)Memory used [KB]: 1663
% 2.93/1.59 % (27294)Time elapsed: 0.011 s
% 2.93/1.59 % (27294)Instructions burned: 16 (million)
% 2.93/1.59 % (27294)------------------------------
% 2.93/1.59 % (27294)------------------------------
% 2.93/1.59 % (27298)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/7Mi)
% 2.93/1.59 % (27280)Instruction limit reached!
% 2.93/1.59 % (27280)------------------------------
% 2.93/1.59 % (27280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (27280)Termination reason: Unknown
% 2.93/1.59 % (27280)Termination phase: Saturation
% 2.93/1.59
% 2.93/1.59 % (27280)Memory used [KB]: 7036
% 2.93/1.59 % (27280)Time elapsed: 0.063 s
% 2.93/1.59 % (27280)Instructions burned: 183 (million)
% 2.93/1.59 % (27280)------------------------------
% 2.93/1.59 % (27280)------------------------------
% 2.93/1.59 % (27298)Instruction limit reached!
% 2.93/1.59 % (27298)------------------------------
% 2.93/1.59 % (27298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (27298)Termination reason: Unknown
% 2.93/1.59 % (27298)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (27298)Memory used [KB]: 1535
% 2.93/1.59 % (27298)Time elapsed: 0.004 s
% 2.93/1.59 % (27298)Instructions burned: 9 (million)
% 2.93/1.59 % (27298)------------------------------
% 2.93/1.59 % (27298)------------------------------
% 2.93/1.60 % (27300)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/4Mi)
% 2.95/1.60 % (27299)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/3Mi)
% 2.95/1.60 % (27300)Instruction limit reached!
% 2.95/1.60 % (27300)------------------------------
% 2.95/1.60 % (27300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (27300)Termination reason: Unknown
% 2.95/1.60 % (27300)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.60 % (27300)Memory used [KB]: 1407
% 2.95/1.60 % (27300)Time elapsed: 0.004 s
% 2.95/1.60 % (27300)Instructions burned: 5 (million)
% 2.95/1.60 % (27300)------------------------------
% 2.95/1.60 % (27300)------------------------------
% 2.95/1.60 % (27302)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/18Mi)
% 2.95/1.60 % (27299)Instruction limit reached!
% 2.95/1.60 % (27299)------------------------------
% 2.95/1.60 % (27299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (27299)Termination reason: Unknown
% 2.95/1.60 % (27299)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.60 % (27299)Memory used [KB]: 1407
% 2.95/1.60 % (27299)Time elapsed: 0.004 s
% 2.95/1.60 % (27299)Instructions burned: 4 (million)
% 2.95/1.60 % (27299)------------------------------
% 2.95/1.60 % (27299)------------------------------
% 2.95/1.61 % (27302)Instruction limit reached!
% 2.95/1.61 % (27302)------------------------------
% 2.95/1.61 % (27302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.61 % (27302)Termination reason: Unknown
% 2.95/1.61 % (27302)Termination phase: shuffling
% 2.95/1.61
% 2.95/1.61 % (27302)Memory used [KB]: 1663
% 2.95/1.61 % (27302)Time elapsed: 0.007 s
% 2.95/1.61 % (27302)Instructions burned: 19 (million)
% 2.95/1.61 % (27302)------------------------------
% 2.95/1.61 % (27302)------------------------------
% 2.95/1.61 % (27304)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/6Mi)
% 2.95/1.61 % (27303)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/710Mi)
% 2.95/1.61 % (27304)Instruction limit reached!
% 2.95/1.61 % (27304)------------------------------
% 2.95/1.61 % (27304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.61 % (27304)Termination reason: Unknown
% 2.95/1.61 % (27304)Termination phase: shuffling
% 2.95/1.61
% 2.95/1.61 % (27304)Memory used [KB]: 1407
% 2.95/1.61 % (27304)Time elapsed: 0.004 s
% 2.95/1.61 % (27304)Instructions burned: 6 (million)
% 2.95/1.61 % (27304)------------------------------
% 2.95/1.61 % (27304)------------------------------
% 2.95/1.61 % (27305)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/902Mi)
% 2.95/1.61 % (27307)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/5Mi)
% 2.95/1.62 % (27307)Instruction limit reached!
% 2.95/1.62 % (27307)------------------------------
% 2.95/1.62 % (27307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.62 % (27307)Termination reason: Unknown
% 2.95/1.62 % (27307)Termination phase: shuffling
% 2.95/1.62
% 2.95/1.62 % (27307)Memory used [KB]: 1407
% 2.95/1.62 % (27307)Time elapsed: 0.003 s
% 2.95/1.62 % (27307)Instructions burned: 8 (million)
% 2.95/1.62 % (27307)------------------------------
% 2.95/1.62 % (27307)------------------------------
% 2.95/1.62 % (27306)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/21Mi)
% 2.95/1.63 % (27310)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/377Mi)
% 2.95/1.63 % (27308)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_27099 for (2999ds/6Mi)
% 2.95/1.63 % (27306)Instruction limit reached!
% 2.95/1.63 % (27306)------------------------------
% 2.95/1.63 % (27306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (27306)Termination reason: Unknown
% 2.95/1.63 % (27306)Termination phase: Property scanning
% 2.95/1.63
% 2.95/1.63 % (27306)Memory used [KB]: 1791
% 2.95/1.63 % (27306)Time elapsed: 0.014 s
% 2.95/1.63 % (27306)Instructions burned: 22 (million)
% 2.95/1.63 % (27306)------------------------------
% 2.95/1.63 % (27306)------------------------------
% 2.95/1.63 % (27308)Instruction limit reached!
% 2.95/1.63 % (27308)------------------------------
% 2.95/1.63 % (27308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (27308)Termination reason: Unknown
% 2.95/1.63 % (27308)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (27308)Memory used [KB]: 1407
% 2.95/1.63 % (27308)Time elapsed: 0.004 s
% 2.95/1.63 % (27308)Instructions burned: 6 (million)
% 2.95/1.63 % (27308)------------------------------
% 2.95/1.63 % (27308)------------------------------
% 2.95/1.65 % (27312)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_27099 for (2998ds/19Mi)
% 2.95/1.65 % (27311)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_27099 for (2998ds/779Mi)
% 2.95/1.65 % (27312)Instruction limit reached!
