%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP033^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:00 AM UTC 2026
% Result : Theorem 2.90s 1.70s
% Output : Refutation 2.90s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP033^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n016.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:24:36 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP033^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer BinaryTree problem prob_508__3255320_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : BinaryTree/prob_508__3255320_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.00 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 327 ( 90 unt; 47 typ; 0 def)
% 0.29/0.45 % Number of atoms : 862 ( 209 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 13 ( 3 avg)
% 0.29/0.45 % Number of connectives : 3488 ( 101 ~; 19 |; 66 &;2845 @)
% 0.29/0.45 % ( 0 <=>; 457 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 21 ( 9 avg)
% 0.29/0.45 % Number of types : 3 ( 2 usr)
% 0.29/0.45 % Number of type conns : 315 ( 315 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 48 ( 45 usr; 5 con; 0-5 aty)
% 0.29/0.45 % Number of variables : 1166 ( 116 ^; 971 !; 34 ?;1166 :)
% 0.29/0.45 % ( 45 !>; 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 Sledgehammer 2021-02-23 16:16:50.765
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (5)
% 0.29/0.45 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
% 0.29/0.45 binary1291135688e_Tree: $tType > $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_t_Product__Type_Oprod,type,
% 0.29/0.45 product_prod: $tType > $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_Int_Oint,type,
% 0.29/0.45 int: $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 (42)
% 0.29/0.45 thf(sy_cl_HOL_Otype,type,
% 0.29/0.45 type:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Obot,type,
% 0.29/0.45 bot:
% 0.29/0.45 !>[A: $tType] : $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] : $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] : $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] : $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] : $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] : $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] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.29/0.45 order_bot:
% 0.29/0.45 !>[A: $tType] : $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] : $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] : $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] : $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] : $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] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.29/0.45 bNF_Ca1785829860lChain:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
% 0.29/0.45 binary210054475elle_T:
% 0.29/0.45 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
% 0.29/0.45 binary1746293266le_Tip:
% 0.29/0.45 !>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Ocase__Tree,type,
% 0.29/0.45 binary536355927e_Tree:
% 0.29/0.45 !>[B: $tType,A: $tType] : ( B > ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B ) > ( binary1291135688e_Tree @ A ) > B ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Orec__Tree,type,
% 0.29/0.45 binary1929596613c_Tree:
% 0.29/0.45 !>[C: $tType,A: $tType] : ( C > ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C ) > ( binary1291135688e_Tree @ A ) > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert,type,
% 0.29/0.45 binary1830089824insert:
% 0.29/0.45 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
% 0.29/0.45 binary827270440e_memb:
% 0.29/0.45 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oremove,type,
% 0.29/0.45 binary997842527remove:
% 0.29/0.45 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Orm,type,
% 0.29/0.45 binary576689334lle_rm:
% 0.29/0.45 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
% 0.29/0.45 binary1653327646_setOf:
% 0.29/0.45 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
% 0.29/0.45 binary1610619414edTree:
% 0.29/0.45 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
% 0.29/0.45 binary231205461t_pred:
% 0.29/0.45 !>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrm,type,
% 0.29/0.45 binary213313527le_wrm:
% 0.29/0.45 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrmrm,type,
% 0.29/0.45 binary1271298290_wrmrm:
% 0.29/0.45 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( product_prod @ ( binary1291135688e_Tree @ A ) @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.29/0.45 bot_bot:
% 0.29/0.45 !>[A: $tType] : A ).
% 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_Orderings_Oordering__top,type,
% 0.29/0.45 ordering_top:
% 0.29/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_OPair,type,
% 0.29/0.45 product_Pair:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.29/0.45 product_case_prod:
% 0.29/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Relation_Oinv__image,type,
% 0.29/0.45 inv_image:
% 0.29/0.45 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ 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_Set_Ois__empty,type,
% 0.29/0.45 is_empty:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member,type,
% 0.29/0.45 member:
% 0.29/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_h,type,
% 0.29/0.45 h: a > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_t1____,type,
% 0.29/0.45 t1: binary1291135688e_Tree @ a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_t2____,type,
% 0.29/0.45 t2: binary1291135688e_Tree @ 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 % Relevant facts (256)
% 0.29/0.45 thf(fact_0_t2nTip,axiom,
% 0.29/0.45 ( t2
% 0.29/0.45 != ( binary1746293266le_Tip @ a ) ) ).
% 0.29/0.45
% 0.29/0.45 % t2nTip
% 0.29/0.45 thf(fact_1_s,axiom,
% 0.29/0.45 binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% 0.29/0.45
% 0.29/0.45 % s
% 0.29/0.45 thf(fact_2_res,axiom,
% 0.29/0.45 ( ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
% 0.29/0.45 = ( binary210054475elle_T @ a @ t1 @ x @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % res
% 0.29/0.45 thf(fact_3_sortLemmaL,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 => ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sortLemmaL
% 0.29/0.45 thf(fact_4_sortLemmaR,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 => ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sortLemmaR
% 0.29/0.45 thf(fact_5_Tree_Oinject,axiom,
% 0.29/0.45 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
% 0.29/0.45 = ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
% 0.29/0.45 = ( ( X21 = Y21 )
% 0.29/0.45 & ( X22 = Y22 )
% 0.29/0.45 & ( X23 = Y23 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.inject
% 0.29/0.45 thf(fact_6_binsert__sorted,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
% 0.29/0.45 ( ( binary1610619414edTree @ A @ H @ T )
% 0.29/0.45 => ( binary1610619414edTree @ A @ H @ ( binary1830089824insert @ A @ H @ X @ T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % binsert_sorted
% 0.29/0.45 thf(fact_7_h1,axiom,
% 0.29/0.45 ( ( ( t1
% 0.29/0.45 != ( binary1746293266le_Tip @ a ) )
% 0.29/0.45 & ( binary1610619414edTree @ a @ h @ t1 ) )
% 0.29/0.45 => ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ t1 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % h1
% 0.29/0.45 thf(fact_8_sortedTree_Osimps_I1_J,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % sortedTree.simps(1)
% 0.29/0.45 thf(fact_9_memb__spec,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
% 0.29/0.45 ( ( binary1610619414edTree @ A @ H @ T )
% 0.29/0.45 => ( ( binary827270440e_memb @ A @ H @ X @ T )
% 0.29/0.45 = ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % memb_spec
% 0.29/0.45 thf(fact_10_sorted__distinct__pred__def,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( binary231205461t_pred @ A )
% 0.29/0.45 = ( ^ [H2: A > int,A2: A,B2: A,T3: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( ( binary1610619414edTree @ A @ H2 @ T3 )
% 0.29/0.45 & ( member @ A @ A2 @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.29/0.45 & ( member @ A @ B2 @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.29/0.45 & ( ( H2 @ A2 )
% 0.29/0.45 = ( H2 @ B2 ) ) )
% 0.29/0.45 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sorted_distinct_pred_def
% 0.29/0.45 thf(fact_11_h2,axiom,
% 0.29/0.45 ( ( ( t2
% 0.29/0.45 != ( binary1746293266le_Tip @ a ) )
% 0.29/0.45 & ( binary1610619414edTree @ a @ h @ t2 ) )
% 0.29/0.45 => ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % h2
% 0.29/0.45 thf(fact_12_sorted__distinct,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,A3: A,B3: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A3 @ B3 @ T ) ).
% 0.29/0.45
% 0.29/0.45 % sorted_distinct
% 0.29/0.45 thf(fact_13__092_060open_062Tip_A_092_060noteq_062_ATip_A_092_060and_062_AsortedTree_Ah_ATip_A_092_060longrightarrow_062_AsortedTree_Ah_A_Iwrm_Ah_ATip_J_092_060close_062,axiom,
% 0.29/0.45 ( ( ( ( binary1746293266le_Tip @ a )
% 0.29/0.45 != ( binary1746293266le_Tip @ a ) )
% 0.29/0.45 & ( binary1610619414edTree @ a @ h @ ( binary1746293266le_Tip @ a ) ) )
% 0.29/0.45 => ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ ( binary1746293266le_Tip @ a ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>Tip \<noteq> Tip \<and> sortedTree h Tip \<longrightarrow> sortedTree h (wrm h Tip)\<close>
% 0.29/0.45 thf(fact_14_Tree_Odistinct_I1_J,axiom,
% 0.29/0.45 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( binary1746293266le_Tip @ A )
% 0.29/0.45 != ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.distinct(1)
% 0.29/0.45 thf(fact_15_binsert_Osimps_I1_J,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,E: A] :
% 0.29/0.45 ( ( binary1830089824insert @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 = ( binary210054475elle_T @ A @ ( binary1746293266le_Tip @ A ) @ E @ ( binary1746293266le_Tip @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % binsert.simps(1)
% 0.29/0.45 thf(fact_16_wrm_Osimps,axiom,
% 0.29/0.45 ! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
% 0.29/0.45 ( ( ( T2
% 0.29/0.45 = ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 => ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 = T1 ) )
% 0.29/0.45 & ( ( T2
% 0.29/0.45 != ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 => ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary213313527le_wrm @ A @ H @ T2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % wrm.simps
% 0.29/0.45 thf(fact_17_Tree_Oinduct,axiom,
% 0.29/0.45 ! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( P @ ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 => ( ! [X1: binary1291135688e_Tree @ A,X2: A,X3: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( P @ X1 )
% 0.29/0.45 => ( ( P @ X3 )
% 0.29/0.45 => ( P @ ( binary210054475elle_T @ A @ X1 @ X2 @ X3 ) ) ) )
% 0.29/0.45 => ( P @ Tree ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.induct
% 0.29/0.45 thf(fact_18_Tree_Oexhaust,axiom,
% 0.29/0.45 ! [A: $tType,Y: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( Y
% 0.29/0.45 != ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 => ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.exhaust
% 0.29/0.45 thf(fact_19_rm__set,axiom,
% 0.29/0.45 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.29/0.45 ( ( ( T
% 0.29/0.45 != ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.29/0.45 => ( member @ A @ ( binary576689334lle_rm @ A @ H @ T ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rm_set
% 0.29/0.45 thf(fact_20_rm_Osimps,axiom,
% 0.29/0.45 ! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
% 0.29/0.45 ( ( ( T2
% 0.29/0.45 = ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 => ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 = X ) )
% 0.29/0.45 & ( ( T2
% 0.29/0.45 != ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 => ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 = ( binary576689334lle_rm @ A @ H @ T2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rm.simps
% 0.29/0.45 thf(fact_21_wrm__set1,axiom,
% 0.29/0.45 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.29/0.45 ( ( ( T
% 0.29/0.45 != ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ ( binary1653327646_setOf @ A @ ( binary213313527le_wrm @ A @ H @ T ) ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % wrm_set1
% 0.29/0.45 thf(fact_22_sortedTree_Osimps_I2_J,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 = ( ( binary1610619414edTree @ A @ H @ T1 )
% 0.29/0.45 & ! [X4: A] :
% 0.29/0.45 ( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T1 ) )
% 0.29/0.45 => ( ord_less @ int @ ( H @ X4 ) @ ( H @ X ) ) )
% 0.29/0.45 & ! [X4: A] :
% 0.29/0.45 ( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T2 ) )
% 0.29/0.45 => ( ord_less @ int @ ( H @ X ) @ ( H @ X4 ) ) )
% 0.29/0.45 & ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sortedTree.simps(2)
% 0.29/0.45 thf(fact_23_remove_Osimps_I1_J,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,E: A] :
% 0.29/0.45 ( ( binary997842527remove @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 = ( binary1746293266le_Tip @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % remove.simps(1)
% 0.29/0.45 thf(fact_24_Tree_Osimps_I4_J,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B] :
% 0.29/0.45 ( ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 = F1 ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.simps(4)
% 0.29/0.45 thf(fact_25_Tree_Osimps_I5_J,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
% 0.29/0.45 = ( F2 @ X21 @ X22 @ X23 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.simps(5)
% 0.29/0.45 thf(fact_26_binsert_Osimps_I2_J,axiom,
% 0.29/0.45 ! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.29/0.45 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 = ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
% 0.29/0.45 & ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.29/0.45 => ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.29/0.45 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
% 0.29/0.45 & ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.29/0.45 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.45 = ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % binsert.simps(2)
% 0.29/0.45 thf(fact_27_setOf_Osimps_I1_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % setOf.simps(1)
% 0.29/0.45 thf(fact_28_Tree_Osimps_I6_J,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,F1: C,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C] :
% 0.29/0.45 ( ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ ( binary1746293266le_Tip @ A ) )
% 0.29/0.45 = F1 ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.simps(6)
% 0.29/0.45 thf(fact_29_Tree_Osimps_I7_J,axiom,
% 0.29/0.45 ! [C: $tType,A: $tType,F1: C,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
% 0.29/0.45 = ( F2 @ X21 @ X22 @ X23 @ ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ X21 ) @ ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ X23 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.simps(7)
% 0.29/0.45 thf(fact_30_Tree_Ocase__distrib,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B,Tree: binary1291135688e_Tree @ A] :
% 0.29/0.45 ( ( H @ ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ Tree ) )
% 0.29/0.45 = ( binary536355927e_Tree @ C @ A @ ( H @ F1 )
% 0.29/0.45 @ ^ [X12: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] : ( H @ ( F2 @ X12 @ X24 @ X32 ) )
% 0.29/0.45 @ Tree ) ) ).
% 0.29/0.45
% 0.29/0.45 % Tree.case_distrib
% 0.29/0.45 thf(fact_31_empty__subsetI,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.29/0.45
% 0.29/0.45 % empty_subsetI
% 0.29/0.45 thf(fact_32_subset__empty,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.45 = ( A4
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_empty
% 0.29/0.45 thf(fact_33_subset__antisym,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.29/0.45 => ( A4 = B4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_antisym
% 0.29/0.45 thf(fact_34_subsetI,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.29/0.45 ( ! [X5: A] :
% 0.29/0.45 ( ( member @ A @ X5 @ A4 )
% 0.29/0.45 => ( member @ A @ X5 @ B4 ) )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetI
% 0.29/0.45 thf(fact_35_empty__Collect__eq,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o] :
% 0.29/0.45 ( ( ( bot_bot @ ( set @ A ) )
% 0.29/0.45 = ( collect @ A @ P ) )
% 0.29/0.45 = ( ! [X4: A] :
% 0.29/0.45 ~ ( P @ X4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % empty_Collect_eq
% 0.29/0.45 thf(fact_36_Collect__empty__eq,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o] :
% 0.29/0.45 ( ( ( collect @ A @ P )
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.45 = ( ! [X4: A] :
% 0.29/0.45 ~ ( P @ X4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_empty_eq
% 0.29/0.45 thf(fact_37_all__not__in__conv,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] :
% 0.29/0.45 ( ( ! [X4: A] :
% 0.29/0.45 ~ ( member @ A @ X4 @ A4 ) )
% 0.29/0.45 = ( A4
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % all_not_in_conv
% 0.29/0.45 thf(fact_38_empty__iff,axiom,
% 0.29/0.45 ! [A: $tType,C2: A] :
% 0.29/0.45 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % empty_iff
% 0.29/0.45 thf(fact_39_bot__apply,axiom,
% 0.29/0.45 ! [C: $tType,D: $tType] :
% 0.29/0.45 ( ( bot @ C )
% 0.29/0.45 => ( ( bot_bot @ ( D > C ) )
% 0.29/0.45 = ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_apply
% 0.29/0.45 thf(fact_40_order__refl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_41_bot__set__def,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bot_bot @ ( set @ A ) )
% 0.29/0.45 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_set_def
% 0.29/0.45 thf(fact_42_le__funD,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B )
% 0.29/0.45 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funD
% 0.29/0.45 thf(fact_43_le__funE,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B )
% 0.29/0.45 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funE
% 0.29/0.45 thf(fact_44_le__funI,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B )
% 0.29/0.45 => ! [F: A > B,G: A > B] :
% 0.29/0.45 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 0.29/0.45 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funI
% 0.29/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A: $tType,A3: A,P: A > $o] :
% 0.29/0.45 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.29/0.45 = ( P @ A3 ) ) ).
% 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 @ ^ [X4: A] : ( member @ A @ X4 @ 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 ( ! [X5: A] :
% 0.29/0.45 ( ( P @ X5 )
% 0.29/0.45 = ( Q @ X5 ) )
% 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,G: A > B] :
% 0.29/0.45 ( ! [X5: A] :
% 0.29/0.45 ( ( F @ X5 )
% 0.29/0.45 = ( G @ X5 ) )
% 0.29/0.45 => ( F = G ) ) ).