% 2.95/1.65 % (27312)------------------------------
% 2.95/1.65 % (27312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (27312)Termination reason: Unknown
% 2.95/1.66 % (27312)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (27312)Memory used [KB]: 1663
% 2.95/1.66 % (27312)Time elapsed: 0.010 s
% 2.95/1.66 % (27312)Instructions burned: 19 (million)
% 2.95/1.66 % (27312)------------------------------
% 2.95/1.66 % (27312)------------------------------
% 2.95/1.67 % (27313)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_27099 for (2998ds/879Mi)
% 2.95/1.68 % (27285)Instruction limit reached!
% 2.95/1.68 % (27285)------------------------------
% 2.95/1.68 % (27285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.68 % (27285)Termination reason: Unknown
% 2.95/1.68 % (27285)Termination phase: Saturation
% 2.95/1.68
% 2.95/1.68 % (27285)Memory used [KB]: 7036
% 2.95/1.68 % (27285)Time elapsed: 0.143 s
% 2.95/1.68 % (27285)Instructions burned: 275 (million)
% 2.95/1.68 % (27285)------------------------------
% 2.95/1.68 % (27285)------------------------------
% 2.95/1.70 % (27314)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_27099 for (2998ds/17Mi)
% 2.95/1.70 % (27314)Instruction limit reached!
% 2.95/1.70 % (27314)------------------------------
% 2.95/1.70 % (27314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (27314)Termination reason: Unknown
% 2.95/1.70 % (27314)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (27314)Memory used [KB]: 1663
% 2.95/1.70 % (27314)Time elapsed: 0.011 s
% 2.95/1.70 % (27314)Instructions burned: 17 (million)
% 2.95/1.70 % (27314)------------------------------
% 2.95/1.70 % (27314)------------------------------
% 2.95/1.72 % (27316)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27099 for (2998ds/3Mi)
% 2.95/1.72 % (27316)Instruction limit reached!
% 2.95/1.72 % (27316)------------------------------
% 2.95/1.72 % (27316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.72 % (27316)Termination reason: Unknown
% 2.95/1.72 % (27316)Termination phase: shuffling
% 2.95/1.72
% 2.95/1.72 % (27316)Memory used [KB]: 1279
% 2.95/1.72 % (27316)Time elapsed: 0.003 s
% 2.95/1.72 % (27316)Instructions burned: 3 (million)
% 2.95/1.72 % (27316)------------------------------
% 2.95/1.72 % (27316)------------------------------
% 2.95/1.72 % (27303)First to succeed.
% 2.95/1.73 % (27303)Refutation found. Thanks to Tanya!
% 2.95/1.73 % SZS status Theorem for DTF2THF_27099
% 2.95/1.73 % SZS output start Proof for DTF2THF_27099
% 2.95/1.73 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.95/1.73 thf(type_def_7, type, set: ($tType) > $tType).
% 2.95/1.73 thf(type_def_8, type, list: ($tType) > $tType).
% 2.95/1.73 thf(type_def_9, type, a: $tType).
% 2.95/1.73 thf(func_def_0, type, list: ($tType) > $tType).
% 2.95/1.73 thf(func_def_1, type, set: ($tType) > $tType).
% 2.95/1.73 thf(func_def_2, type, itself: ($tType) > $tType).
% 2.95/1.73 thf(func_def_3, type, a: $tType).
% 2.95/1.73 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_5, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_6, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_7, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_8, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_9, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_10, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_12, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_13, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_14, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_15, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_16, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_17, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_18, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_19, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_20, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.73 thf(func_def_21, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.95/1.73 thf(func_def_22, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.95/1.73 thf(func_def_23, type, coset: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.95/1.73 thf(func_def_24, type, distinct: !>[X0: $tType]:(list @ X0 > $o)).
% 2.95/1.73 thf(func_def_25, type, linorder_sorted: !>[X0: $tType]:(list @ X0 > $o)).
% 2.95/1.73 thf(func_def_26, type, linord467138063of_set: !>[X0: $tType]:(set @ X0 > list @ X0)).
% 2.95/1.73 thf(func_def_27, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.95/1.73 thf(func_def_28, type, member: !>[X0: $tType]:(list @ X0 > X0 > $o)).
% 2.95/1.73 thf(func_def_29, type, union: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.73 thf(func_def_30, type, merge: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.73 thf(func_def_31, type, merges2133357844_merge: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.73 thf(func_def_32, type, mergesort_remdups: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.95/1.73 thf(func_def_33, type, sorted_by_rel: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > $o)).
% 2.95/1.73 thf(func_def_34, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.95/1.73 thf(func_def_35, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.95/1.73 thf(func_def_36, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.95/1.73 thf(func_def_37, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.95/1.73 thf(func_def_38, type, sorted763967275sorted: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 2.95/1.73 thf(func_def_39, type, sorted873716653sorted: !>[X0: $tType]:(list @ X0 > X0 > $o)).
% 2.95/1.73 thf(func_def_40, type, sorted1061247458sorted: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 2.95/1.73 thf(func_def_41, type, member2: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.95/1.73 thf(func_def_42, type, a2: a).
% 2.95/1.73 thf(func_def_43, type, x: a).
% 2.95/1.73 thf(func_def_44, type, xsa: list @ a).
% 2.95/1.73 thf(func_def_61, type, sP3: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.95/1.73 thf(func_def_62, type, sP4: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.95/1.73 thf(func_def_63, type, sP5: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.95/1.73 thf(func_def_64, type, sP6: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.95/1.73 thf(func_def_65, type, sP7: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.95/1.73 thf(func_def_66, type, sP8: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.95/1.73 thf(func_def_67, type, sP9: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.95/1.73 thf(func_def_68, type, sK10: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_69, type, sK11: !>[X0: $tType]:(set @ X0 > list @ X0)).
% 2.95/1.73 thf(func_def_70, type, sK12: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_71, type, sK13: !>[X0: $tType]:(set @ X0 > list @ X0)).
% 2.95/1.73 thf(func_def_72, type, sK14: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.95/1.73 thf(func_def_73, type, sK15: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.95/1.73 thf(func_def_74, type, sK16: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.95/1.73 thf(func_def_75, type, sK17: a).
% 2.95/1.73 thf(func_def_76, type, sK18: a).
% 2.95/1.73 thf(func_def_77, type, sK19: a).
% 2.95/1.73 thf(func_def_78, type, sK20: !>[X0: $tType]:(set @ X0 > list @ X0 > X0)).
% 2.95/1.73 thf(func_def_79, type, sK21: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_80, type, sK22: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.95/1.73 thf(func_def_81, type, sK23: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_82, type, sK24: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_83, type, sK25: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_84, type, sK26: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_85, type, sK27: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_86, type, sK28: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.95/1.73 thf(func_def_87, type, sK29: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.95/1.73 thf(func_def_88, type, sK30: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.95/1.73 thf(func_def_89, type, sK31: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_90, type, sK32: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_91, type, sK33: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_92, type, sK34: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_93, type, sK35: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_94, type, sK36: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_95, type, sK37: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_96, type, sK38: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_97, type, sK39: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_98, type, sK40: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_99, type, sK41: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_100, type, sK42: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_101, type, sK43: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_102, type, sK44: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.73 thf(func_def_103, type, sK45: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_104, type, sK46: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_105, type, sK47: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_106, type, sK48: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.95/1.73 thf(func_def_107, type, sK49: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_108, type, sK50: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_109, type, sK51: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.95/1.73 thf(func_def_111, type, ph53: !>[X0: $tType]:(X0)).
% 2.95/1.73 thf(f1538,plain,(
% 2.95/1.73 $false),
% 2.95/1.73 inference(avatar_sat_refutation,[status(thm)],[f1387,f1392,f1393,f1398,f1399,f1400,f1401,f1402,f1404,f1408,f1409,f1411,f1412,f1413,f1414,f1514,f1518,f1532,f1537])).
% 2.95/1.73 thf(f1537,plain,(
% 2.95/1.73 ~spl52_1 | ~spl52_2 | spl52_13),
% 2.95/1.73 inference(avatar_contradiction_clause,[status(thm)],[f1536])).
% 2.95/1.73 thf(f1536,plain,(
% 2.95/1.73 $false | (~spl52_1 | ~spl52_2 | spl52_13)),
% 2.95/1.73 inference(subsumption_resolution,[status(thm)],[f1535,f1397])).
% 2.95/1.73 thf(f1397,plain,(
% 2.95/1.73 (x != a2) | spl52_13),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1395])).
% 2.95/1.73 thf(f1395,definition,(
% 2.95/1.73 spl52_13 <=> (x = a2)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_13])],[avatar_definition])).
% 2.95/1.73 thf(f1535,plain,(
% 2.95/1.73 (x = a2) | (~spl52_1 | ~spl52_2)),
% 2.95/1.73 inference(backward_demodulation,[status(thm)],[f1342,f1347])).
% 2.95/1.73 thf(f1347,plain,(
% 2.95/1.73 (a2 = sK19) | ~spl52_2),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1345])).
% 2.95/1.73 thf(f1345,definition,(
% 2.95/1.73 spl52_2 <=> (a2 = sK19)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_2])],[avatar_definition])).
% 2.95/1.73 thf(f1342,plain,(
% 2.95/1.73 (x = sK19) | ~spl52_1),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1341])).
% 2.95/1.73 thf(f1341,definition,(
% 2.95/1.73 spl52_1 <=> (x = sK19)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_1])],[avatar_definition])).
% 2.95/1.73 thf(f1532,plain,(
% 2.95/1.73 ~spl52_10 | ~spl52_11),
% 2.95/1.73 inference(avatar_contradiction_clause,[status(thm)],[f1531])).
% 2.95/1.73 thf(f1531,plain,(
% 2.95/1.73 $false | (~spl52_10 | ~spl52_11)),
% 2.95/1.73 inference(subsumption_resolution,[status(thm)],[f1530,f1164])).
% 2.95/1.73 thf(f1164,plain,(
% 2.95/1.73 ((member2 @ a @ a2 @ (set2 @ a @ xsa)) != $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f834])).
% 2.95/1.73 thf(f834,plain,(
% 2.95/1.73 ((member2 @ a @ a2 @ (set2 @ a @ xsa)) != $true) & ((distinct @ a @ xsa) = $true)),
% 2.95/1.73 inference(flattening,[status(thm)],[f520])).
% 2.95/1.73 thf(f520,plain,(
% 2.95/1.73 ((distinct @ a @ xsa) = $true) & ~((member2 @ a @ a2 @ (set2 @ a @ xsa)) = $true)),
% 2.95/1.73 inference(fool_elimination,[status(thm)],[f519])).
% 2.95/1.73 thf(f519,plain,(
% 2.95/1.73 (distinct @ a @ xsa) & ~(member2 @ a @ a2 @ (set2 @ a @ xsa))),
% 2.95/1.73 inference(rectify,[status(thm)],[f283])).
% 2.95/1.73 thf(f283,axiom,(
% 2.95/1.73 (distinct @ a @ xsa) & ~(member2 @ a @ a2 @ (set2 @ a @ xsa))),
% 2.95/1.73 file('/export/starexec/sandbox2/tmp/tmp.brU5k5c5ot/DTF2THF_27099.p',conj_2)).
% 2.95/1.73 thf(f1530,plain,(
% 2.95/1.73 ((member2 @ a @ a2 @ (set2 @ a @ xsa)) = $true) | (~spl52_10 | ~spl52_11)),
% 2.95/1.73 inference(forward_demodulation,[status(thm)],[f1385,f1382])).
% 2.95/1.73 thf(f1382,plain,(
% 2.95/1.73 (a2 = sK17) | ~spl52_10),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1380])).
% 2.95/1.73 thf(f1380,definition,(
% 2.95/1.73 spl52_10 <=> (a2 = sK17)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_10])],[avatar_definition])).
% 2.95/1.73 thf(f1385,plain,(
% 2.95/1.73 ($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | ~spl52_11),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1384])).
% 2.95/1.73 thf(f1384,definition,(
% 2.95/1.73 spl52_11 <=> ($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa)))),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_11])],[avatar_definition])).
% 2.95/1.73 thf(f1518,plain,(
% 2.95/1.73 ~spl52_7 | ~spl52_8 | spl52_13),
% 2.95/1.73 inference(avatar_contradiction_clause,[status(thm)],[f1517])).
% 2.95/1.73 thf(f1517,plain,(
% 2.95/1.73 $false | (~spl52_7 | ~spl52_8 | spl52_13)),
% 2.95/1.73 inference(subsumption_resolution,[status(thm)],[f1516,f1397])).
% 2.95/1.73 thf(f1516,plain,(
% 2.95/1.73 (x = a2) | (~spl52_7 | ~spl52_8)),
% 2.95/1.73 inference(backward_demodulation,[status(thm)],[f1368,f1371])).
% 2.95/1.73 thf(f1371,plain,(
% 2.95/1.73 (a2 = sK18) | ~spl52_8),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1370])).
% 2.95/1.73 thf(f1370,definition,(
% 2.95/1.73 spl52_8 <=> (a2 = sK18)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_8])],[avatar_definition])).
% 2.95/1.73 thf(f1368,plain,(
% 2.95/1.73 (x = sK18) | ~spl52_7),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1366])).