% 0.29/0.45
% 0.29/0.45 % ext
% 0.29/0.45 thf(fact_49_le__fun__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B )
% 0.29/0.45 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.45 = ( ^ [F3: A > B,G2: A > B] :
% 0.29/0.45 ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_fun_def
% 0.29/0.45 thf(fact_50_order__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( order @ B )
% 0.29/0.45 & ( order @ A ) )
% 0.29/0.45 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.29/0.45 => ( ! [X5: B,Y2: B] :
% 0.29/0.45 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_51_order__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType] :
% 0.29/0.45 ( ( ( order @ C )
% 0.29/0.45 & ( order @ A ) )
% 0.29/0.45 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.29/0.45 => ( ! [X5: A,Y2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.45 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_52_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B )
% 0.29/0.45 & ( ord @ A ) )
% 0.29/0.45 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.29/0.45 => ( ! [X5: B,Y2: B] :
% 0.29/0.45 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_53_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B )
% 0.29/0.45 & ( ord @ A ) )
% 0.29/0.45 => ! [A3: A,B3: A,F: A > B,C2: B] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( ( F @ B3 )
% 0.29/0.45 = C2 )
% 0.29/0.45 => ( ! [X5: A,Y2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_54_eq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.29/0.45 = ( ^ [X4: A,Y4: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.29/0.45 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_iff
% 0.29/0.45 thf(fact_55_antisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ 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_56_linear,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_57_eq__refl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_58_le__cases,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_59_order_Otrans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.29/0.45 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.trans
% 0.29/0.45 thf(fact_60_le__cases3,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_61_antisym__conv,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ 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_62_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 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
% 0.29/0.45 % order_class.order.eq_iff
% 0.29/0.45 thf(fact_63_ord__eq__le__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A )
% 0.29/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.29/0.45 ( ( A3 = B3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.29/0.45 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_trans
% 0.29/0.45 thf(fact_64_ord__le__eq__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A )
% 0.29/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( B3 = C2 )
% 0.29/0.45 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_trans
% 0.29/0.45 thf(fact_65_order__class_Oorder_Oantisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [A3: A,B3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.45 => ( A3 = B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.antisym
% 0.29/0.45 thf(fact_66_order__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_67_dual__order_Orefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.refl
% 0.29/0.45 thf(fact_68_linorder__wlog,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A )
% 0.29/0.45 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.29/0.45 ( ! [A5: A,B5: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.29/0.45 => ( P @ A5 @ B5 ) )
% 0.29/0.45 => ( ! [A5: A,B5: A] :
% 0.29/0.45 ( ( P @ B5 @ A5 )
% 0.29/0.45 => ( P @ A5 @ B5 ) )
% 0.29/0.45 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_wlog
% 0.29/0.45 thf(fact_69_dual__order_Otrans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [B3: A,A3: A,C2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.29/0.45 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.trans
% 0.29/0.45 thf(fact_70_dual__order_Oeq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.29/0.45 = ( ^ [A2: A,B2: 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
% 0.29/0.45 % dual_order.eq_iff
% 0.29/0.45 thf(fact_71_dual__order_Oantisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [B3: A,A3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.45 => ( A3 = B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.antisym
% 0.29/0.45 thf(fact_72_ord__eq__less__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B )
% 0.29/0.45 & ( ord @ A ) )
% 0.29/0.45 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less @ B @ B3 @ C2 )
% 0.29/0.45 => ( ! [X5: B,Y2: B] :
% 0.29/0.45 ( ( ord_less @ B @ X5 @ Y2 )
% 0.29/0.45 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_less_subst
% 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 )
% 0.29/0.45 & ( ord @ A ) )
% 0.29/0.45 => ! [A3: A,B3: A,F: A > B,C2: B] :
% 0.29/0.45 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( ( F @ B3 )
% 0.29/0.45 = C2 )
% 0.29/0.45 => ( ! [X5: A,Y2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.45 => ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_less_eq_subst
% 0.29/0.45 thf(fact_74_order__less__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( order @ B )
% 0.29/0.45 & ( order @ A ) )
% 0.29/0.45 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.29/0.45 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less @ B @ B3 @ C2 )
% 0.29/0.45 => ( ! [X5: B,Y2: B] :
% 0.29/0.45 ( ( ord_less @ B @ X5 @ Y2 )
% 0.29/0.45 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_less_subst1
% 0.29/0.45 thf(fact_75_order__less__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType] :
% 0.29/0.45 ( ( ( order @ C )
% 0.29/0.45 & ( order @ A ) )
% 0.29/0.45 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.29/0.45 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 0.29/0.45 => ( ! [X5: A,Y2: A] :
% 0.29/0.45 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.45 => ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_less_subst2
% 0.29/0.45 thf(fact_76_lt__ex,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( no_bot @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % lt_ex
% 0.29/0.45 thf(fact_77_gt__ex,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( no_top @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.29/0.45
% 0.29/0.45 % gt_ex
% 0.29/0.45 thf(fact_78_neqE,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_79_neq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_80_order_Oasym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [A3: A,B3: A] :
% 0.29/0.45 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.45 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.asym
% 0.29/0.45 thf(fact_81_dense,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_order @ A )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ? [Z3: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Z3 )
% 0.29/0.45 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense
% 0.29/0.45 thf(fact_82_less__imp__neq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ 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_83_less__asym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_84_less__asym_H,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A )
% 0.29/0.45 => ! [A3: A,B3: A] :
% 0.29/0.45 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.45 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_asym'
% 0.29/0.45 thf(fact_85_less__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_86_less__linear,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_87_less__irrefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_88_ord__eq__less__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A )
% 0.29/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.29/0.45 ( ( A3 = B3 )
% 0.29/0.45 => ( ( ord_less @ A @ B3 @ C2 )
% 0.29/0.45 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_less_trans
% 0.29/0.45 thf(fact_89_ord__less__eq__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A )
% 0.29/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.29/0.45 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( B3 = C2 )
% 0.29/0.45 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_less_eq_trans
% 0.29/0.45 thf(fact_90_dual__order_Oasym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [B3: A,A3: A] :
% 0.29/0.45 ( ( ord_less @ A @ B3 @ A3 )
% 0.29/0.45 => ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.asym
% 0.29/0.45 thf(fact_91_less__imp__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ 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_92_less__not__sym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_93_less__induct,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( wellorder @ A )
% 0.29/0.45 => ! [P: A > $o,A3: A] :
% 0.29/0.45 ( ! [X5: A] :
% 0.29/0.45 ( ! [Y5: A] :
% 0.29/0.45 ( ( ord_less @ A @ Y5 @ X5 )
% 0.29/0.45 => ( P @ Y5 ) )
% 0.29/0.45 => ( P @ X5 ) )
% 0.29/0.45 => ( P @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_induct
% 0.29/0.45 thf(fact_94_antisym__conv3,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_95_less__imp__not__eq2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ 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_96_less__imp__triv,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_97_linorder__cases,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_98_dual__order_Oirrefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [A3: A] :
% 0.29/0.45 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.irrefl
% 0.29/0.45 thf(fact_99_order_Ostrict__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.29/0.45 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less @ A @ B3 @ C2 )
% 0.29/0.45 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_trans
% 0.29/0.45 thf(fact_100_less__imp__not__less,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ 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_101_exists__least__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( wellorder @ A )
% 0.29/0.45 => ( ( ^ [P2: A > $o] :
% 0.29/0.45 ? [X6: A] : ( P2 @ X6 ) )
% 0.29/0.45 = ( ^ [P3: A > $o] :
% 0.29/0.45 ? [N: A] :
% 0.29/0.45 ( ( P3 @ N )
% 0.29/0.45 & ! [M: A] :
% 0.29/0.45 ( ( ord_less @ A @ M @ N )
% 0.29/0.45 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % exists_least_iff
% 0.29/0.45 thf(fact_102_linorder__less__wlog,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A )
% 0.29/0.45 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.29/0.45 ( ! [A5: A,B5: A] :
% 0.29/0.45 ( ( ord_less @ A @ A5 @ B5 )
% 0.29/0.45 => ( P @ A5 @ B5 ) )
% 0.29/0.45 => ( ! [A5: A] : ( P @ A5 @ A5 )
% 0.29/0.45 => ( ! [A5: A,B5: A] :
% 0.29/0.45 ( ( P @ B5 @ A5 )
% 0.29/0.45 => ( P @ A5 @ B5 ) )
% 0.29/0.45 => ( P @ A3 @ B3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_less_wlog
% 0.29/0.45 thf(fact_103_dual__order_Ostrict__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [B3: A,A3: A,C2: A] :
% 0.29/0.45 ( ( ord_less @ A @ B3 @ A3 )
% 0.29/0.45 => ( ( ord_less @ A @ C2 @ B3 )
% 0.29/0.45 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_trans
% 0.29/0.45 thf(fact_104_not__less__iff__gr__or__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ 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_105_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [A3: A,B3: A] :
% 0.29/0.45 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.45 => ( A3 != B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_implies_not_eq
% 0.29/0.45 thf(fact_106_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [B3: A,A3: A] :
% 0.29/0.45 ( ( ord_less @ A @ B3 @ A3 )
% 0.29/0.45 => ( A3 != B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_implies_not_eq
% 0.29/0.45 thf(fact_107_bot__fun__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( bot @ B )
% 0.29/0.45 => ( ( bot_bot @ ( A > B ) )
% 0.29/0.45 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_fun_def
% 0.29/0.45 thf(fact_108_emptyE,axiom,
% 0.29/0.45 ! [A: $tType,A3: A] :
% 0.29/0.45 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % emptyE
% 0.29/0.45 thf(fact_109_equals0D,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,A3: A] :
% 0.29/0.45 ( ( A4
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.45 => ~ ( member @ A @ A3 @ A4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % equals0D
% 0.29/0.45 thf(fact_110_equals0I,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] :
% 0.29/0.45 ( ! [Y2: A] :
% 0.29/0.45 ~ ( member @ A @ Y2 @ A4 )
% 0.29/0.45 => ( A4
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equals0I
% 0.29/0.45 thf(fact_111_ex__in__conv,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A] :
% 0.29/0.45 ( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.29/0.45 = ( A4
% 0.29/0.45 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ex_in_conv
% 0.29/0.45 thf(fact_112_in__mono,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.45 => ( ( member @ A @ X @ A4 )
% 0.29/0.45 => ( member @ A @ X @ B4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % in_mono
% 0.29/0.45 thf(fact_113_subsetD,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.45 => ( ( member @ A @ C2 @ A4 )
% 0.29/0.45 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_114_equalityE,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.29/0.45 ( ( A4 = B4 )
% 0.29/0.45 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.45 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityE
% 0.29/0.45 thf(fact_115_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,B6: set @ A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( member @ A @ X4 @ A6 )
% 0.29/0.45 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_116_equalityD1,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.29/0.45 ( ( A4 = B4 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD1
% 0.29/0.45 thf(fact_117_equalityD2,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.29/0.45 ( ( A4 = B4 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD2
% 0.29/0.45 thf(fact_118_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,B6: set @ A] :
% 0.29/0.45 ! [T3: A] :
% 0.29/0.45 ( ( member @ A @ T3 @ A6 )
% 0.29/0.45 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_119_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_120_Collect__mono,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.45 ( ! [X5: A] :
% 0.29/0.45 ( ( P @ X5 )
% 0.29/0.45 => ( Q @ X5 ) )
% 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_121_subset__trans,axiom,
% 0.29/0.45 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_trans
% 0.29/0.45 thf(fact_122_set__eq__subset,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ^ [Y3: set @ A,Z: set @ A] : ( Y3 = Z ) )
% 0.29/0.45 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.29/0.45 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_eq_subset
% 0.29/0.45 thf(fact_123_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 = ( ! [X4: A] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.46 => ( Q @ X4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono_iff
% 0.29/0.46 thf(fact_124_empty__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( collect @ A
% 0.29/0.46 @ ^ [X4: A] : $false ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_def
% 0.29/0.46 thf(fact_125_Collect__subset,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,P: A > $o] :
% 0.29/0.46 ( ord_less_eq @ ( set @ A )
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ A4 )
% 0.29/0.46 & ( P @ X4 ) ) )
% 0.29/0.46 @ A4 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_subset
% 0.29/0.46 thf(fact_126_leD,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % leD
% 0.29/0.46 thf(fact_127_leI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % leI
% 0.29/0.46 thf(fact_128_le__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [X4: A,Y4: A] :
% 0.29/0.46 ( ( ord_less @ A @ X4 @ Y4 )
% 0.29/0.46 | ( X4 = Y4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less
% 0.29/0.46 thf(fact_129_less__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [X4: A,Y4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.29/0.46 & ( X4 != Y4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le
% 0.29/0.46 thf(fact_130_order__le__less__subst1,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( order @ B )
% 0.29/0.46 & ( order @ A ) )
% 0.29/0.46 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less @ B @ B3 @ C2 )
% 0.29/0.46 => ( ! [X5: B,Y2: B] :
% 0.29/0.46 ( ( ord_less @ B @ X5 @ Y2 )
% 0.29/0.46 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst1
% 0.29/0.46 thf(fact_131_order__le__less__subst2,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType] :
% 0.29/0.46 ( ( ( order @ C )
% 0.29/0.46 & ( order @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X5: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst2
% 0.29/0.46 thf(fact_132_order__less__le__subst1,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( order @ B )
% 0.29/0.46 & ( order @ A ) )
% 0.29/0.46 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.29/0.46 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.29/0.46 => ( ! [X5: B,Y2: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst1
% 0.29/0.46 thf(fact_133_order__less__le__subst2,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType] :
% 0.29/0.46 ( ( ( order @ C )
% 0.29/0.46 & ( order @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.29/0.46 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X5: A,Y2: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.46 => ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst2
% 0.29/0.46 thf(fact_134_not__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.29/0.46 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_le
% 0.29/0.46 thf(fact_135_not__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.46 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less
% 0.29/0.46 thf(fact_136_le__neq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( A3 != B3 )
% 0.29/0.46 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_neq_trans
% 0.29/0.46 thf(fact_137_antisym__conv1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv1
% 0.29/0.46 thf(fact_138_antisym__conv2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv2
% 0.29/0.46 thf(fact_139_less__imp__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_le
% 0.29/0.46 thf(fact_140_le__less__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less @ A @ Y @ Z2 )
% 0.29/0.46 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_trans
% 0.29/0.46 thf(fact_141_less__le__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.29/0.46 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le_trans
% 0.29/0.46 thf(fact_142_dense__ge,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( dense_linorder @ A )
% 0.29/0.46 => ! [Z2: A,Y: A] :
% 0.29/0.46 ( ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ X5 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_ge
% 0.29/0.46 thf(fact_143_dense__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( dense_linorder @ A )
% 0.29/0.46 => ! [Y: A,Z2: A] :
% 0.29/0.46 ( ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ X5 @ Z2 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_le
% 0.29/0.46 thf(fact_144_le__less__linear,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_linear
% 0.29/0.46 thf(fact_145_le__imp__less__or__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less @ A @ X @ Y )
% 0.29/0.46 | ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_imp_less_or_eq
% 0.29/0.46 thf(fact_146_less__le__not__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [X4: A,Y4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.29/0.46 & ~ ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le_not_le
% 0.29/0.46 thf(fact_147_not__le__imp__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_le_imp_less
% 0.29/0.46 thf(fact_148_order_Ostrict__trans1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less @ A @ B3 @ C2 )
% 0.29/0.46 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans1
% 0.29/0.46 thf(fact_149_order_Ostrict__trans2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.29/0.46 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans2
% 0.29/0.46 thf(fact_150_order_Oorder__iff__strict,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.46 | ( A2 = B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.order_iff_strict
% 0.29/0.46 thf(fact_151_order_Ostrict__iff__order,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 & ( A2 != B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_iff_order
% 0.29/0.46 thf(fact_152_dual__order_Ostrict__trans1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B3: A,A3: A,C2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( ( ord_less @ A @ C2 @ B3 )
% 0.29/0.46 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans1
% 0.29/0.46 thf(fact_153_dual__order_Ostrict__trans2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B3: A,A3: A,C2: A] :
% 0.29/0.46 ( ( ord_less @ A @ B3 @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.29/0.46 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans2
% 0.29/0.46 thf(fact_154_dense__ge__bounded,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( dense_linorder @ A )
% 0.29/0.46 => ! [Z2: A,X: A,Y: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z2 @ X )
% 0.29/0.46 => ( ! [W: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z2 @ W )
% 0.29/0.46 => ( ( ord_less @ A @ W @ X )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_ge_bounded
% 0.29/0.46 thf(fact_155_dense__le__bounded,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( dense_linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ! [W: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ W )
% 0.29/0.46 => ( ( ord_less @ A @ W @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_le_bounded
% 0.29/0.46 thf(fact_156_order_Ostrict__implies__order,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_implies_order
% 0.29/0.46 thf(fact_157_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.46 | ( A2 = B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.order_iff_strict
% 0.29/0.46 thf(fact_158_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 & ( A2 != B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_iff_order
% 0.29/0.46 thf(fact_159_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B3: A,A3: A] :
% 0.29/0.46 ( ( ord_less @ A @ B3 @ A3 )
% 0.29/0.46 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_implies_order
% 0.29/0.46 thf(fact_160_order_Onot__eq__order__implies__strict,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( A3 != B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.not_eq_order_implies_strict
% 0.29/0.46 thf(fact_161_bot_Oextremum,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum
% 0.29/0.46 thf(fact_162_bot_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.29/0.46 = ( A3
% 0.29/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_unique
% 0.29/0.46 thf(fact_163_bot_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.29/0.46 => ( A3
% 0.29/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_uniqueI
% 0.29/0.46 thf(fact_164_bot_Oextremum__strict,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_strict
% 0.29/0.46 thf(fact_165_bot_Onot__eq__extremum,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 != ( bot_bot @ A ) )
% 0.29/0.46 = ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.not_eq_extremum
% 0.29/0.46 thf(fact_166_subset__emptyI,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A] :
% 0.29/0.46 ( ! [X5: A] :
% 0.29/0.46 ~ ( member @ A @ X5 @ A4 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_emptyI
% 0.29/0.46 thf(fact_167_minf_I8_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ X7 @ Z3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(8)
% 0.29/0.46 thf(fact_168_minf_I6_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ X7 @ Z3 )
% 0.29/0.46 => ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(6)
% 0.29/0.46 thf(fact_169_pinf_I8_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X7 )
% 0.29/0.46 => ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(8)
% 0.29/0.46 thf(fact_170_pinf_I6_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X7 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(6)
% 0.29/0.46 thf(fact_171_verit__comp__simplify1_I3_J,axiom,
% 0.29/0.46 ! [B: $tType] :
% 0.29/0.46 ( ( linorder @ B )
% 0.29/0.46 => ! [B7: B,A7: B] :
% 0.29/0.46 ( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
% 0.29/0.46 = ( ord_less @ B @ A7 @ B7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % verit_comp_simplify1(3)
% 0.29/0.46 thf(fact_172_complete__interval,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( condit1037483654norder @ A )
% 0.29/0.46 => ! [A3: A,B3: A,P: A > $o] :
% 0.29/0.46 ( ( ord_less @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( P @ A3 )
% 0.29/0.46 => ( ~ ( P @ B3 )
% 0.29/0.46 => ? [C4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ C4 )
% 0.29/0.46 & ( ord_less_eq @ A @ C4 @ B3 )
% 0.29/0.46 & ! [X7: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ A3 @ X7 )
% 0.29/0.46 & ( ord_less @ A @ X7 @ C4 ) )
% 0.29/0.46 => ( P @ X7 ) )
% 0.29/0.46 & ! [D2: A] :
% 0.29/0.46 ( ! [X5: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ A3 @ X5 )
% 0.29/0.46 & ( ord_less @ A @ X5 @ D2 ) )
% 0.29/0.46 => ( P @ X5 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % complete_interval
% 0.29/0.46 thf(fact_173_bot_Oordering__top__axioms,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ( ordering_top @ A
% 0.29/0.46 @ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
% 0.29/0.46 @ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 )
% 0.29/0.46 @ ( bot_bot @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.ordering_top_axioms
% 0.29/0.46 thf(fact_174_psubsetI,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.46 => ( ( A4 != B4 )
% 0.29/0.46 => ( ord_less @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubsetI
% 0.29/0.46 thf(fact_175_less__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ( ( ord_less @ ( A > B ) )
% 0.29/0.46 = ( ^ [F3: A > B,G2: A > B] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.29/0.46 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_fun_def
% 0.29/0.46 thf(fact_176_ordering__top_Oextremum,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.29/0.46 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.29/0.46 => ( Less_eq @ A3 @ Top ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering_top.extremum
% 0.29/0.46 thf(fact_177_ordering__top_Oextremum__strict,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.29/0.46 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.29/0.46 => ~ ( Less @ Top @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering_top.extremum_strict
% 0.29/0.46 thf(fact_178_ordering__top_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.29/0.46 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.29/0.46 => ( ( Less_eq @ Top @ A3 )
% 0.29/0.46 = ( A3 = Top ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering_top.extremum_unique
% 0.29/0.46 thf(fact_179_ordering__top_Onot__eq__extremum,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.29/0.46 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.29/0.46 => ( ( A3 != Top )
% 0.29/0.46 = ( Less @ A3 @ Top ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering_top.not_eq_extremum
% 0.29/0.46 thf(fact_180_ordering__top_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.29/0.46 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.29/0.46 => ( ( Less_eq @ Top @ A3 )