% 2.95/1.73 thf(f1366,definition,(
% 2.95/1.73 spl52_7 <=> (x = sK18)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_7])],[avatar_definition])).
% 2.95/1.73 thf(f1514,plain,(
% 2.95/1.73 ~spl52_5 | ~spl52_7 | spl52_12 | spl52_13),
% 2.95/1.73 inference(avatar_contradiction_clause,[status(thm)],[f1513])).
% 2.95/1.73 thf(f1513,plain,(
% 2.95/1.73 $false | (~spl52_5 | ~spl52_7 | spl52_12 | spl52_13)),
% 2.95/1.73 inference(subsumption_resolution,[status(thm)],[f1512,f1397])).
% 2.95/1.73 thf(f1512,plain,(
% 2.95/1.73 (x = a2) | (~spl52_5 | ~spl52_7 | spl52_12)),
% 2.95/1.73 inference(subsumption_resolution,[status(thm)],[f1511,f1391])).
% 2.95/1.73 thf(f1391,plain,(
% 2.95/1.73 ((ord_less @ a @ a2 @ x) != $true) | spl52_12),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1389])).
% 2.95/1.73 thf(f1389,definition,(
% 2.95/1.73 spl52_12 <=> ((ord_less @ a @ a2 @ x) = $true)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_12])],[avatar_definition])).
% 2.95/1.73 thf(f1511,plain,(
% 2.95/1.73 ((ord_less @ a @ a2 @ x) = $true) | (x = a2) | (~spl52_5 | ~spl52_7)),
% 2.95/1.73 inference(subsumption_resolution,[status(thm)],[f1500,f1256])).
% 2.95/1.73 thf(f1256,plain,(
% 2.95/1.73 ((linorder @ a @ type2 @ a) = $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f436])).
% 2.95/1.73 thf(f436,plain,(
% 2.95/1.73 ((linorder @ a @ type2 @ a) = $true)),
% 2.95/1.73 inference(fool_elimination,[status(thm)],[f435])).
% 2.95/1.73 thf(f435,plain,(
% 2.95/1.73 (linorder @ a @ type2 @ a)),
% 2.95/1.73 inference(rectify,[status(thm)],[f280])).
% 2.95/1.73 thf(f280,axiom,(
% 2.95/1.73 (linorder @ a @ type2 @ a)),
% 2.95/1.73 file('/export/starexec/sandbox2/tmp/tmp.brU5k5c5ot/DTF2THF_27099.p',tfree_0)).
% 2.95/1.73 thf(f1500,plain,(
% 2.95/1.73 ((linorder @ a @ type2 @ a) != $true) | (x = a2) | ((ord_less @ a @ a2 @ x) = $true) | (~spl52_5 | ~spl52_7)),
% 2.95/1.73 inference(trivial_inequality_removal,[status(thm)],[f1497])).
% 2.95/1.73 thf(f1497,plain,(
% 2.95/1.73 (x = a2) | ((linorder @ a @ type2 @ a) != $true) | ((ord_less @ a @ a2 @ x) = $true) | ($true != $true) | (~spl52_5 | ~spl52_7)),
% 2.95/1.73 inference(superposition,[status(thm)],[f1097,f1428])).
% 2.95/1.73 thf(f1428,plain,(
% 2.95/1.73 ((ord_less_eq @ a @ a2 @ x) = $true) | (~spl52_5 | ~spl52_7)),
% 2.95/1.73 inference(trivial_inequality_removal,[status(thm)],[f1427])).
% 2.95/1.73 thf(f1427,plain,(
% 2.95/1.73 ($true != $true) | ((ord_less_eq @ a @ a2 @ x) = $true) | (~spl52_5 | ~spl52_7)),
% 2.95/1.73 inference(superposition,[status(thm)],[f1233,f1416])).
% 2.95/1.73 thf(f1416,plain,(
% 2.95/1.73 ($true = (member2 @ a @ x @ (set2 @ a @ xsa))) | (~spl52_5 | ~spl52_7)),
% 2.95/1.73 inference(backward_demodulation,[status(thm)],[f1359,f1368])).
% 2.95/1.73 thf(f1359,plain,(
% 2.95/1.73 ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa))) | ~spl52_5),
% 2.95/1.73 inference(avatar_component_clause,[status(thm)],[f1358])).
% 2.95/1.73 thf(f1358,definition,(
% 2.95/1.73 spl52_5 <=> ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa)))),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_5])],[avatar_definition])).
% 2.95/1.73 thf(f1233,plain,(
% 2.95/1.73 ( ! [X0 : a] : (($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) | ((ord_less_eq @ a @ a2 @ X0) = $true)) )),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f928])).
% 2.95/1.73 thf(f928,plain,(
% 2.95/1.73 ((linorder_sorted @ a @ xsa) = $true) & ! [X0 : a] : (($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) | ((ord_less_eq @ a @ a2 @ X0) = $true))),
% 2.95/1.73 inference(ennf_transformation,[status(thm)],[f390])).
% 2.95/1.73 thf(f390,plain,(
% 2.95/1.73 ((linorder_sorted @ a @ xsa) = $true) & ! [X0 : a] : (($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) => ((ord_less_eq @ a @ a2 @ X0) = $true))),
% 2.95/1.73 inference(fool_elimination,[status(thm)],[f389])).
% 2.95/1.73 thf(f389,plain,(
% 2.95/1.73 ! [X0 : a] : ((member2 @ a @ X0 @ (set2 @ a @ xsa)) => (ord_less_eq @ a @ a2 @ X0)) & (linorder_sorted @ a @ xsa)),
% 2.95/1.73 inference(rectify,[status(thm)],[f282])).
% 2.95/1.73 thf(f282,axiom,(
% 2.95/1.73 ! [X8 : a] : ((member2 @ a @ X8 @ (set2 @ a @ xsa)) => (ord_less_eq @ a @ a2 @ X8)) & (linorder_sorted @ a @ xsa)),
% 2.95/1.73 file('/export/starexec/sandbox2/tmp/tmp.brU5k5c5ot/DTF2THF_27099.p',conj_1)).
% 2.95/1.73 thf(f1097,plain,(
% 2.95/1.73 ( ! [X0 : $tType,X2 : X0,X1 : X0] : (((ord_less_eq @ X0 @ X1 @ X2) != $true) | ((linorder @ X0 @ type2 @ X0) != $true) | (X1 = X2) | ($true = (ord_less @ X0 @ X1 @ X2))) )),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f970])).
% 2.95/1.73 thf(f970,plain,(
% 2.95/1.73 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) != $true) | ! [X1 : X0,X2 : X0] : (((($true != (ord_less @ X0 @ X1 @ X2)) | (X1 != X2)) & ((X1 = X2) | ($true = (ord_less @ X0 @ X1 @ X2)))) | ((ord_less_eq @ X0 @ X1 @ X2) != $true)))),
% 2.95/1.73 inference(nnf_transformation,[status(thm)],[f945])).
% 2.95/1.73 thf(f945,plain,(
% 2.95/1.73 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) != $true) | ! [X1 : X0,X2 : X0] : ((($true != (ord_less @ X0 @ X1 @ X2)) <=> (X1 = X2)) | ((ord_less_eq @ X0 @ X1 @ X2) != $true)))),
% 2.95/1.73 inference(ennf_transformation,[status(thm)],[f843])).
% 2.95/1.73 thf(f843,plain,(
% 2.95/1.73 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) = $true) => ! [X1 : X0,X2 : X0] : (((ord_less_eq @ X0 @ X1 @ X2) = $true) => (($true != (ord_less @ X0 @ X1 @ X2)) <=> (X1 = X2))))),
% 2.95/1.73 inference(flattening,[status(thm)],[f710])).
% 2.95/1.73 thf(f710,plain,(
% 2.95/1.73 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) = $true) => ! [X1 : X0,X2 : X0] : (((ord_less_eq @ X0 @ X1 @ X2) = $true) => ((X1 = X2) <=> ~($true = (ord_less @ X0 @ X1 @ X2)))))),
% 2.95/1.73 inference(fool_elimination,[status(thm)],[f709])).
% 2.95/1.73 thf(f709,plain,(
% 2.95/1.73 ! [X0 : $tType] : ((linorder @ X0 @ type2 @ X0) => ! [X1 : X0,X2 : X0] : ((ord_less_eq @ X0 @ X1 @ X2) => ((X1 = X2) = ~(ord_less @ X0 @ X1 @ X2))))),
% 2.95/1.73 inference(rectify,[status(thm)],[f111])).
% 2.95/1.73 thf(f111,axiom,(
% 2.95/1.73 ! [X0 : $tType] : ((linorder @ X0 @ type2 @ X0) => ! [X5 : X0,X9 : X0] : ((ord_less_eq @ X0 @ X5 @ X9) => ((X5 = X9) = ~(ord_less @ X0 @ X5 @ X9))))),
% 2.95/1.73 file('/export/starexec/sandbox2/tmp/tmp.brU5k5c5ot/DTF2THF_27099.p',fact_110_antisym__conv2)).
% 2.95/1.73 thf(f1414,plain,(
% 2.95/1.73 spl52_2 | ~spl52_3 | ~spl52_1),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1298,f1341,f1349,f1345])).
% 2.95/1.73 thf(f1349,definition,(
% 2.95/1.73 spl52_3 <=> (sP0 = $true)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_3])],[avatar_definition])).
% 2.95/1.73 thf(f1298,plain,(
% 2.95/1.73 (sP0 != $true) | (a2 = sK19) | (x != sK19)),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1138])).
% 2.95/1.73 thf(f1138,plain,(
% 2.95/1.73 (x != sK19) | (x != sK19) | (a2 = sK19) | (sP0 != $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f998])).
% 2.95/1.73 thf(f998,plain,(
% 2.95/1.73 (((((x = sK19) | ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) != $true)) & (a2 != sK19)) | (x = sK19) | ((a2 != sK19) & ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) != $true))) & (((x != sK19) & ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) = $true)) | (a2 = sK19) | ((x != sK19) & ((a2 = sK19) | ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) = $true))))) | (sP0 != $true)),
% 2.95/1.73 inference(skolemisation,[status(esa),new_symbols(skolem,[sK19])],[f996,f997])).
% 2.95/1.73 thf(f997,plain,(
% 2.95/1.73 ? [X0 : a] : (((((x = X0) | ($true != (member2 @ a @ X0 @ (set2 @ a @ xsa)))) & (a2 != X0)) | (x = X0) | ((a2 != X0) & ($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))))) & (((x != X0) & ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa)))) | (a2 = X0) | ((x != X0) & ((a2 = X0) | ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))))))) => (((((x = sK19) | ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) != $true)) & (a2 != sK19)) | (x = sK19) | ((a2 != sK19) & ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) != $true))) & (((x != sK19) & ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) = $true)) | (a2 = sK19) | ((x != sK19) & ((a2 = sK19) | ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) = $true)))))),
% 2.95/1.73 introduced(definition,[],[choice_axiom])).
% 2.95/1.73 thf(f996,plain,(
% 2.95/1.73 ? [X0 : a] : (((((x = X0) | ($true != (member2 @ a @ X0 @ (set2 @ a @ xsa)))) & (a2 != X0)) | (x = X0) | ((a2 != X0) & ($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))))) & (((x != X0) & ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa)))) | (a2 = X0) | ((x != X0) & ((a2 = X0) | ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))))))) | (sP0 != $true)),
% 2.95/1.73 inference(rectify,[status(thm)],[f995])).
% 2.95/1.73 thf(f995,plain,(
% 2.95/1.73 ? [X1 : a] : (((((x = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) != $true)) & (a2 != X1)) | (x = X1) | ((a2 != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) != $true))) & (((x != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true)) | (a2 = X1) | ((x != X1) & ((a2 = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true))))) | (sP0 != $true)),
% 2.95/1.73 inference(flattening,[status(thm)],[f994])).
% 2.95/1.73 thf(f994,plain,(
% 2.95/1.73 ? [X1 : a] : (((((x = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) != $true)) & (a2 != X1)) | ((x = X1) | ((a2 != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) != $true)))) & ((((x != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true)) | (a2 = X1)) | ((x != X1) & ((a2 = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true))))) | (sP0 != $true)),
% 2.95/1.73 inference(nnf_transformation,[status(thm)],[f950])).
% 2.95/1.73 thf(f950,plain,(
% 2.95/1.73 ? [X1 : a] : (((x != X1) & ((a2 = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true))) <~> (((x != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true)) | (a2 = X1))) | ~(sP0 = $true)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.95/1.73 thf(f1413,plain,(
% 2.95/1.73 spl52_11 | ~spl52_9 | ~spl52_10),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1123,f1380,f1376,f1384])).
% 2.95/1.73 thf(f1376,definition,(
% 2.95/1.73 spl52_9 <=> (sP2 = $true)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_9])],[avatar_definition])).