% 0.29/0.46 => ( A3 = Top ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering_top.extremum_uniqueI
% 0.29/0.46 thf(fact_181_not__psubset__empty,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A] :
% 0.29/0.46 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_psubset_empty
% 0.29/0.46 thf(fact_182_subset__iff__psubset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A6 @ B6 )
% 0.29/0.46 | ( A6 = B6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff_psubset_eq
% 0.29/0.46 thf(fact_183_subset__psubset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.46 => ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
% 0.29/0.46 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_psubset_trans
% 0.29/0.46 thf(fact_184_subset__not__subset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.29/0.46 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_not_subset_eq
% 0.29/0.46 thf(fact_185_psubset__subset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.29/0.46 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_subset_trans
% 0.29/0.46 thf(fact_186_psubset__imp__subset,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_imp_subset
% 0.29/0.46 thf(fact_187_psubset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.29/0.46 & ( A6 != B6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_eq
% 0.29/0.46 thf(fact_188_psubsetE,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubsetE
% 0.29/0.46 thf(fact_189_less__eq__set__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.29/0.46 ( ord_less_eq @ ( A > $o )
% 0.29/0.46 @ ^ [X4: A] : ( member @ A @ X4 @ A6 )
% 0.29/0.46 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_set_def
% 0.29/0.46 thf(fact_190_verit__la__disequality,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( A3 = B3 )
% 0.29/0.46 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % verit_la_disequality
% 0.29/0.46 thf(fact_191_ex__gt__or__lt,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( condit1656338222tinuum @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ? [B5: A] :
% 0.29/0.46 ( ( ord_less @ A @ A3 @ B5 )
% 0.29/0.46 | ( ord_less @ A @ B5 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_gt_or_lt
% 0.29/0.46 thf(fact_192_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % verit_comp_simplify1(1)
% 0.29/0.46 thf(fact_193_pinf_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.46 ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z4 @ X5 )
% 0.29/0.46 => ( ( P @ X5 )
% 0.29/0.46 = ( P4 @ X5 ) ) )
% 0.29/0.46 => ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z4 @ X5 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 = ( Q2 @ X5 ) ) )
% 0.29/0.46 => ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X7 )
% 0.29/0.46 => ( ( ( P @ X7 )
% 0.29/0.46 & ( Q @ X7 ) )
% 0.29/0.46 = ( ( P4 @ X7 )
% 0.29/0.46 & ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(1)
% 0.29/0.46 thf(fact_194_pinf_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.46 ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z4 @ X5 )
% 0.29/0.46 => ( ( P @ X5 )
% 0.29/0.46 = ( P4 @ X5 ) ) )
% 0.29/0.46 => ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z4 @ X5 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 = ( Q2 @ X5 ) ) )
% 0.29/0.46 => ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X7 )
% 0.29/0.46 => ( ( ( P @ X7 )
% 0.29/0.46 | ( Q @ X7 ) )
% 0.29/0.46 = ( ( P4 @ X7 )
% 0.29/0.46 | ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(2)
% 0.29/0.46 thf(fact_195_pinf_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X7 )
% 0.29/0.46 => ( X7 != T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(3)
% 0.29/0.46 thf(fact_196_pinf_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X7 )
% 0.29/0.46 => ( X7 != T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(4)
% 0.29/0.46 thf(fact_197_pinf_I5_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X7 )
% 0.29/0.46 => ~ ( ord_less @ A @ X7 @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(5)
% 0.29/0.46 thf(fact_198_pinf_I7_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X7 )
% 0.29/0.46 => ( ord_less @ A @ T @ X7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(7)
% 0.29/0.46 thf(fact_199_pinf_I11_J,axiom,
% 0.29/0.46 ! [C: $tType,D: $tType] :
% 0.29/0.46 ( ( ord @ C )
% 0.29/0.46 => ! [F4: D] :
% 0.29/0.46 ? [Z3: C] :
% 0.29/0.46 ! [X7: C] :
% 0.29/0.46 ( ( ord_less @ C @ Z3 @ X7 )
% 0.29/0.46 => ( F4 = F4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(11)
% 0.29/0.46 thf(fact_200_minf_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.46 ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.46 => ( ( P @ X5 )
% 0.29/0.46 = ( P4 @ X5 ) ) )
% 0.29/0.46 => ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 = ( Q2 @ X5 ) ) )
% 0.29/0.46 => ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ X7 @ Z3 )
% 0.29/0.46 => ( ( ( P @ X7 )
% 0.29/0.46 & ( Q @ X7 ) )
% 0.29/0.46 = ( ( P4 @ X7 )
% 0.29/0.46 & ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(1)
% 0.29/0.46 thf(fact_201_minf_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.46 ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.46 => ( ( P @ X5 )
% 0.29/0.46 = ( P4 @ X5 ) ) )
% 0.29/0.46 => ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 = ( Q2 @ X5 ) ) )
% 0.29/0.46 => ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ X7 @ Z3 )
% 0.29/0.46 => ( ( ( P @ X7 )
% 0.29/0.46 | ( Q @ X7 ) )
% 0.29/0.46 = ( ( P4 @ X7 )
% 0.29/0.46 | ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(2)
% 0.29/0.46 thf(fact_202_minf_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ X7 @ Z3 )
% 0.29/0.46 => ( X7 != T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(3)
% 0.29/0.46 thf(fact_203_minf_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ X7 @ Z3 )
% 0.29/0.46 => ( X7 != T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(4)
% 0.29/0.46 thf(fact_204_minf_I5_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ X7 @ Z3 )
% 0.29/0.46 => ( ord_less @ A @ X7 @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(5)
% 0.29/0.46 thf(fact_205_minf_I7_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X7: A] :
% 0.29/0.46 ( ( ord_less @ A @ X7 @ Z3 )
% 0.29/0.46 => ~ ( ord_less @ A @ T @ X7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(7)
% 0.29/0.46 thf(fact_206_minf_I11_J,axiom,
% 0.29/0.46 ! [C: $tType,D: $tType] :
% 0.29/0.46 ( ( ord @ C )
% 0.29/0.46 => ! [F4: D] :
% 0.29/0.46 ? [Z3: C] :
% 0.29/0.46 ! [X7: C] :
% 0.29/0.46 ( ( ord_less @ C @ X7 @ Z3 )
% 0.29/0.46 => ( F4 = F4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(11)
% 0.29/0.46 thf(fact_207_prop__restrict,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Z5: set @ A,X8: set @ A,P: A > $o] :
% 0.29/0.46 ( ( member @ A @ X @ Z5 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ Z5
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ X8 )
% 0.29/0.46 & ( P @ X4 ) ) ) )
% 0.29/0.46 => ( P @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prop_restrict
% 0.29/0.46 thf(fact_208_Collect__restrict,axiom,
% 0.29/0.46 ! [A: $tType,X8: set @ A,P: A > $o] :
% 0.29/0.46 ( ord_less_eq @ ( set @ A )
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ X8 )
% 0.29/0.46 & ( P @ X4 ) ) )
% 0.29/0.46 @ X8 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_restrict
% 0.29/0.46 thf(fact_209_bot__empty__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.29/0.46 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_empty_eq
% 0.29/0.46 thf(fact_210_Collect__empty__eq__bot,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( ( collect @ A @ P )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( P
% 0.29/0.46 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq_bot
% 0.29/0.46 thf(fact_211_Set_Ois__empty__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( is_empty @ A )
% 0.29/0.46 = ( ^ [A6: set @ A] :
% 0.29/0.46 ( A6
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.is_empty_def
% 0.29/0.46 thf(fact_212_predicate1I,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X5: A] :
% 0.29/0.46 ( ( P @ X5 )
% 0.29/0.46 => ( Q @ X5 ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate1I
% 0.29/0.46 thf(fact_213_less__set__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.29/0.46 ( ord_less @ ( A > $o )
% 0.29/0.46 @ ^ [X4: A] : ( member @ A @ X4 @ A6 )
% 0.29/0.46 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_set_def
% 0.29/0.46 thf(fact_214_rev__predicate1D,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.29/0.46 => ( Q @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_predicate1D
% 0.29/0.46 thf(fact_215_predicate1D,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.29/0.46 => ( ( P @ X )
% 0.29/0.46 => ( Q @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate1D
% 0.29/0.46 thf(fact_216_psubset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.46 => ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
% 0.29/0.46 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_trans
% 0.29/0.46 thf(fact_217_psubsetD,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.29/0.46 => ( ( member @ A @ C2 @ A4 )
% 0.29/0.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubsetD
% 0.29/0.46 thf(fact_218_pred__subset__eq,axiom,
% 0.29/0.46 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > $o )
% 0.29/0.46 @ ^ [X4: A] : ( member @ A @ X4 @ R )
% 0.29/0.46 @ ^ [X4: A] : ( member @ A @ X4 @ S ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.29/0.46
% 0.29/0.46 % pred_subset_eq
% 0.29/0.46 thf(fact_219_subset__Collect__iff,axiom,
% 0.29/0.46 ! [A: $tType,B4: set @ A,A4: set @ A,P: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ A4 )
% 0.29/0.46 & ( P @ X4 ) ) ) )
% 0.29/0.46 = ( ! [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ B4 )
% 0.29/0.46 => ( P @ X4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_Collect_iff
% 0.29/0.46 thf(fact_220_subset__CollectI,axiom,
% 0.29/0.46 ! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.29/0.46 => ( ! [X5: A] :
% 0.29/0.46 ( ( member @ A @ X5 @ B4 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 => ( P @ X5 ) ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A )
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ B4 )
% 0.29/0.46 & ( Q @ X4 ) ) )
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ A4 )
% 0.29/0.46 & ( P @ X4 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_CollectI
% 0.29/0.46 thf(fact_221_conj__subset__def,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X4: A] :
% 0.29/0.46 ( ( P @ X4 )
% 0.29/0.46 & ( Q @ X4 ) ) ) )
% 0.29/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % conj_subset_def
% 0.29/0.46 thf(fact_222_remove_Osimps_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
% 0.29/0.46 ( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.29/0.46 => ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.46 = ( binary210054475elle_T @ A @ ( binary997842527remove @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
% 0.29/0.46 & ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.29/0.46 => ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.29/0.46 => ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.46 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary997842527remove @ A @ H @ E @ T2 ) ) ) )
% 0.29/0.46 & ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.29/0.46 => ( ( ( T1
% 0.29/0.46 = ( binary1746293266le_Tip @ A ) )
% 0.29/0.46 => ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.46 = T2 ) )
% 0.29/0.46 & ( ( T1
% 0.29/0.46 != ( binary1746293266le_Tip @ A ) )
% 0.29/0.46 => ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.29/0.46 = ( product_case_prod @ ( binary1291135688e_Tree @ A ) @ A @ ( binary1291135688e_Tree @ A )
% 0.29/0.46 @ ^ [T1p: binary1291135688e_Tree @ A,R2: A] : ( binary210054475elle_T @ A @ T1p @ R2 @ T2 )
% 0.29/0.46 @ ( binary1271298290_wrmrm @ A @ H @ T1 ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % remove.simps(2)
% 0.29/0.46 thf(fact_223_wrmrm__decomp,axiom,
% 0.29/0.46 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.29/0.46 ( ( T
% 0.29/0.46 != ( binary1746293266le_Tip @ A ) )
% 0.29/0.46 => ( ( binary1271298290_wrmrm @ A @ H @ T )
% 0.29/0.46 = ( product_Pair @ ( binary1291135688e_Tree @ A ) @ A @ ( binary213313527le_wrm @ A @ H @ T ) @ ( binary576689334lle_rm @ A @ H @ T ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % wrmrm_decomp
% 0.29/0.46 thf(fact_224_case__prod__app,axiom,
% 0.29/0.46 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.29/0.46 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.29/0.46 = ( ^ [F3: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
% 0.29/0.46 ( product_case_prod @ B @ C @ A
% 0.29/0.46 @ ^ [L: B,R2: C] : ( F3 @ L @ R2 @ Y4 )
% 0.29/0.46 @ X4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prod_app
% 0.29/0.46 thf(fact_225_case__prod__Pair__iden,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
% 0.29/0.46 = P5 ) ).
% 0.29/0.46
% 0.29/0.46 % case_prod_Pair_iden
% 0.29/0.46 thf(fact_226_bot__empty__eq2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( A > B > $o ) )
% 0.29/0.46 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_empty_eq2
% 0.29/0.46 thf(fact_227_subrelI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.29/0.46 ( ! [X5: A,Y2: B] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y2 ) @ R3 )
% 0.29/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y2 ) @ S2 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subrelI
% 0.29/0.46 thf(fact_228_pred__subset__eq2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B > $o )
% 0.29/0.46 @ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
% 0.29/0.46 @ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.29/0.46
% 0.29/0.46 % pred_subset_eq2
% 0.29/0.46 thf(fact_229_pred__equals__eq2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.29/0.46 ( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) )
% 0.29/0.46 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) ) )
% 0.29/0.46 = ( R = S ) ) ).
% 0.29/0.46
% 0.29/0.46 % pred_equals_eq2
% 0.29/0.46 thf(fact_230_ssubst__Pair__rhs,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R3: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S2 ) @ R )
% 0.29/0.46 => ( ( S3 = S2 )
% 0.29/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S3 ) @ R ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ssubst_Pair_rhs
% 0.29/0.46 thf(fact_231_case__prod__conv,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
% 0.29/0.46 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.29/0.46 = ( F @ A3 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prod_conv
% 0.29/0.46 thf(fact_232_mem__case__prodI2,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
% 0.29/0.46 ( ! [A5: A,B5: B] :
% 0.29/0.46 ( ( P5
% 0.29/0.46 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.29/0.46 => ( member @ C @ Z2 @ ( C2 @ A5 @ B5 ) ) )
% 0.29/0.46 => ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_case_prodI2
% 0.29/0.46 thf(fact_233_predicate2I,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.29/0.46 ( ! [X5: A,Y2: B] :
% 0.29/0.46 ( ( P @ X5 @ Y2 )
% 0.29/0.46 => ( Q @ X5 @ Y2 ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2I
% 0.29/0.46 thf(fact_234_case__prodI,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
% 0.29/0.46 ( ( F @ A3 @ B3 )
% 0.29/0.46 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodI
% 0.29/0.46 thf(fact_235_case__prodI2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,C2: A > B > $o] :
% 0.29/0.46 ( ! [A5: A,B5: B] :
% 0.29/0.46 ( ( P5
% 0.29/0.46 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.29/0.46 => ( C2 @ A5 @ B5 ) )
% 0.29/0.46 => ( product_case_prod @ A @ B @ $o @ C2 @ P5 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodI2
% 0.29/0.46 thf(fact_236_case__prodI2_H,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 0.29/0.46 ( ! [A5: A,B5: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ A5 @ B5 )
% 0.29/0.46 = P5 )
% 0.29/0.46 => ( C2 @ A5 @ B5 @ X ) )
% 0.29/0.46 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodI2'
% 0.29/0.46 thf(fact_237_mem__case__prodI,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B3: C] :
% 0.29/0.46 ( ( member @ A @ Z2 @ ( C2 @ A3 @ B3 ) )
% 0.29/0.46 => ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_case_prodI
% 0.29/0.46 thf(fact_238_Collect__case__prod__mono,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A4: A > B > $o,B4: A > B > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ A4 @ B4 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_case_prod_mono
% 0.29/0.46 thf(fact_239_predicate2D__conj,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
% 0.29/0.46 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 & R )
% 0.29/0.46 => ( R
% 0.29/0.46 & ( ( P @ X @ Y )
% 0.29/0.46 => ( Q @ X @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2D_conj
% 0.29/0.46 thf(fact_240_eq__subset,axiom,
% 0.29/0.46 ! [A: $tType,P: A > A > $o] :
% 0.29/0.46 ( ord_less_eq @ ( A > A > $o )
% 0.29/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.29/0.46 @ ^ [A2: A,B2: A] :
% 0.29/0.46 ( ( P @ A2 @ B2 )
% 0.29/0.46 | ( A2 = B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_subset
% 0.29/0.46 thf(fact_241_predicate2D,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 => ( ( P @ X @ Y )
% 0.29/0.46 => ( Q @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2D
% 0.29/0.46 thf(fact_242_rev__predicate2D,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
% 0.29/0.46 ( ( P @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 => ( Q @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_predicate2D
% 0.29/0.46 thf(fact_243_case__prodD,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.29/0.46 => ( F @ A3 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodD
% 0.29/0.46 thf(fact_244_case__prodE,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C2: A > B > $o,P5: product_prod @ A @ B] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ $o @ C2 @ P5 )
% 0.29/0.46 => ~ ! [X5: A,Y2: B] :
% 0.29/0.46 ( ( P5
% 0.29/0.46 = ( product_Pair @ A @ B @ X5 @ Y2 ) )
% 0.29/0.46 => ~ ( C2 @ X5 @ Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodE
% 0.29/0.46 thf(fact_245_case__prodD_H,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C2: C] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C2 )
% 0.29/0.46 => ( R @ A3 @ B3 @ C2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodD'
% 0.29/0.46 thf(fact_246_case__prodE_H,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P5: product_prod @ A @ B,Z2: C] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ Z2 )
% 0.29/0.46 => ~ ! [X5: A,Y2: B] :
% 0.29/0.46 ( ( P5
% 0.29/0.46 = ( product_Pair @ A @ B @ X5 @ Y2 ) )
% 0.29/0.46 => ~ ( C2 @ X5 @ Y2 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodE'
% 0.29/0.46 thf(fact_247_prod_Ocase__distrib,axiom,
% 0.29/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.29/0.46 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.29/0.46 = ( product_case_prod @ A @ B @ D
% 0.29/0.46 @ ^ [X12: A,X24: B] : ( H @ ( F @ X12 @ X24 ) )
% 0.29/0.46 @ Prod ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod.case_distrib
% 0.29/0.46 thf(fact_248_case__prodE2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
% 0.29/0.46 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
% 0.29/0.46 => ~ ! [X5: B,Y2: C] :
% 0.29/0.46 ( ( Z2
% 0.29/0.46 = ( product_Pair @ B @ C @ X5 @ Y2 ) )
% 0.29/0.46 => ~ ( Q @ ( P @ X5 @ Y2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodE2
% 0.29/0.46 thf(fact_249_case__prod__eta,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ C
% 0.29/0.46 @ ^ [X4: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
% 0.29/0.46 = F ) ).
% 0.29/0.46
% 0.29/0.46 % case_prod_eta
% 0.29/0.46 thf(fact_250_cond__case__prod__eta,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 0.29/0.46 ( ! [X5: A,Y2: B] :
% 0.29/0.46 ( ( F @ X5 @ Y2 )
% 0.29/0.46 = ( G @ ( product_Pair @ A @ B @ X5 @ Y2 ) ) )
% 0.29/0.46 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.29/0.46 = G ) ) ).
% 0.29/0.46
% 0.29/0.46 % cond_case_prod_eta
% 0.29/0.46 thf(fact_251_less__by__empty,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_by_empty
% 0.29/0.46 thf(fact_252_relChain__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.29/0.46 = ( ^ [R2: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.29/0.46 ! [I: A,J: A] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % relChain_def
% 0.29/0.46 thf(fact_253_split__part,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ $o
% 0.29/0.46 @ ^ [A2: A,B2: B] :
% 0.29/0.46 ( P
% 0.29/0.46 & ( Q @ A2 @ B2 ) ) )
% 0.29/0.46 = ( ^ [Ab: product_prod @ A @ B] :
% 0.29/0.46 ( P
% 0.29/0.46 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % split_part
% 0.29/0.46 thf(fact_254_prod_Odisc__eq__case,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.29/0.46 ( product_case_prod @ A @ B @ $o
% 0.29/0.46 @ ^ [Uu: A,Uv: B] : $true
% 0.29/0.46 @ Prod ) ).
% 0.29/0.46
% 0.29/0.46 % prod.disc_eq_case
% 0.29/0.46 thf(fact_255_inv__image__def,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( inv_image @ B @ A )
% 0.29/0.46 = ( ^ [R2: set @ ( product_prod @ B @ B ),F3: A > B] :
% 0.29/0.46 ( collect @ ( product_prod @ A @ A )
% 0.29/0.46 @ ( product_case_prod @ A @ A @ $o
% 0.29/0.46 @ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inv_image_def
% 0.29/0.46
% 0.29/0.46 % Type constructors (23)
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( order_bot @ A9 )
% 0.29/0.46 => ( order_bot @ ( 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 )
% 0.29/0.46 => ( preorder @ ( 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 )
% 0.29/0.46 => ( order @ ( 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 )
% 0.29/0.46 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( bot @ A9 )
% 0.29/0.46 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.29/0.46 condit1037483654norder @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.29/0.46 preorder @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.29/0.46 linorder @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.29/0.46 no_top @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.29/0.46 no_bot @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.29/0.46 order @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.29/0.46 ord @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_4,axiom,
% 0.29/0.46 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.29/0.46 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.29/0.46 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.29/0.46 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
% 0.29/0.46 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_9,axiom,
% 0.29/0.46 order_bot @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.29/0.46 preorder @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
% 0.29/0.46 linorder @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 0.29/0.46 order @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 0.29/0.46 ord @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Obot_14,axiom,
% 0.29/0.46 bot @ $o ).
% 0.29/0.46
% 0.29/0.46 % Conjectures (1)
% 0.29/0.46 thf(conj_0,conjecture,
% 0.29/0.46 binary1610619414edTree @ a @ h @ t1 ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 0.30/1.51 ---- Embedded in THF ---
% 0.30/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.51 %%% Generated on Tue Feb 24 08:24:37 EST 2026
% 0.30/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.51 %%% Logic specification used:
% 0.30/1.51 %%% thf(spec, logic, $$dhol).
% 0.30/1.51
% 0.30/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.yadyxb1T5r/DTF2DTF28889.p
% 0.30/1.51 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree, type, binary1291135688e_Tree: ($tType > $tType)).
% 0.30/1.51 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 0.30/1.51 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.30/1.51 thf(ty_t_Int_Oint, type, int: $tType).
% 0.30/1.51 thf(ty_tf_a, type, a: $tType).
% 0.30/1.51 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT, type, binary210054475elle_T: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip, type, binary1746293266le_Tip: (!> [A:$tType]: ((binary1291135688e_Tree @ A)))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Ocase__Tree, type, binary536355927e_Tree: (!> [B:$tType,A:$tType]: ((B > (((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > B))) > ((binary1291135688e_Tree @ A) > B)))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Orec__Tree, type, binary1929596613c_Tree: (!> [C:$tType,A:$tType]: ((C > (((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > (C > (C > C))))) > ((binary1291135688e_Tree @ A) > C)))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert, type, binary1830089824insert: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb, type, binary827270440e_memb: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > $o)))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oremove, type, binary997842527remove: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Orm, type, binary576689334lle_rm: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > A))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf, type, binary1653327646_setOf: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (set @ A))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree, type, binary1610619414edTree: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > $o))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred, type, binary231205461t_pred: (!> [A:$tType]: (((A > int) > (A > (A > ((binary1291135688e_Tree @ A) > $o))))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrm, type, binary213313527le_wrm: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A)))))).
% 0.30/1.51 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrmrm, type, binary1271298290_wrmrm: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > ((product_prod @ (binary1291135688e_Tree @ A)) @ A)))))).
% 0.30/1.51 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 0.30/1.51 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.30/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.30/1.51 thf(sy_c_Orderings_Oordering__top, type, ordering_top: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > (A > $o)))))).
% 0.30/1.51 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 0.30/1.51 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 0.30/1.51 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 0.30/1.51 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.30/1.51 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.30/1.51 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.30/1.51 thf(sy_v_h, type, h: (a > int)).
% 0.30/1.51 thf(sy_v_t1____, type, t1: (binary1291135688e_Tree @ a)).
% 0.30/1.51 thf(sy_v_t2____, type, t2: (binary1291135688e_Tree @ a)).
% 0.30/1.51 thf(sy_v_x____, type, x: a).
% 0.30/1.51 thf(fact_0_t2nTip, axiom, (t2 != (binary1746293266le_Tip @ a))).
% 0.30/1.51 thf(fact_1_s, axiom, (((binary1610619414edTree @ a) @ h) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2))).
% 0.30/1.51 thf(fact_2_res, axiom, ((((binary213313527le_wrm @ a) @ h) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2)) = ((((binary210054475elle_T @ a) @ t1) @ x) @ (((binary213313527le_wrm @ a) @ h) @ t2)))).
% 0.30/1.51 thf(fact_3_sortLemmaL, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X:A,T2:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) => (((binary1610619414edTree @ A) @ H) @ T1))))).
% 0.30/1.51 thf(fact_4_sortLemmaR, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X:A,T2:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) => (((binary1610619414edTree @ A) @ H) @ T2))))).
% 0.30/1.51 thf(fact_5_Tree_Oinject, axiom, (! [A:$tType,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A),Y21:(binary1291135688e_Tree @ A),Y22:A,Y23:(binary1291135688e_Tree @ A)]: (((((((binary210054475elle_T @ A) @ X21) @ X22) @ X23) = ((((binary210054475elle_T @ A) @ Y21) @ Y22) @ Y23)) = ((X21 = Y21) & ((X22 = Y22) & (X23 = Y23))))))).
% 0.30/1.51 thf(fact_6_binsert__sorted, axiom, (! [A:$tType,H:(A > int),T:(binary1291135688e_Tree @ A),X:A]: (((((binary1610619414edTree @ A) @ H) @ T) => (((binary1610619414edTree @ A) @ H) @ ((((binary1830089824insert @ A) @ H) @ X) @ T)))))).
% 0.30/1.51 thf(fact_7_h1, axiom, (((t1 != (binary1746293266le_Tip @ a)) & (((binary1610619414edTree @ a) @ h) @ t1)) => (((binary1610619414edTree @ a) @ h) @ (((binary213313527le_wrm @ a) @ h) @ t1)))).
% 0.30/1.51 thf(fact_8_sortedTree_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int)]: ((((binary1610619414edTree @ A) @ H) @ (binary1746293266le_Tip @ A))))).
% 0.30/1.51 thf(fact_9_memb__spec, axiom, (! [A:$tType,H:(A > int),T:(binary1291135688e_Tree @ A),X:A]: (((((binary1610619414edTree @ A) @ H) @ T) => (((((binary827270440e_memb @ A) @ H) @ X) @ T) = (((member @ A) @ X) @ ((binary1653327646_setOf @ A) @ T))))))).
% 0.30/1.51 thf(fact_10_sorted__distinct__pred__def, axiom, (! [A:$tType]: (((binary231205461t_pred @ A) = (^ [H2:(A > int),A2:A,B2:A,T3:(binary1291135688e_Tree @ A)]: ((((((binary1610619414edTree @ A) @ H2) @ T3) & ((((member @ A) @ A2) @ ((binary1653327646_setOf @ A) @ T3)) & ((((member @ A) @ B2) @ ((binary1653327646_setOf @ A) @ T3)) & ((H2 @ A2) = (H2 @ B2))))) => (A2 = B2)))))))).
% 0.30/1.51 thf(fact_11_h2, axiom, (((t2 != (binary1746293266le_Tip @ a)) & (((binary1610619414edTree @ a) @ h) @ t2)) => (((binary1610619414edTree @ a) @ h) @ (((binary213313527le_wrm @ a) @ h) @ t2)))).
% 0.30/1.51 thf(fact_12_sorted__distinct, axiom, (! [A:$tType,H:(A > int),A3:A,B3:A,T:(binary1291135688e_Tree @ A)]: ((((((binary231205461t_pred @ A) @ H) @ A3) @ B3) @ T)))).
% 0.30/1.51 thf(fact_13__092_060open_062Tip_A_092_060noteq_062_ATip_A_092_060and_062_AsortedTree_Ah_ATip_A_092_060longrightarrow_062_AsortedTree_Ah_A_Iwrm_Ah_ATip_J_092_060close_062, axiom, ((((binary1746293266le_Tip @ a) != (binary1746293266le_Tip @ a)) & (((binary1610619414edTree @ a) @ h) @ (binary1746293266le_Tip @ a))) => (((binary1610619414edTree @ a) @ h) @ (((binary213313527le_wrm @ a) @ h) @ (binary1746293266le_Tip @ a))))).
% 0.30/1.51 thf(fact_14_Tree_Odistinct_I1_J, axiom, (! [A:$tType,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: (((binary1746293266le_Tip @ A) != ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23))))).
% 0.30/1.51 thf(fact_15_binsert_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int),E:A]: ((((((binary1830089824insert @ A) @ H) @ E) @ (binary1746293266le_Tip @ A)) = ((((binary210054475elle_T @ A) @ (binary1746293266le_Tip @ A)) @ E) @ (binary1746293266le_Tip @ A)))))).
% 0.30/1.51 thf(fact_16_wrm_Osimps, axiom, (! [A:$tType,T2:(binary1291135688e_Tree @ A),H:(A > int),T1:(binary1291135688e_Tree @ A),X:A]: ((((T2 = (binary1746293266le_Tip @ A)) => ((((binary213313527le_wrm @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = T1)) & ((T2 != (binary1746293266le_Tip @ A)) => ((((binary213313527le_wrm @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ X) @ (((binary213313527le_wrm @ A) @ H) @ T2)))))))).
% 0.30/1.51 thf(fact_17_Tree_Oinduct, axiom, (! [A:$tType,P:((binary1291135688e_Tree @ A) > $o),Tree:(binary1291135688e_Tree @ A)]: (((P @ (binary1746293266le_Tip @ A)) => ((! [X1:(binary1291135688e_Tree @ A),X2:A,X3:(binary1291135688e_Tree @ A)]: (((P @ X1) => ((P @ X3) => (P @ ((((binary210054475elle_T @ A) @ X1) @ X2) @ X3)))))) => (P @ Tree)))))).
% 0.30/1.51 thf(fact_18_Tree_Oexhaust, axiom, (! [A:$tType,Y:(binary1291135688e_Tree @ A)]: (((Y != (binary1746293266le_Tip @ A)) => (~ ((! [X212:(binary1291135688e_Tree @ A),X222:A,X232:(binary1291135688e_Tree @ A)]: ((Y != ((((binary210054475elle_T @ A) @ X212) @ X222) @ X232)))))))))).
% 0.30/1.51 thf(fact_19_rm__set, axiom, (! [A:$tType,T:(binary1291135688e_Tree @ A),H:(A > int)]: ((((T != (binary1746293266le_Tip @ A)) & (((binary1610619414edTree @ A) @ H) @ T)) => (((member @ A) @ (((binary576689334lle_rm @ A) @ H) @ T)) @ ((binary1653327646_setOf @ A) @ T)))))).
% 0.30/1.51 thf(fact_20_rm_Osimps, axiom, (! [A:$tType,T2:(binary1291135688e_Tree @ A),H:(A > int),T1:(binary1291135688e_Tree @ A),X:A]: ((((T2 = (binary1746293266le_Tip @ A)) => ((((binary576689334lle_rm @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = X)) & ((T2 != (binary1746293266le_Tip @ A)) => ((((binary576689334lle_rm @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = (((binary576689334lle_rm @ A) @ H) @ T2))))))).
% 0.30/1.51 thf(fact_21_wrm__set1, axiom, (! [A:$tType,T:(binary1291135688e_Tree @ A),H:(A > int)]: ((((T != (binary1746293266le_Tip @ A)) & (((binary1610619414edTree @ A) @ H) @ T)) => (((ord_less_eq @ (set @ A)) @ ((binary1653327646_setOf @ A) @ (((binary213313527le_wrm @ A) @ H) @ T))) @ ((binary1653327646_setOf @ A) @ T)))))).
% 0.30/1.51 thf(fact_22_sortedTree_Osimps_I2_J, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X:A,T2:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary1610619414edTree @ A) @ H) @ T1) & ((! [X4:A]: (((((member @ A) @ X4) @ ((binary1653327646_setOf @ A) @ T1)) => (((ord_less @ int) @ (H @ X4)) @ (H @ X))))) & ((! [X4:A]: (((((member @ A) @ X4) @ ((binary1653327646_setOf @ A) @ T2)) => (((ord_less @ int) @ (H @ X)) @ (H @ X4))))) & (((binary1610619414edTree @ A) @ H) @ T2)))))))).
% 0.30/1.51 thf(fact_23_remove_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int),E:A]: ((((((binary997842527remove @ A) @ H) @ E) @ (binary1746293266le_Tip @ A)) = (binary1746293266le_Tip @ A))))).
% 0.30/1.51 thf(fact_24_Tree_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F2:((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > B)))]: (((((((binary536355927e_Tree @ B) @ A) @ F1) @ F2) @ (binary1746293266le_Tip @ A)) = F1)))).
% 0.30/1.51 thf(fact_25_Tree_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F2:((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > B))),X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: (((((((binary536355927e_Tree @ B) @ A) @ F1) @ F2) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23)) = (((F2 @ X21) @ X22) @ X23))))).
% 0.30/1.51 thf(fact_26_binsert_Osimps_I2_J, axiom, (! [A:$tType,H:(A > int),E:A,X:A,T1:(binary1291135688e_Tree @ A),T2:(binary1291135688e_Tree @ A)]: ((((((ord_less @ int) @ (H @ E)) @ (H @ X)) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ ((((binary1830089824insert @ A) @ H) @ E) @ T1)) @ X) @ T2))) & ((~ ((((ord_less @ int) @ (H @ E)) @ (H @ X)))) => (((((ord_less @ int) @ (H @ X)) @ (H @ E)) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ X) @ ((((binary1830089824insert @ A) @ H) @ E) @ T2)))) & ((~ ((((ord_less @ int) @ (H @ X)) @ (H @ E)))) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ E) @ T2))))))))).
% 0.30/1.51 thf(fact_27_setOf_Osimps_I1_J, axiom, (! [A:$tType]: ((((binary1653327646_setOf @ A) @ (binary1746293266le_Tip @ A)) = (bot_bot @ (set @ A)))))).
% 0.30/1.51 thf(fact_28_Tree_Osimps_I6_J, axiom, (! [A:$tType,C:$tType,F1:C,F2:((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > (C > (C > C)))))]: (((((((binary1929596613c_Tree @ C) @ A) @ F1) @ F2) @ (binary1746293266le_Tip @ A)) = F1)))).
% 0.30/1.51 thf(fact_29_Tree_Osimps_I7_J, axiom, (! [C:$tType,A:$tType,F1:C,F2:((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > (C > (C > C))))),X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: (((((((binary1929596613c_Tree @ C) @ A) @ F1) @ F2) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23)) = (((((F2 @ X21) @ X22) @ X23) @ (((((binary1929596613c_Tree @ C) @ A) @ F1) @ F2) @ X21)) @ (((((binary1929596613c_Tree @ C) @ A) @ F1) @ F2) @ X23)))))).
% 0.30/1.51 thf(fact_30_Tree_Ocase__distrib, axiom, (! [C:$tType,B:$tType,A:$tType,H:(B > C),F1:B,F2:((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > B))),Tree:(binary1291135688e_Tree @ A)]: (((H @ (((((binary536355927e_Tree @ B) @ A) @ F1) @ F2) @ Tree)) = (((((binary536355927e_Tree @ C) @ A) @ (H @ F1)) @ (^ [X12:(binary1291135688e_Tree @ A),X24:A,X32:(binary1291135688e_Tree @ A)]: ((H @ (((F2 @ X12) @ X24) @ X32))))) @ Tree))))).
% 0.30/1.51 thf(fact_31_empty__subsetI, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A4)))).
% 0.30/1.51 thf(fact_32_subset__empty, axiom, (! [A:$tType,A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = (A4 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_33_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A4) => (A4 = B4)))))).
% 0.30/1.51 thf(fact_34_subsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A4) => (((member @ A) @ X5) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 0.30/1.51 thf(fact_35_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 0.30/1.51 thf(fact_36_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 0.30/1.51 thf(fact_37_all__not__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A4))))) = (A4 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_38_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 0.30/1.51 thf(fact_39_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X4:D]: ((bot_bot @ C)))))))).
% 0.30/1.51 thf(fact_40_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 0.30/1.51 thf(fact_41_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 0.30/1.51 thf(fact_42_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 0.30/1.51 thf(fact_43_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 0.30/1.51 thf(fact_44_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X5:A]: ((((ord_less_eq @ B) @ (F @ X5)) @ (G @ X5)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 0.30/1.51 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 0.30/1.51 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A4)))) = A4)))).
% 0.30/1.51 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.30/1.51 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X5:A]: (((F @ X5) = (G @ X5)))) => (F = G))))).
% 0.30/1.51 thf(fact_49_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F3 @ X4)) @ (G2 @ X4))))))))))).
% 0.30/1.51 thf(fact_50_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X5:B,Y2:B]: (((((ord_less_eq @ B) @ X5) @ Y2) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_51_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => (((ord_less_eq @ C) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C2)))))))))).
% 0.30/1.51 thf(fact_52_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C2:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X5:B,Y2:B]: (((((ord_less_eq @ B) @ X5) @ Y2) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_53_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C2) => ((! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => (((ord_less_eq @ B) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C2)))))))))).
% 0.30/1.51 thf(fact_54_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X4))))))))).
% 0.30/1.51 thf(fact_55_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 0.30/1.51 thf(fact_56_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_57_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 0.30/1.51 thf(fact_58_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_59_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 0.30/1.51 thf(fact_60_le__cases3, axiom, (! [A:$tType]: (((linorder @ 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))))))))))))))))).
% 0.30/1.51 thf(fact_61_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 0.30/1.51 thf(fact_62_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) & (((ord_less_eq @ A) @ B2) @ A2))))))))).
% 0.30/1.51 thf(fact_63_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C2:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 0.30/1.51 thf(fact_64_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 0.30/1.51 thf(fact_65_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 0.30/1.51 thf(fact_66_order__trans, axiom, (! [A:$tType]: (((preorder @ 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))))))))).
% 0.30/1.51 thf(fact_67_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 0.30/1.51 thf(fact_68_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A3) @ B3))))))))).
% 0.30/1.51 thf(fact_69_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 0.30/1.51 thf(fact_70_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A2:A,B2:A]: (((((ord_less_eq @ A) @ B2) @ A2) & (((ord_less_eq @ A) @ A2) @ B2))))))))).
% 0.30/1.51 thf(fact_71_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 0.30/1.51 thf(fact_72_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C2:B]: (((A3 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X5:B,Y2:B]: (((((ord_less @ B) @ X5) @ Y2) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_73_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A3) @ B3) => (((F @ B3) = C2) => ((! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (((ord_less @ B) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less @ B) @ (F @ A3)) @ C2)))))))))).
% 0.30/1.51 thf(fact_74_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C2:B]: (((((ord_less @ A) @ A3) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X5:B,Y2:B]: (((((ord_less @ B) @ X5) @ Y2) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_75_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C2) => ((! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (((ord_less @ C) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 0.30/1.51 thf(fact_76_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X:A]: ((? [Y2:A]: ((((ord_less @ A) @ Y2) @ X))))))))).
% 0.30/1.51 thf(fact_77_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X:A]: ((? [X_1:A]: ((((ord_less @ A) @ X) @ X_1))))))))).
% 0.30/1.51 thf(fact_78_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 0.30/1.51 thf(fact_79_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 0.30/1.51 thf(fact_80_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (~ ((((ord_less @ A) @ B3) @ A3)))))))))).
% 0.30/1.51 thf(fact_81_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y))))))))))).
% 0.30/1.51 thf(fact_82_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 0.30/1.51 thf(fact_83_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 0.30/1.51 thf(fact_84_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (~ ((((ord_less @ A) @ B3) @ A3)))))))))).
% 0.30/1.51 thf(fact_85_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 0.30/1.51 thf(fact_86_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 0.30/1.51 thf(fact_87_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 0.30/1.51 thf(fact_88_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C2:A]: (((A3 = B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.30/1.51 thf(fact_89_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C2:A]: (((((ord_less @ A) @ A3) @ B3) => ((B3 = C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.30/1.51 thf(fact_90_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (~ ((((ord_less @ A) @ A3) @ B3)))))))))).
% 0.30/1.51 thf(fact_91_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 0.30/1.51 thf(fact_92_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 0.30/1.51 thf(fact_93_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A3:A]: (((! [X5:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X5) => (P @ Y5)))) => (P @ X5)))) => (P @ A3)))))))).
% 0.30/1.51 thf(fact_94_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 0.30/1.51 thf(fact_95_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 0.30/1.51 thf(fact_96_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 0.30/1.51 thf(fact_97_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 0.30/1.51 thf(fact_98_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 0.30/1.51 thf(fact_99_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C2:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.30/1.51 thf(fact_100_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 0.30/1.51 thf(fact_101_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X6:A]: ((P2 @ X6))))) = (^ [P3:(A > $o)]: ((? [N:A]: (((P3 @ N) & (! [M:A]: (((((ord_less @ A) @ M) @ N) => (~ ((P3 @ M)))))))))))))))).
% 0.30/1.51 thf(fact_102_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B5:A]: (((((ord_less @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A]: (((P @ A5) @ A5))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A3) @ B3)))))))))).
% 0.30/1.51 thf(fact_103_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C2:A]: (((((ord_less @ A) @ B3) @ A3) => ((((ord_less @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A3))))))))).
% 0.30/1.51 thf(fact_104_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 0.30/1.51 thf(fact_105_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (A3 != B3)))))))).
% 0.30/1.51 thf(fact_106_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (A3 != B3)))))))).
% 0.30/1.51 thf(fact_107_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X4:A]: ((bot_bot @ B)))))))).
% 0.30/1.51 thf(fact_108_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 0.30/1.51 thf(fact_109_equals0D, axiom, (! [A:$tType,A4:(set @ A),A3:A]: (((A4 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A4))))))).
% 0.30/1.51 thf(fact_110_equals0I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A4))))) => (A4 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_111_ex__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ A4))) = (A4 != (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_112_in__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))))).
% 0.30/1.51 thf(fact_113_subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ B4)))))).
% 0.30/1.51 thf(fact_114_equalityE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))))).
% 0.30/1.51 thf(fact_115_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A6) => (((member @ A) @ X4) @ B6)))))))))).
% 0.30/1.51 thf(fact_116_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 0.30/1.51 thf(fact_117_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A4))))).
% 0.30/1.51 thf(fact_118_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A6) => (((member @ A) @ T3) @ B6)))))))))).
% 0.30/1.51 thf(fact_119_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 0.30/1.51 thf(fact_120_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 0.30/1.51 thf(fact_121_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C3) => (((ord_less_eq @ (set @ A)) @ A4) @ C3)))))).
% 0.30/1.51 thf(fact_122_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y3:(set @ A),Z:(set @ A)]: ((Y3 = Z))) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))).
% 0.30/1.51 thf(fact_123_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 0.30/1.51 thf(fact_124_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X4:A]: ($false))))))).
% 0.30/1.51 thf(fact_125_Collect__subset, axiom, (! [A:$tType,A4:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A4) & (P @ X4)))))) @ A4)))).
% 0.30/1.51 thf(fact_126_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 0.30/1.51 thf(fact_127_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_128_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) | (X4 = Y4))))))))).
% 0.30/1.51 thf(fact_129_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (X4 != Y4))))))))).
% 0.30/1.51 thf(fact_130_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X5:B,Y2:B]: (((((ord_less @ B) @ X5) @ Y2) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_131_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C2) => ((! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => (((ord_less_eq @ C) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 0.30/1.51 thf(fact_132_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C2:B]: (((((ord_less @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X5:B,Y2:B]: (((((ord_less_eq @ B) @ X5) @ Y2) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_133_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (((ord_less @ C) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 0.30/1.51 thf(fact_134_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_135_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_136_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((A3 != B3) => (((ord_less @ A) @ A3) @ B3))))))))).
% 0.30/1.51 thf(fact_137_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 0.30/1.51 thf(fact_138_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 0.30/1.51 thf(fact_139_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 0.30/1.51 thf(fact_140_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 0.30/1.51 thf(fact_141_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 0.30/1.51 thf(fact_142_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,Y:A]: (((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (((ord_less_eq @ A) @ Y) @ X5)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 0.30/1.51 thf(fact_143_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y:A,Z2:A]: (((! [X5:A]: (((((ord_less @ A) @ X5) @ Y) => (((ord_less_eq @ A) @ X5) @ Z2)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 0.30/1.51 thf(fact_144_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_145_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 0.30/1.51 thf(fact_146_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (~ ((((ord_less_eq @ A) @ Y4) @ X4))))))))))).
% 0.30/1.51 thf(fact_147_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 0.30/1.51 thf(fact_148_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.30/1.51 thf(fact_149_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C2:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.30/1.51 thf(fact_150_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) | (A2 = B2))))))))).
% 0.30/1.51 thf(fact_151_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) & (A2 != B2))))))))).
% 0.30/1.51 thf(fact_152_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A3))))))))).
% 0.30/1.51 thf(fact_153_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C2:A]: (((((ord_less @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A3))))))))).
% 0.30/1.51 thf(fact_154_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ 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))))))))).
% 0.30/1.51 thf(fact_155_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 0.30/1.51 thf(fact_156_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 0.30/1.51 thf(fact_157_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) | (A2 = B2))))))))).
% 0.30/1.51 thf(fact_158_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) & (A2 != B2))))))))).
% 0.30/1.51 thf(fact_159_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 0.30/1.51 thf(fact_160_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((A3 != B3) => ((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less @ A) @ A3) @ B3))))))))).
% 0.30/1.51 thf(fact_161_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 0.30/1.51 thf(fact_162_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 0.30/1.51 thf(fact_163_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 0.30/1.51 thf(fact_164_bot_Oextremum__strict, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ (bot_bot @ A)))))))))).
% 0.30/1.51 thf(fact_165_bot_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((A3 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A3)))))))).