% 2.95/1.73 thf(f1123,plain,(
% 2.95/1.73 ($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (a2 != sK17) | (sP2 != $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f988])).
% 2.95/1.73 thf(f988,plain,(
% 2.95/1.73 ((x = a2) & ((($true != (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (($true != (member2 @ a @ sK17 @ (set2 @ a @ xsa))) & (a2 != sK17)) | (a2 = sK17)) & (($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | ((($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (a2 = sK17)) & (a2 != sK17))))) | (sP2 != $true)),
% 2.95/1.73 inference(skolemisation,[status(esa),new_symbols(skolem,[sK17])],[f986,f987])).
% 2.95/1.73 thf(f987,plain,(
% 2.95/1.73 ? [X0 : a] : ((($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) & (a2 != X0)) | (a2 = X0)) & (($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | ((($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0)) & (a2 != X0)))) => ((($true != (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (($true != (member2 @ a @ sK17 @ (set2 @ a @ xsa))) & (a2 != sK17)) | (a2 = sK17)) & (($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | ((($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (a2 = sK17)) & (a2 != sK17))))),
% 2.95/1.73 introduced(definition,[],[choice_axiom])).
% 2.95/1.73 thf(f986,plain,(
% 2.95/1.73 ((x = a2) & ? [X0 : a] : ((($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) & (a2 != X0)) | (a2 = X0)) & (($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | ((($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0)) & (a2 != X0))))) | (sP2 != $true)),
% 2.95/1.73 inference(flattening,[status(thm)],[f985])).
% 2.95/1.73 thf(f985,plain,(
% 2.95/1.73 ((x = a2) & ? [X0 : a] : ((($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) | ((($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) & (a2 != X0)) | (a2 = X0))) & (($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | ((($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0)) & (a2 != X0))))) | (sP2 != $true)),
% 2.95/1.73 inference(nnf_transformation,[status(thm)],[f952])).
% 2.95/1.73 thf(f952,plain,(
% 2.95/1.73 ((x = a2) & ? [X0 : a] : (((($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0)) & (a2 != X0)) <~> ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))))) | ~(sP2 = $true)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.95/1.73 thf(f1412,plain,(
% 2.95/1.73 ~spl52_3 | spl52_1 | ~spl52_2),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1299,f1345,f1341,f1349])).
% 2.95/1.73 thf(f1299,plain,(
% 2.95/1.73 (a2 != sK19) | (x = sK19) | (sP0 != $true)),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1140])).
% 2.95/1.73 thf(f1140,plain,(
% 2.95/1.73 (a2 != sK19) | (a2 != sK19) | (sP0 != $true) | (x = sK19)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f998])).
% 2.95/1.73 thf(f1411,plain,(
% 2.95/1.73 spl52_2 | ~spl52_3 | spl52_4),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1300,f1353,f1349,f1345])).
% 2.95/1.73 thf(f1353,definition,(
% 2.95/1.73 spl52_4 <=> ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) = $true)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_4])],[avatar_definition])).
% 2.95/1.73 thf(f1300,plain,(
% 2.95/1.73 (sP0 != $true) | ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) = $true) | (a2 = sK19)),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1135])).
% 2.95/1.73 thf(f1135,plain,(
% 2.95/1.73 ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) = $true) | (a2 = sK19) | ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) = $true) | (a2 = sK19) | (sP0 != $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f998])).
% 2.95/1.73 thf(f1409,plain,(
% 2.95/1.73 ~spl52_5 | spl52_7 | ~spl52_6),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1308,f1362,f1366,f1358])).
% 2.95/1.73 thf(f1362,definition,(
% 2.95/1.73 spl52_6 <=> ($true = sP1)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[spl52_6])],[avatar_definition])).
% 2.95/1.73 thf(f1308,plain,(
% 2.95/1.73 ($true != (member2 @ a @ sK18 @ (set2 @ a @ xsa))) | ($true != sP1) | (x = sK18)),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1131])).
% 2.95/1.73 thf(f1131,plain,(
% 2.95/1.73 ($true != sP1) | ($true != (member2 @ a @ sK18 @ (set2 @ a @ xsa))) | (x = sK18) | ($true != (member2 @ a @ sK18 @ (set2 @ a @ xsa)))),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f993])).
% 2.95/1.73 thf(f993,plain,(
% 2.95/1.73 (((ord_less @ a @ a2 @ x) != $true) & ((((a2 != sK18) & ($true != (member2 @ a @ sK18 @ (set2 @ a @ xsa)))) | (x = sK18) | (($true != (member2 @ a @ sK18 @ (set2 @ a @ xsa))) & (a2 != sK18))) & ((((a2 = sK18) | ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa)))) & (x != sK18)) | ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa))) | (a2 = sK18)))) | ($true != sP1)),
% 2.95/1.73 inference(skolemisation,[status(esa),new_symbols(skolem,[sK18])],[f991,f992])).
% 2.95/1.73 thf(f992,plain,(
% 2.95/1.73 ? [X0 : a] : ((((a2 != X0) & ($true != (member2 @ a @ X0 @ (set2 @ a @ xsa)))) | (x = X0) | (($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) & (a2 != X0))) & ((((a2 = X0) | ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa)))) & (x != X0)) | ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0))) => ((((a2 != sK18) & ($true != (member2 @ a @ sK18 @ (set2 @ a @ xsa)))) | (x = sK18) | (($true != (member2 @ a @ sK18 @ (set2 @ a @ xsa))) & (a2 != sK18))) & ((((a2 = sK18) | ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa)))) & (x != sK18)) | ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa))) | (a2 = sK18)))),
% 2.95/1.73 introduced(definition,[],[choice_axiom])).
% 2.95/1.73 thf(f991,plain,(
% 2.95/1.73 (((ord_less @ a @ a2 @ x) != $true) & ? [X0 : a] : ((((a2 != X0) & ($true != (member2 @ a @ X0 @ (set2 @ a @ xsa)))) | (x = X0) | (($true != (member2 @ a @ X0 @ (set2 @ a @ xsa))) & (a2 != X0))) & ((((a2 = X0) | ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa)))) & (x != X0)) | ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0)))) | ($true != sP1)),
% 2.95/1.73 inference(rectify,[status(thm)],[f990])).
% 2.95/1.73 thf(f990,plain,(
% 2.95/1.73 (((ord_less @ a @ a2 @ x) != $true) & ? [X2 : a] : ((((a2 != X2) & ($true != (member2 @ a @ X2 @ (set2 @ a @ xsa)))) | (x = X2) | (($true != (member2 @ a @ X2 @ (set2 @ a @ xsa))) & (a2 != X2))) & ((((a2 = X2) | ($true = (member2 @ a @ X2 @ (set2 @ a @ xsa)))) & (x != X2)) | ($true = (member2 @ a @ X2 @ (set2 @ a @ xsa))) | (a2 = X2)))) | ($true != sP1)),
% 2.95/1.73 inference(flattening,[status(thm)],[f989])).