% 0.30/1.51 thf(fact_166_subset__emptyI, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X5:A]: ((~ ((((member @ A) @ X5) @ A4))))) => (((ord_less_eq @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_167_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X7)))))))))))))).
% 0.30/1.51 thf(fact_168_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((ord_less_eq @ A) @ X7) @ T)))))))))))).
% 0.30/1.51 thf(fact_169_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (((ord_less_eq @ A) @ T) @ X7)))))))))))).
% 0.30/1.51 thf(fact_170_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (~ ((((ord_less_eq @ A) @ X7) @ T)))))))))))))).
% 0.30/1.51 thf(fact_171_verit__comp__simplify1_I3_J, axiom, (! [B:$tType]: (((linorder @ B) => (! [B7:B,A7:B]: (((~ ((((ord_less_eq @ B) @ B7) @ A7))) = (((ord_less @ B) @ A7) @ B7)))))))).
% 0.30/1.51 thf(fact_172_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A3:A,B3:A,P:(A > $o)]: (((((ord_less @ A) @ A3) @ B3) => ((P @ A3) => ((~ ((P @ B3))) => (? [C4:A]: (((((ord_less_eq @ A) @ A3) @ C4) & ((((ord_less_eq @ A) @ C4) @ B3) & ((! [X7:A]: ((((((ord_less_eq @ A) @ A3) @ X7) & (((ord_less @ A) @ X7) @ C4)) => (P @ X7)))) & (! [D2:A]: (((! [X5:A]: ((((((ord_less_eq @ A) @ A3) @ X5) & (((ord_less @ A) @ X5) @ D2)) => (P @ X5)))) => (((ord_less_eq @ A) @ D2) @ C4)))))))))))))))))).
% 0.30/1.51 thf(fact_173_bot_Oordering__top__axioms, axiom, (! [A:$tType]: (((order_bot @ A) => ((((ordering_top @ A) @ (^ [X4:A,Y4:A]: ((((ord_less_eq @ A) @ Y4) @ X4)))) @ (^ [X4:A,Y4:A]: ((((ord_less @ A) @ Y4) @ X4)))) @ (bot_bot @ A)))))).
% 0.30/1.51 thf(fact_174_psubsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((A4 != B4) => (((ord_less @ (set @ A)) @ A4) @ B4)))))).
% 0.30/1.51 thf(fact_175_less__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F3))))))))))).
% 0.30/1.51 thf(fact_176_ordering__top_Oextremum, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => ((Less_eq @ A3) @ Top))))).
% 0.30/1.51 thf(fact_177_ordering__top_Oextremum__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (~ (((Less @ Top) @ A3))))))).
% 0.30/1.51 thf(fact_178_ordering__top_Oextremum__unique, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (((Less_eq @ Top) @ A3) = (A3 = Top)))))).
% 0.30/1.51 thf(fact_179_ordering__top_Onot__eq__extremum, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => ((A3 != Top) = ((Less @ A3) @ Top)))))).
% 0.30/1.51 thf(fact_180_ordering__top_Oextremum__uniqueI, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (((Less_eq @ Top) @ A3) => (A3 = Top)))))).
% 0.30/1.51 thf(fact_181_not__psubset__empty, axiom, (! [A:$tType,A4:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A4) @ (bot_bot @ (set @ A)))))))).
% 0.30/1.51 thf(fact_182_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B6) | (A6 = B6)))))))).
% 0.30/1.51 thf(fact_183_subset__psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less @ (set @ A)) @ B4) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 0.30/1.51 thf(fact_184_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))))).
% 0.30/1.51 thf(fact_185_psubset__subset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 0.30/1.51 thf(fact_186_psubset__imp__subset, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 0.30/1.51 thf(fact_187_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (A6 != B6)))))))).
% 0.30/1.51 thf(fact_188_psubsetE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B4) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))).
% 0.30/1.51 thf(fact_189_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A6)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ B6)))))))))).
% 0.30/1.51 thf(fact_190_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B3:A]: (((A3 = B3) | ((~ ((((ord_less_eq @ A) @ A3) @ B3))) | (~ ((((ord_less_eq @ A) @ B3) @ A3))))))))))).
% 0.30/1.51 thf(fact_191_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A3:A]: ((? [B5:A]: (((((ord_less @ A) @ A3) @ B5) | (((ord_less @ A) @ B5) @ A3)))))))))).
% 0.30/1.51 thf(fact_192_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 0.30/1.51 thf(fact_193_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P4 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (((P @ X7) & (Q @ X7)) = ((P4 @ X7) & (Q2 @ X7)))))))))))))))).
% 0.30/1.51 thf(fact_194_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P4 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (((P @ X7) | (Q @ X7)) = ((P4 @ X7) | (Q2 @ X7)))))))))))))))).
% 0.30/1.51 thf(fact_195_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (X7 != T)))))))))))).
% 0.30/1.51 thf(fact_196_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (X7 != T)))))))))))).
% 0.30/1.51 thf(fact_197_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (~ ((((ord_less @ A) @ X7) @ T)))))))))))))).
% 0.30/1.51 thf(fact_198_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (((ord_less @ A) @ T) @ X7)))))))))))).
% 0.30/1.51 thf(fact_199_pinf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F4:D]: ((? [Z3:C]: ((! [X7:C]: (((((ord_less @ C) @ Z3) @ X7) => (F4 = F4)))))))))))).
% 0.30/1.51 thf(fact_200_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P4 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((P @ X7) & (Q @ X7)) = ((P4 @ X7) & (Q2 @ X7)))))))))))))))).
% 0.30/1.51 thf(fact_201_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P4 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((P @ X7) | (Q @ X7)) = ((P4 @ X7) | (Q2 @ X7)))))))))))))))).
% 0.30/1.51 thf(fact_202_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (X7 != T)))))))))))).
% 0.30/1.51 thf(fact_203_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (X7 != T)))))))))))).
% 0.30/1.51 thf(fact_204_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((ord_less @ A) @ X7) @ T)))))))))))).
% 0.30/1.51 thf(fact_205_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (~ ((((ord_less @ A) @ T) @ X7)))))))))))))).
% 0.30/1.51 thf(fact_206_minf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F4:D]: ((? [Z3:C]: ((! [X7:C]: (((((ord_less @ C) @ X7) @ Z3) => (F4 = F4)))))))))))).
% 0.30/1.51 thf(fact_207_prop__restrict, axiom, (! [A:$tType,X:A,Z5:(set @ A),X8:(set @ A),P:(A > $o)]: (((((member @ A) @ X) @ Z5) => ((((ord_less_eq @ (set @ A)) @ Z5) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ X8) & (P @ X4)))))) => (P @ X)))))).
% 0.30/1.51 thf(fact_208_Collect__restrict, axiom, (! [A:$tType,X8:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ X8) & (P @ X4)))))) @ X8)))).
% 0.30/1.51 thf(fact_209_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (bot_bot @ (set @ A))))))))).
% 0.30/1.51 thf(fact_210_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 0.30/1.51 thf(fact_211_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A6:(set @ A)]: ((A6 = (bot_bot @ (set @ A))))))))).
% 0.30/1.51 thf(fact_212_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 0.30/1.51 thf(fact_213_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A6)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ B6)))))))))).
% 0.30/1.51 thf(fact_214_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X)))))).
% 0.30/1.51 thf(fact_215_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X) => (Q @ X)))))).
% 0.30/1.51 thf(fact_216_psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B4) => ((((ord_less @ (set @ A)) @ B4) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 0.30/1.51 thf(fact_217_psubsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ B4)))))).
% 0.30/1.51 thf(fact_218_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ R)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 0.30/1.51 thf(fact_219_subset__Collect__iff, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => ((((ord_less_eq @ (set @ A)) @ B4) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A4) & (P @ X4)))))) = (! [X4:A]: (((((member @ A) @ X4) @ B4) => (P @ X4))))))))).
% 0.30/1.51 thf(fact_220_subset__CollectI, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A),Q:(A > $o),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => ((! [X5:A]: (((((member @ A) @ X5) @ B4) => ((Q @ X5) => (P @ X5))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ B4) & (Q @ X4)))))) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A4) & (P @ X4))))))))))).
% 0.30/1.51 thf(fact_221_conj__subset__def, axiom, (! [A:$tType,A4:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ (^ [X4:A]: (((P @ X4) & (Q @ X4)))))) = ((((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ P)) & (((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ Q))))))).
% 0.30/1.51 thf(fact_222_remove_Osimps_I2_J, axiom, (! [A:$tType,H:(A > int),E:A,X:A,T1:(binary1291135688e_Tree @ A),T2:(binary1291135688e_Tree @ A)]: ((((((ord_less @ int) @ (H @ E)) @ (H @ X)) => (((((binary997842527remove @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ ((((binary997842527remove @ A) @ H) @ E) @ T1)) @ X) @ T2))) & ((~ ((((ord_less @ int) @ (H @ E)) @ (H @ X)))) => (((((ord_less @ int) @ (H @ X)) @ (H @ E)) => (((((binary997842527remove @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ X) @ ((((binary997842527remove @ A) @ H) @ E) @ T2)))) & ((~ ((((ord_less @ int) @ (H @ X)) @ (H @ E)))) => (((T1 = (binary1746293266le_Tip @ A)) => (((((binary997842527remove @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = T2)) & ((T1 != (binary1746293266le_Tip @ A)) => (((((binary997842527remove @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = (((((product_case_prod @ (binary1291135688e_Tree @ A)) @ A) @ (binary1291135688e_Tree @ A)) @ (^ [T1p:(binary1291135688e_Tree @ A),R2:A]: (((((binary210054475elle_T @ A) @ T1p) @ R2) @ T2)))) @ (((binary1271298290_wrmrm @ A) @ H) @ T1)))))))))))).
% 0.30/1.51 thf(fact_223_wrmrm__decomp, axiom, (! [A:$tType,T:(binary1291135688e_Tree @ A),H:(A > int)]: (((T != (binary1746293266le_Tip @ A)) => ((((binary1271298290_wrmrm @ A) @ H) @ T) = ((((product_Pair @ (binary1291135688e_Tree @ A)) @ A) @ (((binary213313527le_wrm @ A) @ H) @ T)) @ (((binary576689334lle_rm @ A) @ H) @ T))))))).
% 0.30/1.51 thf(fact_224_case__prod__app, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D > A)) = (^ [F3:(B > (C > (D > A))),X4:((product_prod @ B) @ C),Y4:D]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [L:B,R2:C]: ((((F3 @ L) @ R2) @ Y4)))) @ X4))))))).
% 0.30/1.51 thf(fact_225_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P5:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P5) = P5)))).
% 0.30/1.51 thf(fact_226_bot__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((bot_bot @ (A > (B > $o))) = (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))))))))).
% 0.30/1.51 thf(fact_227_subrelI, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((! [X5:A,Y2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y2)) @ R3) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y2)) @ S2)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ S2))))).
% 0.30/1.51 thf(fact_228_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ R)))) @ (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ S)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 0.30/1.51 thf(fact_229_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: ((((^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ R))) = (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ S)))) = (R = S))))).
% 0.30/1.51 thf(fact_230_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R3:A,S2:B,R:(set @ ((product_prod @ A) @ B)),S3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R3) @ S2)) @ R) => ((S3 = S2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R3) @ S3)) @ R)))))).
% 0.30/1.51 thf(fact_231_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A3:B,B3:C]: (((((((product_case_prod @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((F @ A3) @ B3))))).
% 0.30/1.51 thf(fact_232_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P5:((product_prod @ A) @ B),Z2:C,C2:(A > (B > (set @ C)))]: (((! [A5:A,B5:B]: (((P5 = ((((product_Pair @ A) @ B) @ A5) @ B5)) => (((member @ C) @ Z2) @ ((C2 @ A5) @ B5))))) => (((member @ C) @ Z2) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C2) @ P5)))))).
% 0.30/1.51 thf(fact_233_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X5:A,Y2:B]: ((((P @ X5) @ Y2) => ((Q @ X5) @ Y2)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 0.30/1.51 thf(fact_234_case__prodI, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A3:A,B3:B]: ((((F @ A3) @ B3) => (((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 0.30/1.51 thf(fact_235_case__prodI2, axiom, (! [B:$tType,A:$tType,P5:((product_prod @ A) @ B),C2:(A > (B > $o))]: (((! [A5:A,B5:B]: (((P5 = ((((product_Pair @ A) @ B) @ A5) @ B5)) => ((C2 @ A5) @ B5)))) => (((((product_case_prod @ A) @ B) @ $o) @ C2) @ P5))))).
% 0.30/1.51 thf(fact_236_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P5:((product_prod @ A) @ B),C2:(A > (B > (C > $o))),X:C]: (((! [A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A5) @ B5) = P5) => (((C2 @ A5) @ B5) @ X)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P5) @ X))))).
% 0.30/1.51 thf(fact_237_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),A3:B,B3:C]: (((((member @ A) @ Z2) @ ((C2 @ A3) @ B3)) => (((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ ((((product_Pair @ B) @ C) @ A3) @ B3))))))).
% 0.30/1.51 thf(fact_238_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A4:(A > (B > $o)),B4:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A4) @ B4) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A4))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B4))))))).
% 0.30/1.51 thf(fact_239_predicate2D__conj, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),R:$o,X:A,Y:B]: ((((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) & R) => (R & (((P @ X) @ Y) => ((Q @ X) @ Y))))))).
% 0.30/1.51 thf(fact_240_eq__subset, axiom, (! [A:$tType,P:(A > (A > $o))]: ((((ord_less_eq @ (A > (A > $o))) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ (^ [A2:A,B2:A]: ((((P @ A2) @ B2) | (A2 = B2)))))))).
% 0.30/1.51 thf(fact_241_predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),X:A,Y:B]: (((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => (((P @ X) @ Y) => ((Q @ X) @ Y)))))).
% 0.30/1.51 thf(fact_242_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X:A,Y:B,Q:(A > (B > $o))]: ((((P @ X) @ Y) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X) @ Y)))))).
% 0.30/1.51 thf(fact_243_case__prodD, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A3:A,B3:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) => ((F @ A3) @ B3))))).
% 0.30/1.51 thf(fact_244_case__prodE, axiom, (! [A:$tType,B:$tType,C2:(A > (B > $o)),P5:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C2) @ P5) => (~ ((! [X5:A,Y2:B]: (((P5 = ((((product_Pair @ A) @ B) @ X5) @ Y2)) => (~ (((C2 @ X5) @ Y2)))))))))))).
% 0.30/1.51 thf(fact_245_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R:(A > (B > (C > $o))),A3:A,B3:B,C2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ C2) => (((R @ A3) @ B3) @ C2))))).
% 0.30/1.51 thf(fact_246_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(A > (B > (C > $o))),P5:((product_prod @ A) @ B),Z2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P5) @ Z2) => (~ ((! [X5:A,Y2:B]: (((P5 = ((((product_Pair @ A) @ B) @ X5) @ Y2)) => (~ ((((C2 @ X5) @ Y2) @ Z2)))))))))))).
% 0.30/1.51 thf(fact_247_prod_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H:(C > D),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (((((product_case_prod @ A) @ B) @ D) @ (^ [X12:A,X24:B]: ((H @ ((F @ X12) @ X24))))) @ Prod))))).
% 0.30/1.51 thf(fact_248_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P:(B > (C > A)),Z2:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P) @ Z2)) => (~ ((! [X5:B,Y2:C]: (((Z2 = ((((product_Pair @ B) @ C) @ X5) @ Y2)) => (~ ((Q @ ((P @ X5) @ Y2))))))))))))).
% 0.30/1.51 thf(fact_249_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X4:A,Y4:B]: ((F @ ((((product_Pair @ A) @ B) @ X4) @ Y4))))) = F)))).
% 0.30/1.51 thf(fact_250_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C)),G:(((product_prod @ A) @ B) > C)]: (((! [X5:A,Y2:B]: ((((F @ X5) @ Y2) = (G @ ((((product_Pair @ A) @ B) @ X5) @ Y2))))) => (((((product_case_prod @ A) @ B) @ C) @ F) = G))))).
% 0.30/1.51 thf(fact_251_less__by__empty, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A)),B4:(set @ ((product_prod @ A) @ A))]: (((A4 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A4) @ B4))))).
% 0.30/1.51 thf(fact_252_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R2:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R2) => (((ord_less_eq @ B) @ (As @ I)) @ (As @ J)))))))))))).
% 0.30/1.51 thf(fact_253_split__part, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A2:A,B2:B]: ((P & ((Q @ A2) @ B2))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))).
% 0.30/1.51 thf(fact_254_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 0.30/1.51 thf(fact_255_inv__image__def, axiom, (! [A:$tType,B:$tType]: ((((inv_image @ B) @ A) = (^ [R2:(set @ ((product_prod @ B) @ B)),F3:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X4:A,Y4:A]: ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F3 @ X4)) @ (F3 @ Y4))) @ R2))))))))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A8:$tType,A9:$tType]: (((order_bot @ A9) => (order_bot @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ int)).
% 0.30/1.51 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, (preorder @ int)).
% 0.30/1.51 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 0.30/1.51 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, (no_top @ int)).
% 0.30/1.51 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, (no_bot @ int)).
% 0.30/1.51 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, (order @ int)).
% 0.30/1.51 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, (ord @ int)).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oorder__bot_4, axiom, (! [A8:$tType]: ((order_bot @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_5, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Obot_8, axiom, (! [A8:$tType]: ((bot @ (set @ A8))))).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__bot_9, axiom, (order_bot @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_10, axiom, (preorder @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder_11, axiom, (linorder @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_12, axiom, (order @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oord_13, axiom, (ord @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Obot_14, axiom, (bot @ $o)).
% 0.30/1.51 thf(conj_0, conjecture, (((binary1610619414edTree @ a) @ h) @ t1)).
% 0.30/1.51 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.yadyxb1T5r/DTF2DTF28889.p
% 0.30/1.51 ------------------------
% 0.30/1.55 ---- Cleaned THF ---
% 0.30/1.55 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
% 0.30/1.55 binary1291135688e_Tree: $tType > $tType ).
% 0.30/1.55
% 0.30/1.55 thf(ty_t_Product__Type_Oprod,type,
% 0.30/1.55 product_prod: $tType > $tType > $tType ).
% 0.30/1.55
% 0.30/1.55 thf(ty_t_Set_Oset,type,
% 0.30/1.55 set: $tType > $tType ).
% 0.30/1.55
% 0.30/1.55 thf(ty_t_Int_Oint,type,
% 0.30/1.55 int: $tType ).
% 0.30/1.55
% 0.30/1.55 thf(ty_tf_a,type,
% 0.30/1.55 a: $tType ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_HOL_Otype,type,
% 0.30/1.55 type:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Obot,type,
% 0.30/1.55 bot:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Oord,type,
% 0.30/1.55 ord:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Oorder,type,
% 0.30/1.55 order:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Ono__bot,type,
% 0.30/1.55 no_bot:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Ono__top,type,
% 0.30/1.55 no_top:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/1.55 linorder:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/1.55 preorder:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.30/1.55 order_bot:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Owellorder,type,
% 0.30/1.55 wellorder:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Odense__order,type,
% 0.30/1.55 dense_order:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.30/1.55 dense_linorder:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.30/1.55 condit1656338222tinuum:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.30/1.55 condit1037483654norder:
% 0.30/1.55 !>[A: $tType] : $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.30/1.55 bNF_Ca1785829860lChain:
% 0.30/1.55 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
% 0.30/1.55 binary210054475elle_T:
% 0.30/1.55 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
% 0.30/1.55 binary1746293266le_Tip:
% 0.30/1.55 !>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Ocase__Tree,type,
% 0.30/1.55 binary536355927e_Tree:
% 0.30/1.55 !>[B: $tType,A: $tType] : ( B > ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B ) > ( binary1291135688e_Tree @ A ) > B ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Orec__Tree,type,
% 0.30/1.55 binary1929596613c_Tree:
% 0.30/1.55 !>[C: $tType,A: $tType] : ( C > ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C ) > ( binary1291135688e_Tree @ A ) > C ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert,type,
% 0.30/1.55 binary1830089824insert:
% 0.30/1.55 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
% 0.30/1.55 binary827270440e_memb:
% 0.30/1.55 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oremove,type,
% 0.30/1.55 binary997842527remove:
% 0.30/1.55 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Orm,type,
% 0.30/1.55 binary576689334lle_rm:
% 0.30/1.55 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > A ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
% 0.30/1.55 binary1653327646_setOf:
% 0.30/1.55 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
% 0.30/1.55 binary1610619414edTree:
% 0.30/1.55 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
% 0.30/1.55 binary231205461t_pred:
% 0.30/1.55 !>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrm,type,
% 0.30/1.55 binary213313527le_wrm:
% 0.30/1.55 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrmrm,type,
% 0.30/1.55 binary1271298290_wrmrm:
% 0.30/1.55 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( product_prod @ ( binary1291135688e_Tree @ A ) @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/1.55 bot_bot:
% 0.30/1.55 !>[A: $tType] : A ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/1.55 ord_less:
% 0.30/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/1.55 ord_less_eq:
% 0.30/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Oordering__top,type,
% 0.30/1.55 ordering_top:
% 0.30/1.55 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Product__Type_OPair,type,
% 0.30/1.55 product_Pair:
% 0.30/1.55 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.30/1.55 product_case_prod:
% 0.30/1.55 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Relation_Oinv__image,type,
% 0.30/1.55 inv_image:
% 0.30/1.55 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Set_OCollect,type,
% 0.30/1.55 collect:
% 0.30/1.55 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Set_Ois__empty,type,
% 0.30/1.55 is_empty:
% 0.30/1.55 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_member,type,
% 0.30/1.55 member:
% 0.30/1.55 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/1.55
% 0.30/1.55 thf(sy_v_h,type,
% 0.30/1.55 h: a > int ).
% 0.30/1.55
% 0.30/1.55 thf(sy_v_t1____,type,
% 0.30/1.55 t1: binary1291135688e_Tree @ a ).
% 0.30/1.55
% 0.30/1.55 thf(sy_v_t2____,type,
% 0.30/1.55 t2: binary1291135688e_Tree @ a ).
% 0.30/1.55
% 0.30/1.55 thf(sy_v_x____,type,
% 0.30/1.55 x: a ).
% 0.30/1.55
% 0.30/1.55 thf(fact_0_t2nTip,axiom,
% 0.30/1.55 ( t2
% 0.30/1.55 != ( binary1746293266le_Tip @ a ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_1_s,axiom,
% 0.30/1.55 binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_2_res,axiom,
% 0.30/1.55 ( ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
% 0.30/1.55 = ( binary210054475elle_T @ a @ t1 @ x @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_3_sortLemmaL,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 => ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_4_sortLemmaR,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 => ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_5_Tree_Oinject,axiom,
% 0.30/1.55 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
% 0.30/1.55 = ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
% 0.30/1.55 = ( ( X21 = Y21 )
% 0.30/1.55 & ( X22 = Y22 )
% 0.30/1.55 & ( X23 = Y23 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_6_binsert__sorted,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
% 0.30/1.55 ( ( binary1610619414edTree @ A @ H @ T )
% 0.30/1.55 => ( binary1610619414edTree @ A @ H @ ( binary1830089824insert @ A @ H @ X @ T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_7_h1,axiom,
% 0.30/1.55 ( ( ( t1
% 0.30/1.55 != ( binary1746293266le_Tip @ a ) )
% 0.30/1.55 & ( binary1610619414edTree @ a @ h @ t1 ) )
% 0.30/1.55 => ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ t1 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_8_sortedTree_Osimps_I1_J,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_9_memb__spec,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
% 0.30/1.55 ( ( binary1610619414edTree @ A @ H @ T )
% 0.30/1.55 => ( ( binary827270440e_memb @ A @ H @ X @ T )
% 0.30/1.55 = ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_10_sorted__distinct__pred__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( binary231205461t_pred @ A )
% 0.30/1.55 = ( ^ [H2: A > int,A2: A,B2: A,T3: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( ( binary1610619414edTree @ A @ H2 @ T3 )
% 0.30/1.55 & ( member @ A @ A2 @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.30/1.55 & ( member @ A @ B2 @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.30/1.55 & ( ( H2 @ A2 )
% 0.30/1.55 = ( H2 @ B2 ) ) )
% 0.30/1.55 => ( A2 = B2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_11_h2,axiom,
% 0.30/1.55 ( ( ( t2
% 0.30/1.55 != ( binary1746293266le_Tip @ a ) )
% 0.30/1.55 & ( binary1610619414edTree @ a @ h @ t2 ) )
% 0.30/1.55 => ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_12_sorted__distinct,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,A3: A,B3: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A3 @ B3 @ T ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_13__092_060open_062Tip_A_092_060noteq_062_ATip_A_092_060and_062_AsortedTree_Ah_ATip_A_092_060longrightarrow_062_AsortedTree_Ah_A_Iwrm_Ah_ATip_J_092_060close_062,axiom,
% 0.30/1.55 ( ( ( ( binary1746293266le_Tip @ a )
% 0.30/1.55 != ( binary1746293266le_Tip @ a ) )
% 0.30/1.55 & ( binary1610619414edTree @ a @ h @ ( binary1746293266le_Tip @ a ) ) )
% 0.30/1.55 => ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ ( binary1746293266le_Tip @ a ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_14_Tree_Odistinct_I1_J,axiom,
% 0.30/1.55 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( binary1746293266le_Tip @ A )
% 0.30/1.55 != ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_15_binsert_Osimps_I1_J,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,E: A] :
% 0.30/1.55 ( ( binary1830089824insert @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 = ( binary210054475elle_T @ A @ ( binary1746293266le_Tip @ A ) @ E @ ( binary1746293266le_Tip @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_16_wrm_Osimps,axiom,
% 0.30/1.55 ! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
% 0.30/1.55 ( ( ( T2
% 0.30/1.55 = ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = T1 ) )
% 0.30/1.55 & ( ( T2
% 0.30/1.55 != ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary213313527le_wrm @ A @ H @ T2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_17_Tree_Oinduct,axiom,
% 0.30/1.55 ! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( P @ ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ( ! [X1: binary1291135688e_Tree @ A,X2: A,X3: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( P @ X1 )
% 0.30/1.55 => ( ( P @ X3 )
% 0.30/1.55 => ( P @ ( binary210054475elle_T @ A @ X1 @ X2 @ X3 ) ) ) )
% 0.30/1.55 => ( P @ Tree ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_18_Tree_Oexhaust,axiom,
% 0.30/1.55 ! [A: $tType,Y: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( Y
% 0.30/1.55 != ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( Y
% 0.30/1.55 != ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_19_rm__set,axiom,
% 0.30/1.55 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.30/1.55 ( ( ( T
% 0.30/1.55 != ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.30/1.55 => ( member @ A @ ( binary576689334lle_rm @ A @ H @ T ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_20_rm_Osimps,axiom,
% 0.30/1.55 ! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
% 0.30/1.55 ( ( ( T2
% 0.30/1.55 = ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = X ) )
% 0.30/1.55 & ( ( T2
% 0.30/1.55 != ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( binary576689334lle_rm @ A @ H @ T2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_21_wrm__set1,axiom,
% 0.30/1.55 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.30/1.55 ( ( ( T
% 0.30/1.55 != ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ ( binary1653327646_setOf @ A @ ( binary213313527le_wrm @ A @ H @ T ) ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_22_sortedTree_Osimps_I2_J,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( ( binary1610619414edTree @ A @ H @ T1 )
% 0.30/1.55 & ! [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T1 ) )
% 0.30/1.55 => ( ord_less @ int @ ( H @ X4 ) @ ( H @ X ) ) )
% 0.30/1.55 & ! [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T2 ) )
% 0.30/1.55 => ( ord_less @ int @ ( H @ X ) @ ( H @ X4 ) ) )
% 0.30/1.55 & ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_23_remove_Osimps_I1_J,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,E: A] :
% 0.30/1.55 ( ( binary997842527remove @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 = ( binary1746293266le_Tip @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_24_Tree_Osimps_I4_J,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B] :
% 0.30/1.55 ( ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 = F1 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_25_Tree_Osimps_I5_J,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
% 0.30/1.55 = ( F2 @ X21 @ X22 @ X23 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_26_binsert_Osimps_I2_J,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.30/1.55 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
% 0.30/1.55 & ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.30/1.55 => ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.30/1.55 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
% 0.30/1.55 & ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.30/1.55 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_27_setOf_Osimps_I1_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_28_Tree_Osimps_I6_J,axiom,
% 0.30/1.55 ! [A: $tType,C: $tType,F1: C,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C] :
% 0.30/1.55 ( ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 = F1 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_29_Tree_Osimps_I7_J,axiom,
% 0.30/1.55 ! [C: $tType,A: $tType,F1: C,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
% 0.30/1.55 = ( F2 @ X21 @ X22 @ X23 @ ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ X21 ) @ ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ X23 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_30_Tree_Ocase__distrib,axiom,
% 0.30/1.55 ! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B,Tree: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( H @ ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ Tree ) )
% 0.30/1.55 = ( binary536355927e_Tree @ C @ A @ ( H @ F1 )
% 0.30/1.55 @ ^ [X12: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] : ( H @ ( F2 @ X12 @ X24 @ X32 ) )
% 0.30/1.55 @ Tree ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_31_empty__subsetI,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_32_subset__empty,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 = ( A4
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_33_subset__antisym,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.30/1.55 => ( A4 = B4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_34_subsetI,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ( member @ A @ X5 @ A4 )
% 0.30/1.55 => ( member @ A @ X5 @ B4 ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_35_empty__Collect__eq,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o] :
% 0.30/1.55 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/1.55 = ( collect @ A @ P ) )
% 0.30/1.55 = ( ! [X4: A] :
% 0.30/1.55 ~ ( P @ X4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_36_Collect__empty__eq,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o] :
% 0.30/1.55 ( ( ( collect @ A @ P )
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 = ( ! [X4: A] :
% 0.30/1.55 ~ ( P @ X4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_37_all__not__in__conv,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] :
% 0.30/1.55 ( ( ! [X4: A] :
% 0.30/1.55 ~ ( member @ A @ X4 @ A4 ) )
% 0.30/1.55 = ( A4
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_38_empty__iff,axiom,
% 0.30/1.55 ! [A: $tType,C2: A] :
% 0.30/1.55 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_39_bot__apply,axiom,
% 0.30/1.55 ! [C: $tType,D: $tType] :
% 0.30/1.55 ( ( bot @ C )
% 0.30/1.55 => ( ( bot_bot @ ( D > C ) )
% 0.30/1.55 = ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_40_order__refl,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_41_bot__set__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( bot_bot @ ( set @ A ) )
% 0.30/1.55 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_42_le__funD,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ! [F: A > B,G: A > B,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/1.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_43_le__funE,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ! [F: A > B,G: A > B,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/1.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_44_le__funI,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ! [F: A > B,G: A > B] :
% 0.30/1.55 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 0.30/1.55 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/1.55 ! [A: $tType,A3: A,P: A > $o] :
% 0.30/1.55 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.30/1.55 = ( P @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] :
% 0.30/1.55 ( ( collect @ A
% 0.30/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.30/1.55 = A4 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_47_Collect__cong,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ( P @ X5 )
% 0.30/1.55 = ( Q @ X5 ) )
% 0.30/1.55 => ( ( collect @ A @ P )
% 0.30/1.55 = ( collect @ A @ Q ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_48_ext,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ( F @ X5 )
% 0.30/1.55 = ( G @ X5 ) )
% 0.30/1.55 => ( F = G ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_49_le__fun__def,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/1.55 = ( ^ [F3: A > B,G2: A > B] :
% 0.30/1.55 ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_50_order__subst1,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( order @ B )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.30/1.55 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.30/1.55 => ( ! [X5: B,Y2: B] :
% 0.30/1.55 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_51_order__subst2,axiom,
% 0.30/1.55 ! [A: $tType,C: $tType] :
% 0.30/1.55 ( ( ( order @ C )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.30/1.55 => ( ! [X5: A,Y2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_52_ord__eq__le__subst,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( ord @ B )
% 0.30/1.55 & ( ord @ A ) )
% 0.30/1.55 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.30/1.55 ( ( A3
% 0.30/1.55 = ( F @ B3 ) )
% 0.30/1.55 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.30/1.55 => ( ! [X5: B,Y2: B] :
% 0.30/1.55 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_53_ord__le__eq__subst,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( ord @ B )
% 0.30/1.55 & ( ord @ A ) )
% 0.30/1.55 => ! [A3: A,B3: A,F: A > B,C2: B] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ( F @ B3 )
% 0.30/1.55 = C2 )
% 0.30/1.55 => ( ! [X5: A,Y2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_54_eq__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.30/1.55 = ( ^ [X4: A,Y4: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/1.55 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_55_antisym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_56_linear,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_57_eq__refl,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( X = Y )
% 0.30/1.55 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_58_le__cases,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_59_order_Otrans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.30/1.55 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_60_le__cases3,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z2: A] :
% 0.30/1.55 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.30/1.55 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.30/1.55 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.30/1.55 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/1.55 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.30/1.55 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_61_antisym__conv,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [Y: A,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 = ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_62_order__class_Oorder_Oeq__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.30/1.55 = ( ^ [A2: A,B2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 & ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_63_ord__eq__le__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord @ A )
% 0.30/1.55 => ! [A3: A,B3: A,C2: A] :
% 0.30/1.55 ( ( A3 = B3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.30/1.55 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_64_ord__le__eq__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord @ A )
% 0.30/1.55 => ! [A3: A,B3: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( B3 = C2 )
% 0.30/1.55 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_65_order__class_Oorder_Oantisym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/1.55 => ( A3 = B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_66_order__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.30/1.55 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_67_dual__order_Orefl,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_68_linorder__wlog,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.30/1.55 ( ! [A5: A,B5: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.30/1.55 => ( P @ A5 @ B5 ) )
% 0.30/1.55 => ( ! [A5: A,B5: A] :
% 0.30/1.55 ( ( P @ B5 @ A5 )
% 0.30/1.55 => ( P @ A5 @ B5 ) )
% 0.30/1.55 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_69_dual__order_Otrans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B3: A,A3: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.30/1.55 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_70_dual__order_Oeq__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.30/1.55 = ( ^ [A2: A,B2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/1.55 & ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_71_dual__order_Oantisym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B3: A,A3: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( A3 = B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_72_ord__eq__less__subst,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( ord @ B )
% 0.30/1.55 & ( ord @ A ) )
% 0.30/1.55 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.30/1.55 ( ( A3
% 0.30/1.55 = ( F @ B3 ) )
% 0.30/1.55 => ( ( ord_less @ B @ B3 @ C2 )
% 0.30/1.55 => ( ! [X5: B,Y2: B] :
% 0.30/1.55 ( ( ord_less @ B @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_73_ord__less__eq__subst,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( ord @ B )
% 0.30/1.55 & ( ord @ A ) )
% 0.30/1.55 => ! [A3: A,B3: A,F: A > B,C2: B] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ( F @ B3 )
% 0.30/1.55 = C2 )
% 0.30/1.55 => ( ! [X5: A,Y2: A] :
% 0.30/1.55 ( ( ord_less @ A @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_74_order__less__subst1,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( order @ B )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.30/1.55 => ( ( ord_less @ B @ B3 @ C2 )
% 0.30/1.55 => ( ! [X5: B,Y2: B] :
% 0.30/1.55 ( ( ord_less @ B @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_75_order__less__subst2,axiom,
% 0.30/1.55 ! [A: $tType,C: $tType] :
% 0.30/1.55 ( ( ( order @ C )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 0.30/1.55 => ( ! [X5: A,Y2: A] :
% 0.30/1.55 ( ( ord_less @ A @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_76_lt__ex,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( no_bot @ A )
% 0.30/1.55 => ! [X: A] :
% 0.30/1.55 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_77_gt__ex,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( no_top @ A )
% 0.30/1.55 => ! [X: A] :
% 0.30/1.55 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_78_neqE,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( X != Y )
% 0.30/1.55 => ( ~ ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_79_neq__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( X != Y )
% 0.30/1.55 = ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_80_order_Oasym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_81_dense,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( dense_order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ? [Z3: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Z3 )
% 0.30/1.55 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_82_less__imp__neq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( X != Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_83_less__asym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_84_less__asym_H,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [A3: A,B3: A] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_85_less__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z2: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less @ A @ Y @ Z2 )
% 0.30/1.55 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_86_less__linear,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 | ( X = Y )
% 0.30/1.55 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_87_less__irrefl,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A] :
% 0.30/1.55 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_88_ord__eq__less__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord @ A )
% 0.30/1.55 => ! [A3: A,B3: A,C2: A] :
% 0.30/1.55 ( ( A3 = B3 )
% 0.30/1.55 => ( ( ord_less @ A @ B3 @ C2 )
% 0.30/1.55 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_89_ord__less__eq__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord @ A )
% 0.30/1.55 => ! [A3: A,B3: A,C2: A] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( B3 = C2 )
% 0.30/1.55 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_90_dual__order_Oasym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B3: A,A3: A] :
% 0.30/1.55 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/1.55 => ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_91_less__imp__not__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( X != Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_92_less__not__sym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_93_less__induct,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( wellorder @ A )
% 0.30/1.55 => ! [P: A > $o,A3: A] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ! [Y5: A] :
% 0.30/1.55 ( ( ord_less @ A @ Y5 @ X5 )
% 0.30/1.55 => ( P @ Y5 ) )
% 0.30/1.55 => ( P @ X5 ) )
% 0.30/1.55 => ( P @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_94_antisym__conv3,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [Y: A,X: A] :
% 0.30/1.55 ( ~ ( ord_less @ A @ Y @ X )
% 0.30/1.55 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/1.55 = ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_95_less__imp__not__eq2,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( Y != X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_96_less__imp__triv,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,P: $o] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less @ A @ Y @ X )
% 0.30/1.55 => P ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_97_linorder__cases,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ( X != Y )
% 0.30/1.55 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_98_dual__order_Oirrefl,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A] :
% 0.30/1.55 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_99_order_Ostrict__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A,C2: A] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less @ A @ B3 @ C2 )
% 0.30/1.55 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_100_less__imp__not__less,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_101_exists__least__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( wellorder @ A )
% 0.30/1.55 => ( ( ^ [P2: A > $o] :
% 0.30/1.55 ? [X6: A] : ( P2 @ X6 ) )
% 0.30/1.55 = ( ^ [P3: A > $o] :
% 0.30/1.55 ? [N: A] :
% 0.30/1.55 ( ( P3 @ N )
% 0.30/1.55 & ! [M: A] :
% 0.30/1.55 ( ( ord_less @ A @ M @ N )
% 0.30/1.55 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_102_linorder__less__wlog,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.30/1.55 ( ! [A5: A,B5: A] :
% 0.30/1.55 ( ( ord_less @ A @ A5 @ B5 )
% 0.30/1.55 => ( P @ A5 @ B5 ) )
% 0.30/1.55 => ( ! [A5: A] : ( P @ A5 @ A5 )
% 0.30/1.55 => ( ! [A5: A,B5: A] :
% 0.30/1.55 ( ( P @ B5 @ A5 )
% 0.30/1.55 => ( P @ A5 @ B5 ) )
% 0.30/1.55 => ( P @ A3 @ B3 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_103_dual__order_Ostrict__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B3: A,A3: A,C2: A] :
% 0.30/1.55 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/1.55 => ( ( ord_less @ A @ C2 @ B3 )
% 0.30/1.55 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_104_not__less__iff__gr__or__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/1.55 = ( ( ord_less @ A @ Y @ X )
% 0.30/1.55 | ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_105_order_Ostrict__implies__not__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( A3 != B3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_106_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B3: A,A3: A] :
% 0.30/1.55 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/1.55 => ( A3 != B3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_107_bot__fun__def,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( bot @ B )
% 0.30/1.55 => ( ( bot_bot @ ( A > B ) )
% 0.30/1.55 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_108_emptyE,axiom,
% 0.30/1.55 ! [A: $tType,A3: A] :
% 0.30/1.55 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_109_equals0D,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,A3: A] :
% 0.30/1.55 ( ( A4
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 => ~ ( member @ A @ A3 @ A4 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_110_equals0I,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] :
% 0.30/1.55 ( ! [Y2: A] :
% 0.30/1.55 ~ ( member @ A @ Y2 @ A4 )
% 0.30/1.55 => ( A4
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_111_ex__in__conv,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] :
% 0.30/1.55 ( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.30/1.55 = ( A4
% 0.30/1.55 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_112_in__mono,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( member @ A @ X @ A4 )