% 2.95/1.73 thf(f989,plain,(
% 2.95/1.73 (((ord_less @ a @ a2 @ x) != $true) & ? [X2 : a] : (((((a2 != X2) & ($true != (member2 @ a @ X2 @ (set2 @ a @ xsa)))) | (x = X2)) | (($true != (member2 @ a @ X2 @ (set2 @ a @ xsa))) & (a2 != X2))) & ((((a2 = X2) | ($true = (member2 @ a @ X2 @ (set2 @ a @ xsa)))) & (x != X2)) | (($true = (member2 @ a @ X2 @ (set2 @ a @ xsa))) | (a2 = X2))))) | ($true != sP1)),
% 2.95/1.73 inference(nnf_transformation,[status(thm)],[f951])).
% 2.95/1.73 thf(f951,plain,(
% 2.95/1.73 (((ord_less @ a @ a2 @ x) != $true) & ? [X2 : a] : ((($true = (member2 @ a @ X2 @ (set2 @ a @ xsa))) | (a2 = X2)) <~> (((a2 = X2) | ($true = (member2 @ a @ X2 @ (set2 @ a @ xsa)))) & (x != X2)))) | ~($true = sP1)),
% 2.95/1.73 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.95/1.73 thf(f1408,plain,(
% 2.95/1.73 ~spl52_4 | ~spl52_3 | spl52_1),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1309,f1341,f1349,f1353])).
% 2.95/1.73 thf(f1309,plain,(
% 2.95/1.73 (sP0 != $true) | ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) != $true) | (x = sK19)),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1141])).
% 2.95/1.73 thf(f1141,plain,(
% 2.95/1.73 ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) != $true) | ((member2 @ a @ sK19 @ (set2 @ a @ xsa)) != $true) | (x = sK19) | (sP0 != $true) | (x = sK19)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f998])).
% 2.95/1.73 thf(f1404,plain,(
% 2.95/1.73 spl52_10 | ~spl52_9 | spl52_11),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1319,f1384,f1376,f1380])).
% 2.95/1.73 thf(f1319,plain,(
% 2.95/1.73 (sP2 != $true) | ($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (a2 = sK17)),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1124])).
% 2.95/1.73 thf(f1124,plain,(
% 2.95/1.73 (a2 = sK17) | (sP2 != $true) | ($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | ($true = (member2 @ a @ sK17 @ (set2 @ a @ xsa)))),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f988])).
% 2.95/1.73 thf(f1402,plain,(
% 2.95/1.73 spl52_6 | spl52_9 | spl52_3),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1145,f1349,f1376,f1362])).
% 2.95/1.73 thf(f1145,plain,(
% 2.95/1.73 ($true = sP1) | (sP2 = $true) | (sP0 = $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f953])).
% 2.95/1.73 thf(f953,plain,(
% 2.95/1.73 (sP2 = $true) | ((($true = sP1) | ((sP0 = $true) & ((ord_less @ a @ a2 @ x) = $true))) & (x != a2))),
% 2.95/1.73 inference(definition_folding,[status(thm)],[f877,f952,f951,f950])).
% 2.95/1.73 thf(f877,plain,(
% 2.95/1.73 ((x = a2) & ? [X0 : a] : (((($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0)) & (a2 != X0)) <~> ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))))) | (((((ord_less @ a @ a2 @ x) != $true) & ? [X2 : a] : ((($true = (member2 @ a @ X2 @ (set2 @ a @ xsa))) | (a2 = X2)) <~> (((a2 = X2) | ($true = (member2 @ a @ X2 @ (set2 @ a @ xsa)))) & (x != X2)))) | (? [X1 : a] : (((x != X1) & ((a2 = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true))) <~> (((x != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true)) | (a2 = X1))) & ((ord_less @ a @ a2 @ x) = $true))) & (x != a2))),
% 2.95/1.73 inference(ennf_transformation,[status(thm)],[f833])).
% 2.95/1.73 thf(f833,plain,(
% 2.95/1.73 ~((! [X0 : a] : (((($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0)) & (a2 != X0)) <=> ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa)))) | (x != a2)) & ((x = a2) | ((((ord_less @ a @ a2 @ x) = $true) | ! [X2 : a] : ((($true = (member2 @ a @ X2 @ (set2 @ a @ xsa))) | (a2 = X2)) <=> (((a2 = X2) | ($true = (member2 @ a @ X2 @ (set2 @ a @ xsa)))) & (x != X2)))) & (((ord_less @ a @ a2 @ x) != $true) | ! [X1 : a] : ((((x != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true)) | (a2 = X1)) <=> ((x != X1) & ((a2 = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true))))))))),
% 2.95/1.73 inference(flattening,[status(thm)],[f482])).
% 2.95/1.73 thf(f482,plain,(
% 2.95/1.73 ~((! [X0 : a] : (((($true = (member2 @ a @ X0 @ (set2 @ a @ xsa))) | (a2 = X0)) & (a2 != X0)) <=> ($true = (member2 @ a @ X0 @ (set2 @ a @ xsa)))) | (x != a2)) & ((x = a2) | ((! [X1 : a] : ((((x != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true)) | (a2 = X1)) <=> ((x != X1) & ((a2 = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) = $true)))) | ~((ord_less @ a @ a2 @ x) = $true)) & (((ord_less @ a @ a2 @ x) = $true) | ! [X2 : a] : ((($true = (member2 @ a @ X2 @ (set2 @ a @ xsa))) | (a2 = X2)) <=> (((a2 = X2) | ($true = (member2 @ a @ X2 @ (set2 @ a @ xsa)))) & (x != X2)))))))),
% 2.95/1.73 inference(fool_elimination,[status(thm)],[f481])).
% 2.95/1.73 thf(f481,plain,(
% 2.95/1.73 ~(((x != a2) | ! [X0 : a] : ((a2 != X0) & ((member2 @ a @ X0 @ (set2 @ a @ xsa)) | (a2 = X0)) = (member2 @ a @ X0 @ (set2 @ a @ xsa)))) & ((x = a2) | ((! [X1 : a] : ((x != X1) & ((member2 @ a @ X1 @ (set2 @ a @ xsa)) | (a2 = X1)) = (a2 = X1) | ((member2 @ a @ X1 @ (set2 @ a @ xsa)) & (x != X1))) | ~(ord_less @ a @ a2 @ x)) & ((ord_less @ a @ a2 @ x) | ! [X2 : a] : ((member2 @ a @ X2 @ (set2 @ a @ xsa)) | (a2 = X2) = ((a2 = X2) | (member2 @ a @ X2 @ (set2 @ a @ xsa))) & (x != X2))))))),
% 2.95/1.73 inference(rectify,[status(thm)],[f285])).
% 2.95/1.73 thf(f285,negated_conjecture,(
% 2.95/1.73 ~(((x != a2) | ! [X17 : a] : ((a2 != X17) & ((member2 @ a @ X17 @ (set2 @ a @ xsa)) | (a2 = X17)) = (member2 @ a @ X17 @ (set2 @ a @ xsa)))) & ((x = a2) | ((! [X17 : a] : ((x != X17) & ((member2 @ a @ X17 @ (set2 @ a @ xsa)) | (a2 = X17)) = (a2 = X17) | ((member2 @ a @ X17 @ (set2 @ a @ xsa)) & (x != X17))) | ~(ord_less @ a @ a2 @ x)) & ((ord_less @ a @ a2 @ x) | ! [X17 : a] : ((member2 @ a @ X17 @ (set2 @ a @ xsa)) | (a2 = X17) = ((a2 = X17) | (member2 @ a @ X17 @ (set2 @ a @ xsa))) & (x != X17))))))),
% 2.95/1.73 inference(negated_conjecture,[status(cth)],[f284])).
% 2.95/1.73 thf(f284,conjecture,(
% 2.95/1.73 ((x != a2) | ! [X17 : a] : ((a2 != X17) & ((member2 @ a @ X17 @ (set2 @ a @ xsa)) | (a2 = X17)) = (member2 @ a @ X17 @ (set2 @ a @ xsa)))) & ((x = a2) | ((! [X17 : a] : ((x != X17) & ((member2 @ a @ X17 @ (set2 @ a @ xsa)) | (a2 = X17)) = (a2 = X17) | ((member2 @ a @ X17 @ (set2 @ a @ xsa)) & (x != X17))) | ~(ord_less @ a @ a2 @ x)) & ((ord_less @ a @ a2 @ x) | ! [X17 : a] : ((member2 @ a @ X17 @ (set2 @ a @ xsa)) | (a2 = X17) = ((a2 = X17) | (member2 @ a @ X17 @ (set2 @ a @ xsa))) & (x != X17)))))),
% 2.95/1.73 file('/export/starexec/sandbox2/tmp/tmp.brU5k5c5ot/DTF2THF_27099.p',conj_3)).
% 2.95/1.73 thf(f1401,plain,(
% 2.95/1.73 spl52_6 | spl52_12 | spl52_9),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1144,f1376,f1389,f1362])).
% 2.95/1.73 thf(f1144,plain,(
% 2.95/1.73 ($true = sP1) | ((ord_less @ a @ a2 @ x) = $true) | (sP2 = $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f953])).
% 2.95/1.73 thf(f1400,plain,(
% 2.95/1.73 ~spl52_9 | spl52_13),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1127,f1395,f1376])).
% 2.95/1.73 thf(f1127,plain,(
% 2.95/1.73 (x = a2) | (sP2 != $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f988])).
% 2.95/1.73 thf(f1399,plain,(
% 2.95/1.73 ~spl52_6 | spl52_8 | spl52_5),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1321,f1358,f1370,f1362])).
% 2.95/1.73 thf(f1321,plain,(
% 2.95/1.73 ($true != sP1) | (a2 = sK18) | ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa)))),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1129])).
% 2.95/1.73 thf(f1129,plain,(
% 2.95/1.73 ($true != sP1) | ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa))) | (a2 = sK18) | (a2 = sK18) | ($true = (member2 @ a @ sK18 @ (set2 @ a @ xsa)))),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f993])).
% 2.95/1.73 thf(f1398,plain,(
% 2.95/1.73 spl52_9 | ~spl52_13),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1143,f1395,f1376])).
% 2.95/1.73 thf(f1143,plain,(
% 2.95/1.73 (x != a2) | (sP2 = $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f953])).
% 2.95/1.73 thf(f1393,plain,(
% 2.95/1.73 ~spl52_8 | spl52_7 | ~spl52_6),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1326,f1362,f1366,f1370])).
% 2.95/1.73 thf(f1326,plain,(
% 2.95/1.73 ($true != sP1) | (x = sK18) | (a2 != sK18)),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1132])).
% 2.95/1.73 thf(f1132,plain,(
% 2.95/1.73 ($true != sP1) | (a2 != sK18) | (a2 != sK18) | (x = sK18)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f993])).
% 2.95/1.73 thf(f1392,plain,(
% 2.95/1.73 ~spl52_12 | ~spl52_6),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1134,f1362,f1389])).
% 2.95/1.73 thf(f1134,plain,(
% 2.95/1.73 ((ord_less @ a @ a2 @ x) != $true) | ($true != sP1)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f993])).
% 2.95/1.73 thf(f1387,plain,(
% 2.95/1.73 ~spl52_9 | spl52_10 | ~spl52_11),
% 2.95/1.73 inference(avatar_split_clause,[status(thm)],[f1329,f1384,f1380,f1376])).
% 2.95/1.73 thf(f1329,plain,(
% 2.95/1.73 ($true != (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (a2 = sK17) | (sP2 != $true)),
% 2.95/1.73 inference(duplicate_literal_removal,[status(thm)],[f1126])).
% 2.95/1.73 thf(f1126,plain,(
% 2.95/1.73 ($true != (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (a2 = sK17) | ($true != (member2 @ a @ sK17 @ (set2 @ a @ xsa))) | (sP2 != $true)),
% 2.95/1.73 inference(cnf_transformation,[status(thm)],[f988])).
% 2.95/1.73 % SZS output end Proof for DTF2THF_27099
% 2.95/1.73 % (27303)------------------------------
% 2.95/1.73 % (27303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.73 % (27303)Termination reason: Refutation
% 2.95/1.73
% 2.95/1.73 % (27303)Memory used [KB]: 7036
% 2.95/1.73 % (27303)Time elapsed: 0.120 s
% 2.95/1.73 % (27303)Instructions burned: 214 (million)
% 2.95/1.73 % (27303)------------------------------
% 2.95/1.73 % (27303)------------------------------
% 2.95/1.73 % (27275)Success in time 0.204 s
%------------------------------------------------------------------------------