% 0.30/1.55 => ( member @ A @ X @ B4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_113_subsetD,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( member @ A @ C2 @ A4 )
% 0.30/1.55 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_114_equalityE,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ( A4 = B4 )
% 0.30/1.55 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_115_subset__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.30/1.55 ! [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A6 )
% 0.30/1.55 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_116_equalityD1,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ( A4 = B4 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_117_equalityD2,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ( A4 = B4 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_118_subset__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.30/1.55 ! [T3: A] :
% 0.30/1.55 ( ( member @ A @ T3 @ A6 )
% 0.30/1.55 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_119_subset__refl,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_120_Collect__mono,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ( P @ X5 )
% 0.30/1.55 => ( Q @ X5 ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_121_subset__trans,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_122_set__eq__subset,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ^ [Y3: set @ A,Z: set @ A] : ( Y3 = Z ) )
% 0.30/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.30/1.55 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_123_Collect__mono__iff,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/1.55 = ( ! [X4: A] :
% 0.30/1.55 ( ( P @ X4 )
% 0.30/1.55 => ( Q @ X4 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_124_empty__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( bot_bot @ ( set @ A ) )
% 0.30/1.55 = ( collect @ A
% 0.30/1.55 @ ^ [X4: A] : $false ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_125_Collect__subset,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,P: A > $o] :
% 0.30/1.55 ( ord_less_eq @ ( set @ A )
% 0.30/1.55 @ ( collect @ A
% 0.30/1.55 @ ^ [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A4 )
% 0.30/1.55 & ( P @ X4 ) ) )
% 0.30/1.55 @ A4 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_126_leD,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [Y: A,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_127_leI,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_128_le__less,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less_eq @ A )
% 0.30/1.55 = ( ^ [X4: A,Y4: A] :
% 0.30/1.55 ( ( ord_less @ A @ X4 @ Y4 )
% 0.30/1.55 | ( X4 = Y4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_129_less__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less @ A )
% 0.30/1.55 = ( ^ [X4: A,Y4: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/1.55 & ( X4 != Y4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_130_order__le__less__subst1,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( order @ B )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.30/1.55 => ( ( ord_less @ B @ B3 @ C2 )
% 0.30/1.55 => ( ! [X5: B,Y2: B] :
% 0.30/1.55 ( ( ord_less @ B @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_131_order__le__less__subst2,axiom,
% 0.30/1.55 ! [A: $tType,C: $tType] :
% 0.30/1.55 ( ( ( order @ C )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 0.30/1.55 => ( ! [X5: A,Y2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_132_order__less__le__subst1,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( order @ B )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.30/1.55 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.30/1.55 => ( ! [X5: B,Y2: B] :
% 0.30/1.55 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_133_order__less__le__subst2,axiom,
% 0.30/1.55 ! [A: $tType,C: $tType] :
% 0.30/1.55 ( ( ( order @ C )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.30/1.55 => ( ! [X5: A,Y2: A] :
% 0.30/1.55 ( ( ord_less @ A @ X5 @ Y2 )
% 0.30/1.55 => ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.30/1.55 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_134_not__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.30/1.55 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_135_not__less,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/1.55 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_136_le__neq__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( A3 != B3 )
% 0.30/1.55 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_137_antisym__conv1,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 = ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_138_antisym__conv2,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/1.55 = ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_139_less__imp__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_140_le__less__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less @ A @ Y @ Z2 )
% 0.30/1.55 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_141_less__le__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z2: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.30/1.55 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_142_dense__ge,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( dense_linorder @ A )
% 0.30/1.55 => ! [Z2: A,Y: A] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z2 @ X5 )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ X5 ) )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_143_dense__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( dense_linorder @ A )
% 0.30/1.55 => ! [Y: A,Z2: A] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ X5 @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ X5 @ Z2 ) )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_144_le__less__linear,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_145_le__imp__less__or__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 | ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_146_less__le__not__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ( ( ord_less @ A )
% 0.30/1.55 = ( ^ [X4: A,Y4: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/1.55 & ~ ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_147_not__le__imp__less,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [Y: A,X: A] :
% 0.30/1.55 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_148_order_Ostrict__trans1,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less @ A @ B3 @ C2 )
% 0.30/1.55 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_149_order_Ostrict__trans2,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A,C2: A] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.30/1.55 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_150_order_Oorder__iff__strict,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less_eq @ A )
% 0.30/1.55 = ( ^ [A2: A,B2: A] :
% 0.30/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/1.55 | ( A2 = B2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_151_order_Ostrict__iff__order,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less @ A )
% 0.30/1.55 = ( ^ [A2: A,B2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 & ( A2 != B2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_152_dual__order_Ostrict__trans1,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B3: A,A3: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/1.55 => ( ( ord_less @ A @ C2 @ B3 )
% 0.30/1.55 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_153_dual__order_Ostrict__trans2,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B3: A,A3: A,C2: A] :
% 0.30/1.55 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.30/1.55 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_154_dense__ge__bounded,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( dense_linorder @ A )
% 0.30/1.55 => ! [Z2: A,X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z2 @ X )
% 0.30/1.55 => ( ! [W: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z2 @ W )
% 0.30/1.55 => ( ( ord_less @ A @ W @ X )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_155_dense__le__bounded,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( dense_linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z2: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ! [W: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ W )
% 0.30/1.55 => ( ( ord_less @ A @ W @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_156_order_Ostrict__implies__order,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_157_dual__order_Oorder__iff__strict,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less_eq @ A )
% 0.30/1.55 = ( ^ [B2: A,A2: A] :
% 0.30/1.55 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/1.55 | ( A2 = B2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_158_dual__order_Ostrict__iff__order,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less @ A )
% 0.30/1.55 = ( ^ [B2: A,A2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/1.55 & ( A2 != B2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_159_dual__order_Ostrict__implies__order,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B3: A,A3: A] :
% 0.30/1.55 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/1.55 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_160_order_Onot__eq__order__implies__strict,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A,B3: A] :
% 0.30/1.55 ( ( A3 != B3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_161_bot_Oextremum,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_162_bot_Oextremum__unique,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ! [A3: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/1.55 = ( A3
% 0.30/1.55 = ( bot_bot @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_163_bot_Oextremum__uniqueI,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ! [A3: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/1.55 => ( A3
% 0.30/1.55 = ( bot_bot @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_164_bot_Oextremum__strict,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ! [A3: A] :
% 0.30/1.55 ~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_165_bot_Onot__eq__extremum,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ! [A3: A] :
% 0.30/1.55 ( ( A3
% 0.30/1.55 != ( bot_bot @ A ) )
% 0.30/1.55 = ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_166_subset__emptyI,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ~ ( member @ A @ X5 @ A4 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_167_minf_I8_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_168_minf_I6_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/1.55 => ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_169_pinf_I8_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/1.55 => ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_170_pinf_I6_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_171_verit__comp__simplify1_I3_J,axiom,
% 0.30/1.55 ! [B: $tType] :
% 0.30/1.55 ( ( linorder @ B )
% 0.30/1.55 => ! [B7: B,A7: B] :
% 0.30/1.55 ( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
% 0.30/1.55 = ( ord_less @ B @ A7 @ B7 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_172_complete__interval,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( condit1037483654norder @ A )
% 0.30/1.55 => ! [A3: A,B3: A,P: A > $o] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/1.55 => ( ( P @ A3 )
% 0.30/1.55 => ( ~ ( P @ B3 )
% 0.30/1.55 => ? [C4: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A3 @ C4 )
% 0.30/1.55 & ( ord_less_eq @ A @ C4 @ B3 )
% 0.30/1.55 & ! [X7: A] :
% 0.30/1.55 ( ( ( ord_less_eq @ A @ A3 @ X7 )
% 0.30/1.55 & ( ord_less @ A @ X7 @ C4 ) )
% 0.30/1.55 => ( P @ X7 ) )
% 0.30/1.55 & ! [D2: A] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ( ( ord_less_eq @ A @ A3 @ X5 )
% 0.30/1.55 & ( ord_less @ A @ X5 @ D2 ) )
% 0.30/1.55 => ( P @ X5 ) )
% 0.30/1.55 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_173_bot_Oordering__top__axioms,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ( ordering_top @ A
% 0.30/1.55 @ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
% 0.30/1.55 @ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 )
% 0.30/1.55 @ ( bot_bot @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_174_psubsetI,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( A4 != B4 )
% 0.30/1.55 => ( ord_less @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_175_less__fun__def,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ( ( ord_less @ ( A > B ) )
% 0.30/1.55 = ( ^ [F3: A > B,G2: A > B] :
% 0.30/1.55 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.30/1.55 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_176_ordering__top_Oextremum,axiom,
% 0.30/1.55 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/1.55 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/1.55 => ( Less_eq @ A3 @ Top ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_177_ordering__top_Oextremum__strict,axiom,
% 0.30/1.55 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/1.55 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/1.55 => ~ ( Less @ Top @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_178_ordering__top_Oextremum__unique,axiom,
% 0.30/1.55 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/1.55 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/1.55 => ( ( Less_eq @ Top @ A3 )
% 0.30/1.55 = ( A3 = Top ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_179_ordering__top_Onot__eq__extremum,axiom,
% 0.30/1.55 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/1.55 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/1.55 => ( ( A3 != Top )
% 0.30/1.55 = ( Less @ A3 @ Top ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_180_ordering__top_Oextremum__uniqueI,axiom,
% 0.30/1.55 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/1.55 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/1.55 => ( ( Less_eq @ Top @ A3 )
% 0.30/1.55 => ( A3 = Top ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_181_not__psubset__empty,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A] :
% 0.30/1.55 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_182_subset__iff__psubset__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) @ A6 @ B6 )
% 0.30/1.55 | ( A6 = B6 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_183_subset__psubset__trans,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
% 0.30/1.55 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_184_subset__not__subset__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.30/1.55 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_185_psubset__subset__trans,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.30/1.55 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_186_psubset__imp__subset,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_187_psubset__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.30/1.55 & ( A6 != B6 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_188_psubsetE,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_189_less__eq__set__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.30/1.55 ( ord_less_eq @ ( A > $o )
% 0.30/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ A6 )
% 0.30/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_190_verit__la__disequality,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [A3: A,B3: A] :
% 0.30/1.55 ( ( A3 = B3 )
% 0.30/1.55 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/1.55 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_191_ex__gt__or__lt,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( condit1656338222tinuum @ A )
% 0.30/1.55 => ! [A3: A] :
% 0.30/1.55 ? [B5: A] :
% 0.30/1.55 ( ( ord_less @ A @ A3 @ B5 )
% 0.30/1.55 | ( ord_less @ A @ B5 @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_192_verit__comp__simplify1_I1_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: A] :
% 0.30/1.55 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_193_pinf_I1_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/1.55 ( ? [Z4: A] :
% 0.30/1.55 ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z4 @ X5 )
% 0.30/1.55 => ( ( P @ X5 )
% 0.30/1.55 = ( P4 @ X5 ) ) )
% 0.30/1.55 => ( ? [Z4: A] :
% 0.30/1.55 ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z4 @ X5 )
% 0.30/1.55 => ( ( Q @ X5 )
% 0.30/1.55 = ( Q2 @ X5 ) ) )
% 0.30/1.55 => ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/1.55 => ( ( ( P @ X7 )
% 0.30/1.55 & ( Q @ X7 ) )
% 0.30/1.55 = ( ( P4 @ X7 )
% 0.30/1.55 & ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_194_pinf_I2_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/1.55 ( ? [Z4: A] :
% 0.30/1.55 ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z4 @ X5 )
% 0.30/1.55 => ( ( P @ X5 )
% 0.30/1.55 = ( P4 @ X5 ) ) )
% 0.30/1.55 => ( ? [Z4: A] :
% 0.30/1.55 ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z4 @ X5 )
% 0.30/1.55 => ( ( Q @ X5 )
% 0.30/1.55 = ( Q2 @ X5 ) ) )
% 0.30/1.55 => ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/1.55 => ( ( ( P @ X7 )
% 0.30/1.55 | ( Q @ X7 ) )
% 0.30/1.55 = ( ( P4 @ X7 )
% 0.30/1.55 | ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_195_pinf_I3_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/1.55 => ( X7 != T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_196_pinf_I4_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/1.55 => ( X7 != T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_197_pinf_I5_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/1.55 => ~ ( ord_less @ A @ X7 @ T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_198_pinf_I7_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/1.55 => ( ord_less @ A @ T @ X7 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_199_pinf_I11_J,axiom,
% 0.30/1.55 ! [C: $tType,D: $tType] :
% 0.30/1.55 ( ( ord @ C )
% 0.30/1.55 => ! [F4: D] :
% 0.30/1.55 ? [Z3: C] :
% 0.30/1.55 ! [X7: C] :
% 0.30/1.55 ( ( ord_less @ C @ Z3 @ X7 )
% 0.30/1.55 => ( F4 = F4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_200_minf_I1_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/1.55 ( ? [Z4: A] :
% 0.30/1.55 ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ X5 @ Z4 )
% 0.30/1.55 => ( ( P @ X5 )
% 0.30/1.55 = ( P4 @ X5 ) ) )
% 0.30/1.55 => ( ? [Z4: A] :
% 0.30/1.55 ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ X5 @ Z4 )
% 0.30/1.55 => ( ( Q @ X5 )
% 0.30/1.55 = ( Q2 @ X5 ) ) )
% 0.30/1.55 => ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/1.55 => ( ( ( P @ X7 )
% 0.30/1.55 & ( Q @ X7 ) )
% 0.30/1.55 = ( ( P4 @ X7 )
% 0.30/1.55 & ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_201_minf_I2_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/1.55 ( ? [Z4: A] :
% 0.30/1.55 ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ X5 @ Z4 )
% 0.30/1.55 => ( ( P @ X5 )
% 0.30/1.55 = ( P4 @ X5 ) ) )
% 0.30/1.55 => ( ? [Z4: A] :
% 0.30/1.55 ! [X5: A] :
% 0.30/1.55 ( ( ord_less @ A @ X5 @ Z4 )
% 0.30/1.55 => ( ( Q @ X5 )
% 0.30/1.55 = ( Q2 @ X5 ) ) )
% 0.30/1.55 => ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/1.55 => ( ( ( P @ X7 )
% 0.30/1.55 | ( Q @ X7 ) )
% 0.30/1.55 = ( ( P4 @ X7 )
% 0.30/1.55 | ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_202_minf_I3_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/1.55 => ( X7 != T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_203_minf_I4_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/1.55 => ( X7 != T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_204_minf_I5_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/1.55 => ( ord_less @ A @ X7 @ T ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_205_minf_I7_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [T: A] :
% 0.30/1.55 ? [Z3: A] :
% 0.30/1.55 ! [X7: A] :
% 0.30/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/1.55 => ~ ( ord_less @ A @ T @ X7 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_206_minf_I11_J,axiom,
% 0.30/1.55 ! [C: $tType,D: $tType] :
% 0.30/1.55 ( ( ord @ C )
% 0.30/1.55 => ! [F4: D] :
% 0.30/1.55 ? [Z3: C] :
% 0.30/1.55 ! [X7: C] :
% 0.30/1.55 ( ( ord_less @ C @ X7 @ Z3 )
% 0.30/1.55 => ( F4 = F4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_207_prop__restrict,axiom,
% 0.30/1.55 ! [A: $tType,X: A,Z5: set @ A,X8: set @ A,P: A > $o] :
% 0.30/1.55 ( ( member @ A @ X @ Z5 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ Z5
% 0.30/1.55 @ ( collect @ A
% 0.30/1.55 @ ^ [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ X8 )
% 0.30/1.55 & ( P @ X4 ) ) ) )
% 0.30/1.55 => ( P @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_208_Collect__restrict,axiom,
% 0.30/1.55 ! [A: $tType,X8: set @ A,P: A > $o] :
% 0.30/1.55 ( ord_less_eq @ ( set @ A )
% 0.30/1.55 @ ( collect @ A
% 0.30/1.55 @ ^ [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ X8 )
% 0.30/1.55 & ( P @ X4 ) ) )
% 0.30/1.55 @ X8 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_209_bot__empty__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( bot_bot @ ( A > $o ) )
% 0.30/1.55 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_210_Collect__empty__eq__bot,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o] :
% 0.30/1.55 ( ( ( collect @ A @ P )
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 = ( P
% 0.30/1.55 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_211_Set_Ois__empty__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( is_empty @ A )
% 0.30/1.55 = ( ^ [A6: set @ A] :
% 0.30/1.55 ( A6
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_212_predicate1I,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.55 ( ! [X5: A] :
% 0.30/1.55 ( ( P @ X5 )
% 0.30/1.55 => ( Q @ X5 ) )
% 0.30/1.55 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_213_less__set__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.30/1.55 ( ord_less @ ( A > $o )
% 0.30/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ A6 )
% 0.30/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_214_rev__predicate1D,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 0.30/1.55 ( ( P @ X )
% 0.30/1.55 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/1.55 => ( Q @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_215_predicate1D,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/1.55 => ( ( P @ X )
% 0.30/1.55 => ( Q @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_216_psubset__trans,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
% 0.30/1.55 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_217_psubsetD,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.30/1.55 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 => ( ( member @ A @ C2 @ A4 )
% 0.30/1.55 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_218_pred__subset__eq,axiom,
% 0.30/1.55 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( A > $o )
% 0.30/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ R )
% 0.30/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ S ) )
% 0.30/1.55 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_219_subset__Collect__iff,axiom,
% 0.30/1.55 ! [A: $tType,B4: set @ A,A4: set @ A,P: A > $o] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4
% 0.30/1.55 @ ( collect @ A
% 0.30/1.55 @ ^ [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A4 )
% 0.30/1.55 & ( P @ X4 ) ) ) )
% 0.30/1.55 = ( ! [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ B4 )
% 0.30/1.55 => ( P @ X4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_220_subset__CollectI,axiom,
% 0.30/1.55 ! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.30/1.55 => ( ! [X5: A] :
% 0.30/1.55 ( ( member @ A @ X5 @ B4 )
% 0.30/1.55 => ( ( Q @ X5 )
% 0.30/1.55 => ( P @ X5 ) ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A )
% 0.30/1.55 @ ( collect @ A
% 0.30/1.55 @ ^ [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ B4 )
% 0.30/1.55 & ( Q @ X4 ) ) )
% 0.30/1.55 @ ( collect @ A
% 0.30/1.55 @ ^ [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A4 )
% 0.30/1.55 & ( P @ X4 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_221_conj__subset__def,axiom,
% 0.30/1.55 ! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4
% 0.30/1.55 @ ( collect @ A
% 0.30/1.55 @ ^ [X4: A] :
% 0.30/1.55 ( ( P @ X4 )
% 0.30/1.55 & ( Q @ X4 ) ) ) )
% 0.30/1.55 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
% 0.30/1.55 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_222_remove_Osimps_I2_J,axiom,
% 0.30/1.55 ! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
% 0.30/1.55 ( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.30/1.55 => ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( binary210054475elle_T @ A @ ( binary997842527remove @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
% 0.30/1.55 & ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.30/1.55 => ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.30/1.55 => ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary997842527remove @ A @ H @ E @ T2 ) ) ) )
% 0.30/1.55 & ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.30/1.55 => ( ( ( T1
% 0.30/1.55 = ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = T2 ) )
% 0.30/1.55 & ( ( T1
% 0.30/1.55 != ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.30/1.55 = ( product_case_prod @ ( binary1291135688e_Tree @ A ) @ A @ ( binary1291135688e_Tree @ A )
% 0.30/1.55 @ ^ [T1p: binary1291135688e_Tree @ A,R2: A] : ( binary210054475elle_T @ A @ T1p @ R2 @ T2 )
% 0.30/1.55 @ ( binary1271298290_wrmrm @ A @ H @ T1 ) ) ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_223_wrmrm__decomp,axiom,
% 0.30/1.55 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.30/1.55 ( ( T
% 0.30/1.55 != ( binary1746293266le_Tip @ A ) )
% 0.30/1.55 => ( ( binary1271298290_wrmrm @ A @ H @ T )
% 0.30/1.55 = ( product_Pair @ ( binary1291135688e_Tree @ A ) @ A @ ( binary213313527le_wrm @ A @ H @ T ) @ ( binary576689334lle_rm @ A @ H @ T ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_224_case__prod__app,axiom,
% 0.30/1.55 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.30/1.55 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.30/1.55 = ( ^ [F3: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
% 0.30/1.55 ( product_case_prod @ B @ C @ A
% 0.30/1.55 @ ^ [L: B,R2: C] : ( F3 @ L @ R2 @ Y4 )
% 0.30/1.55 @ X4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_225_case__prod__Pair__iden,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
% 0.30/1.55 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
% 0.30/1.55 = P5 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_226_bot__empty__eq2,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( bot_bot @ ( A > B > $o ) )
% 0.30/1.55 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_227_subrelI,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.30/1.55 ( ! [X5: A,Y2: B] :
% 0.30/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y2 ) @ R3 )
% 0.30/1.55 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y2 ) @ S2 ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_228_pred__subset__eq2,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.30/1.56 ( ( ord_less_eq @ ( A > B > $o )
% 0.30/1.56 @ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
% 0.30/1.56 @ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) )
% 0.30/1.56 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_229_pred__equals__eq2,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.30/1.56 ( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) )
% 0.30/1.56 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) ) )
% 0.30/1.56 = ( R = S ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_230_ssubst__Pair__rhs,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,R3: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
% 0.30/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S2 ) @ R )
% 0.30/1.56 => ( ( S3 = S2 )
% 0.30/1.56 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S3 ) @ R ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_231_case__prod__conv,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
% 0.30/1.56 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.30/1.56 = ( F @ A3 @ B3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_232_mem__case__prodI2,axiom,
% 0.30/1.56 ! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
% 0.30/1.56 ( ! [A5: A,B5: B] :
% 0.30/1.56 ( ( P5
% 0.30/1.56 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.30/1.56 => ( member @ C @ Z2 @ ( C2 @ A5 @ B5 ) ) )
% 0.30/1.56 => ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P5 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_233_predicate2I,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.30/1.56 ( ! [X5: A,Y2: B] :
% 0.30/1.56 ( ( P @ X5 @ Y2 )
% 0.30/1.56 => ( Q @ X5 @ Y2 ) )
% 0.30/1.56 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_234_case__prodI,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
% 0.30/1.56 ( ( F @ A3 @ B3 )
% 0.30/1.56 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_235_case__prodI2,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,C2: A > B > $o] :
% 0.30/1.56 ( ! [A5: A,B5: B] :
% 0.30/1.56 ( ( P5
% 0.30/1.56 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.30/1.56 => ( C2 @ A5 @ B5 ) )
% 0.30/1.56 => ( product_case_prod @ A @ B @ $o @ C2 @ P5 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_236_case__prodI2_H,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 0.30/1.56 ( ! [A5: A,B5: B] :
% 0.30/1.56 ( ( ( product_Pair @ A @ B @ A5 @ B5 )
% 0.30/1.56 = P5 )
% 0.30/1.56 => ( C2 @ A5 @ B5 @ X ) )
% 0.30/1.56 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ X ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_237_mem__case__prodI,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B3: C] :
% 0.30/1.56 ( ( member @ A @ Z2 @ ( C2 @ A3 @ B3 ) )
% 0.30/1.56 => ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_238_Collect__case__prod__mono,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,A4: A > B > $o,B4: A > B > $o] :
% 0.30/1.56 ( ( ord_less_eq @ ( A > B > $o ) @ A4 @ B4 )
% 0.30/1.56 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_239_predicate2D__conj,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
% 0.30/1.56 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/1.56 & R )
% 0.30/1.56 => ( R
% 0.30/1.56 & ( ( P @ X @ Y )
% 0.30/1.56 => ( Q @ X @ Y ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_240_eq__subset,axiom,
% 0.30/1.56 ! [A: $tType,P: A > A > $o] :
% 0.30/1.56 ( ord_less_eq @ ( A > A > $o )
% 0.30/1.56 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.30/1.56 @ ^ [A2: A,B2: A] :
% 0.30/1.56 ( ( P @ A2 @ B2 )
% 0.30/1.56 | ( A2 = B2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_241_predicate2D,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
% 0.30/1.56 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/1.56 => ( ( P @ X @ Y )
% 0.30/1.56 => ( Q @ X @ Y ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_242_rev__predicate2D,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
% 0.30/1.56 ( ( P @ X @ Y )
% 0.30/1.56 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/1.56 => ( Q @ X @ Y ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_243_case__prodD,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
% 0.30/1.56 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.30/1.56 => ( F @ A3 @ B3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_244_case__prodE,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,C2: A > B > $o,P5: product_prod @ A @ B] :
% 0.30/1.56 ( ( product_case_prod @ A @ B @ $o @ C2 @ P5 )
% 0.30/1.56 => ~ ! [X5: A,Y2: B] :
% 0.30/1.56 ( ( P5
% 0.30/1.56 = ( product_Pair @ A @ B @ X5 @ Y2 ) )
% 0.30/1.56 => ~ ( C2 @ X5 @ Y2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_245_case__prodD_H,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C2: C] :
% 0.30/1.56 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C2 )
% 0.30/1.56 => ( R @ A3 @ B3 @ C2 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_246_case__prodE_H,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P5: product_prod @ A @ B,Z2: C] :
% 0.30/1.56 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ Z2 )
% 0.30/1.56 => ~ ! [X5: A,Y2: B] :
% 0.30/1.56 ( ( P5
% 0.30/1.56 = ( product_Pair @ A @ B @ X5 @ Y2 ) )
% 0.30/1.56 => ~ ( C2 @ X5 @ Y2 @ Z2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_247_prod_Ocase__distrib,axiom,
% 0.30/1.56 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.30/1.56 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.30/1.56 = ( product_case_prod @ A @ B @ D
% 0.30/1.56 @ ^ [X12: A,X24: B] : ( H @ ( F @ X12 @ X24 ) )
% 0.30/1.56 @ Prod ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_248_case__prodE2,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
% 0.30/1.56 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
% 0.30/1.56 => ~ ! [X5: B,Y2: C] :
% 0.30/1.56 ( ( Z2
% 0.30/1.56 = ( product_Pair @ B @ C @ X5 @ Y2 ) )
% 0.30/1.56 => ~ ( Q @ ( P @ X5 @ Y2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_249_case__prod__eta,axiom,
% 0.30/1.56 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.30/1.56 ( ( product_case_prod @ A @ B @ C
% 0.30/1.56 @ ^ [X4: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
% 0.30/1.56 = F ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_250_cond__case__prod__eta,axiom,
% 0.30/1.56 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 0.30/1.56 ( ! [X5: A,Y2: B] :
% 0.30/1.56 ( ( F @ X5 @ Y2 )
% 0.30/1.56 = ( G @ ( product_Pair @ A @ B @ X5 @ Y2 ) ) )
% 0.30/1.56 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.30/1.56 = G ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_251_less__by__empty,axiom,
% 0.30/1.56 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
% 0.30/1.56 ( ( A4
% 0.30/1.56 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/1.56 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B4 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_252_relChain__def,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType] :
% 0.30/1.56 ( ( ord @ B )
% 0.30/1.56 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.30/1.56 = ( ^ [R2: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.30/1.56 ! [I: A,J: A] :
% 0.30/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
% 0.30/1.56 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_253_split__part,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 0.30/1.56 ( ( product_case_prod @ A @ B @ $o
% 0.30/1.56 @ ^ [A2: A,B2: B] :
% 0.30/1.56 ( P
% 0.30/1.56 & ( Q @ A2 @ B2 ) ) )
% 0.30/1.56 = ( ^ [Ab: product_prod @ A @ B] :
% 0.30/1.56 ( P
% 0.30/1.56 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_254_prod_Odisc__eq__case,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/1.56 ( product_case_prod @ A @ B @ $o
% 0.30/1.56 @ ^ [Uu: A,Uv: B] : $true
% 0.30/1.56 @ Prod ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_255_inv__image__def,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType] :
% 0.30/1.56 ( ( inv_image @ B @ A )
% 0.30/1.56 = ( ^ [R2: set @ ( product_prod @ B @ B ),F3: A > B] :
% 0.30/1.56 ( collect @ ( product_prod @ A @ A )
% 0.30/1.56 @ ( product_case_prod @ A @ A @ $o
% 0.30/1.56 @ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.30/1.56 ! [A8: $tType,A9: $tType] :
% 0.30/1.56 ( ( order_bot @ A9 )
% 0.30/1.56 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/1.56 ! [A8: $tType,A9: $tType] :
% 0.30/1.56 ( ( preorder @ A9 )
% 0.30/1.56 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/1.56 ! [A8: $tType,A9: $tType] :
% 0.30/1.56 ( ( order @ A9 )
% 0.30/1.56 => ( order @ ( A8 > A9 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/1.56 ! [A8: $tType,A9: $tType] :
% 0.30/1.56 ( ( ord @ A9 )
% 0.30/1.56 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/1.56 ! [A8: $tType,A9: $tType] :
% 0.30/1.56 ( ( bot @ A9 )
% 0.30/1.56 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.30/1.56 condit1037483654norder @ int ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.30/1.56 preorder @ int ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.30/1.56 linorder @ int ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.30/1.56 no_top @ int ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.30/1.56 no_bot @ int ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.30/1.56 order @ int ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.30/1.56 ord @ int ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Set_Oset___Orderings_Oorder__bot_4,axiom,
% 0.30/1.56 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.30/1.56 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.30/1.56 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.30/1.56 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
% 0.30/1.56 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_HOL_Obool___Orderings_Oorder__bot_9,axiom,
% 0.30/1.56 order_bot @ $o ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.30/1.56 preorder @ $o ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
% 0.30/1.56 linorder @ $o ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 0.30/1.56 order @ $o ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 0.30/1.56 ord @ $o ).
% 0.30/1.56
% 0.30/1.56 thf(tcon_HOL_Obool___Orderings_Obot_14,axiom,
% 0.30/1.56 bot @ $o ).
% 0.30/1.56
% 0.30/1.56 thf(conj_0,conjecture,
% 0.30/1.56 binary1610619414edTree @ a @ h @ t1 ).
% 0.30/1.56 ------------------------
% 0.30/1.58 % (29167)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/2Mi)
% 0.30/1.58 % (29169)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_28889 for (2999ds/275Mi)
% 0.30/1.58 % (29165)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_28889 for (2999ds/4Mi)
% 0.30/1.58 % (29167)Instruction limit reached!
% 0.30/1.58 % (29167)------------------------------
% 0.30/1.58 % (29167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.58 % (29167)Termination reason: Unknown
% 0.30/1.58 % (29167)Termination phase: shuffling
% 0.30/1.58
% 0.30/1.58 % (29167)Memory used [KB]: 1407
% 0.30/1.58 % (29167)Time elapsed: 0.003 s
% 0.30/1.58 % (29167)Instructions burned: 3 (million)
% 0.30/1.58 % (29167)------------------------------
% 0.30/1.58 % (29167)------------------------------
% 0.30/1.58 % (29165)Instruction limit reached!
% 0.30/1.58 % (29165)------------------------------
% 0.30/1.58 % (29165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.58 % (29165)Termination reason: Unknown
% 0.30/1.58 % (29165)Termination phase: shuffling
% 0.30/1.58
% 0.30/1.58 % (29165)Memory used [KB]: 1407
% 0.30/1.58 % (29165)Time elapsed: 0.003 s
% 0.30/1.58 % (29165)Instructions burned: 4 (million)
% 0.30/1.58 % (29165)------------------------------
% 0.30/1.58 % (29165)------------------------------
% 0.30/1.59 % (29164)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/183Mi)
% 0.30/1.59 % (29166)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_28889 for (2999ds/27Mi)
% 0.30/1.59 % (29168)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_28889 for (2999ds/2Mi)
% 0.30/1.59 % (29170)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/18Mi)
% 0.30/1.59 % (29168)Instruction limit reached!
% 0.30/1.59 % (29168)------------------------------
% 0.30/1.59 % (29168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.59 % (29168)Termination reason: Unknown
% 0.30/1.59 % (29168)Termination phase: shuffling
% 0.30/1.59
% 0.30/1.59 % (29168)Memory used [KB]: 1407
% 0.30/1.59 % (29168)Time elapsed: 0.004 s
% 0.30/1.59 % (29168)Instructions burned: 2 (million)
% 0.30/1.59 % (29168)------------------------------
% 0.30/1.59 % (29168)------------------------------
% 0.30/1.59 % (29171)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/3Mi)
% 0.30/1.59 % (29172)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_28889 for (2999ds/37Mi)
% 0.30/1.59 % (29171)Instruction limit reached!
% 0.30/1.59 % (29171)------------------------------
% 0.30/1.59 % (29171)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.59 % (29171)Termination reason: Unknown
% 0.30/1.59 % (29171)Termination phase: shuffling
% 0.30/1.59
% 0.30/1.59 % (29171)Memory used [KB]: 1407
% 0.30/1.59 % (29171)Time elapsed: 0.003 s
% 0.30/1.59 % (29171)Instructions burned: 5 (million)
% 0.30/1.59 % (29171)------------------------------
% 0.30/1.59 % (29171)------------------------------
% 2.90/1.60 % (29170)Instruction limit reached!
% 2.90/1.60 % (29170)------------------------------
% 2.90/1.60 % (29170)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.60 % (29170)Termination reason: Unknown
% 2.90/1.60 % (29170)Termination phase: shuffling
% 2.90/1.60
% 2.90/1.60 % (29170)Memory used [KB]: 1791
% 2.90/1.60 % (29170)Time elapsed: 0.015 s
% 2.90/1.60 % (29170)Instructions burned: 19 (million)
% 2.90/1.60 % (29170)------------------------------
% 2.90/1.60 % (29170)------------------------------
% 2.90/1.60 % (29174)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_28889 for (2999ds/3Mi)
% 2.90/1.60 % (29174)Instruction limit reached!
% 2.90/1.60 % (29174)------------------------------
% 2.90/1.60 % (29174)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.60 % (29174)Termination reason: Unknown
% 2.90/1.60 % (29174)Termination phase: shuffling
% 2.90/1.60
% 2.90/1.60 % (29174)Memory used [KB]: 1407
% 2.90/1.60 % (29174)Time elapsed: 0.002 s
% 2.90/1.60 % (29174)Instructions burned: 3 (million)
% 2.90/1.60 % (29174)------------------------------
% 2.90/1.60 % (29174)------------------------------
% 2.90/1.61 % (29166)Instruction limit reached!
% 2.90/1.61 % (29166)------------------------------
% 2.90/1.61 % (29166)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (29166)Termination reason: Unknown
% 2.90/1.61 % (29166)Termination phase: Property scanning
% 2.90/1.61
% 2.90/1.61 % (29166)Memory used [KB]: 1791
% 2.90/1.61 % (29166)Time elapsed: 0.019 s
% 2.90/1.61 % (29166)Instructions burned: 28 (million)
% 2.90/1.61 % (29166)------------------------------
% 2.90/1.61 % (29166)------------------------------
% 2.90/1.61 % (29172)Instruction limit reached!
% 2.90/1.61 % (29172)------------------------------
% 2.90/1.61 % (29172)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (29172)Termination reason: Unknown
% 2.90/1.61 % (29172)Termination phase: shuffling
% 2.90/1.61
% 2.90/1.61 % (29172)Memory used [KB]: 2174
% 2.90/1.61 % (29172)Time elapsed: 0.013 s
% 2.90/1.61 % (29172)Instructions burned: 39 (million)
% 2.90/1.61 % (29172)------------------------------
% 2.90/1.61 % (29172)------------------------------
% 2.90/1.61 % (29169)Refutation not found, incomplete strategy
% 2.90/1.61 % (29169)------------------------------
% 2.90/1.61 % (29169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (29169)Termination reason: Refutation not found, incomplete strategy
% 2.90/1.61
% 2.90/1.61
% 2.90/1.61 % (29169)Memory used [KB]: 6780
% 2.90/1.61 % (29169)Time elapsed: 0.025 s
% 2.90/1.61 % (29169)Instructions burned: 72 (million)
% 2.90/1.61 % (29169)------------------------------
% 2.90/1.61 % (29169)------------------------------
% 2.90/1.61 % (29173)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_28889 for (2999ds/15Mi)
% 2.90/1.61 % (29176)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/7Mi)
% 2.90/1.62 % (29178)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_28889 for (2999ds/3Mi)
% 2.90/1.62 % (29179)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_28889 for (2999ds/3Mi)
% 2.90/1.62 % (29176)Instruction limit reached!
% 2.90/1.62 % (29176)------------------------------
% 2.90/1.62 % (29176)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.62 % (29176)Termination reason: Unknown
% 2.90/1.62 % (29176)Termination phase: shuffling
% 2.90/1.62
% 2.90/1.62 % (29176)Memory used [KB]: 1535
% 2.90/1.62 % (29176)Time elapsed: 0.004 s
% 2.90/1.62 % (29176)Instructions burned: 8 (million)
% 2.90/1.62 % (29176)------------------------------
% 2.90/1.62 % (29176)------------------------------
% 2.90/1.62 % (29178)Instruction limit reached!
% 2.90/1.62 % (29178)------------------------------
% 2.90/1.62 % (29178)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.62 % (29178)Termination reason: Unknown
% 2.90/1.62 % (29178)Termination phase: shuffling
% 2.90/1.62
% 2.90/1.62 % (29178)Memory used [KB]: 1407
% 2.90/1.62 % (29178)Time elapsed: 0.003 s
% 2.90/1.62 % (29178)Instructions burned: 3 (million)
% 2.90/1.62 % (29178)------------------------------
% 2.90/1.62 % (29178)------------------------------
% 2.90/1.62 % (29179)Instruction limit reached!
% 2.90/1.62 % (29179)------------------------------
% 2.90/1.62 % (29179)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.62 % (29179)Termination reason: Unknown
% 2.90/1.62 % (29179)Termination phase: shuffling
% 2.90/1.62
% 2.90/1.62 % (29179)Memory used [KB]: 1407
% 2.90/1.62 % (29179)Time elapsed: 0.003 s
% 2.90/1.62 % (29179)Instructions burned: 3 (million)
% 2.90/1.62 % (29179)------------------------------
% 2.90/1.62 % (29179)------------------------------
% 2.90/1.62 % (29173)Instruction limit reached!
% 2.90/1.62 % (29173)------------------------------
% 2.90/1.62 % (29173)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.62 % (29173)Termination reason: Unknown
% 2.90/1.62 % (29173)Termination phase: shuffling
% 2.90/1.62
% 2.90/1.62 % (29173)Memory used [KB]: 1791
% 2.90/1.62 % (29173)Time elapsed: 0.012 s
% 2.90/1.62 % (29173)Instructions burned: 16 (million)
% 2.90/1.62 % (29173)------------------------------
% 2.90/1.62 % (29173)------------------------------
% 2.90/1.62 % (29177)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/16Mi)
% 2.90/1.62 % (29175)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_28889 for (2999ds/1041Mi)
% 2.90/1.63 % (29177)Instruction limit reached!
% 2.90/1.63 % (29177)------------------------------
% 2.90/1.63 % (29177)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.63 % (29177)Termination reason: Unknown
% 2.90/1.63 % (29177)Termination phase: shuffling
% 2.90/1.63
% 2.90/1.63 % (29177)Memory used [KB]: 1791
% 2.90/1.63 % (29177)Time elapsed: 0.009 s
% 2.90/1.63 % (29177)Instructions burned: 18 (million)
% 2.90/1.63 % (29177)------------------------------
% 2.90/1.63 % (29177)------------------------------
% 2.90/1.63 % (29180)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/7Mi)
% 2.90/1.63 % (29181)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/3Mi)
% 2.90/1.63 % (29182)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/4Mi)
% 2.90/1.63 % (29181)Instruction limit reached!
% 2.90/1.63 % (29181)------------------------------
% 2.90/1.63 % (29181)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.63 % (29181)Termination reason: Unknown
% 2.90/1.63 % (29181)Termination phase: shuffling
% 2.90/1.63
% 2.90/1.63 % (29181)Memory used [KB]: 1407
% 2.90/1.63 % (29181)Time elapsed: 0.002 s
% 2.90/1.63 % (29181)Instructions burned: 4 (million)
% 2.90/1.63 % (29181)------------------------------
% 2.90/1.63 % (29181)------------------------------
% 2.90/1.63 % (29180)Instruction limit reached!
% 2.90/1.63 % (29180)------------------------------
% 2.90/1.63 % (29180)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.63 % (29180)Termination reason: Unknown
% 2.90/1.63 % (29180)Termination phase: shuffling
% 2.90/1.63
% 2.90/1.63 % (29180)Memory used [KB]: 1535
% 2.90/1.63 % (29180)Time elapsed: 0.004 s
% 2.90/1.63 % (29180)Instructions burned: 8 (million)
% 2.90/1.63 % (29180)------------------------------
% 2.90/1.63 % (29180)------------------------------
% 2.90/1.63 % (29182)Instruction limit reached!
% 2.90/1.63 % (29182)------------------------------
% 2.90/1.63 % (29182)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.63 % (29182)Termination reason: Unknown
% 2.90/1.63 % (29182)Termination phase: shuffling
% 2.90/1.63
% 2.90/1.63 % (29182)Memory used [KB]: 1535
% 2.90/1.63 % (29182)Time elapsed: 0.003 s
% 2.90/1.63 % (29182)Instructions burned: 6 (million)
% 2.90/1.63 % (29182)------------------------------
% 2.90/1.63 % (29182)------------------------------
% 2.90/1.63 % (29183)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/18Mi)
% 2.90/1.64 % (29184)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/710Mi)
% 2.90/1.64 % (29186)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_28889 for (2999ds/902Mi)
% 2.90/1.64 % (29185)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/6Mi)
% 2.90/1.64 % (29187)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_28889 for (2999ds/21Mi)
% 2.90/1.64 % (29185)Instruction limit reached!
% 2.90/1.64 % (29185)------------------------------
% 2.90/1.64 % (29185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.64 % (29185)Termination reason: Unknown
% 2.90/1.64 % (29185)Termination phase: shuffling
% 2.90/1.64
% 2.90/1.64 % (29185)Memory used [KB]: 1535
% 2.90/1.64 % (29185)Time elapsed: 0.004 s
% 2.90/1.64 % (29185)Instructions burned: 6 (million)
% 2.90/1.64 % (29185)------------------------------
% 2.90/1.64 % (29185)------------------------------
% 2.90/1.64 % (29183)Instruction limit reached!
% 2.90/1.64 % (29183)------------------------------
% 2.90/1.64 % (29183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.64 % (29183)Termination reason: Unknown
% 2.90/1.64 % (29183)Termination phase: shuffling
% 2.90/1.64
% 2.90/1.64 % (29183)Memory used [KB]: 1791
% 2.90/1.64 % (29183)Time elapsed: 0.011 s
% 2.90/1.64 % (29183)Instructions burned: 18 (million)
% 2.90/1.64 % (29183)------------------------------
% 2.90/1.64 % (29183)------------------------------
% 2.90/1.65 % (29187)Instruction limit reached!
% 2.90/1.65 % (29187)------------------------------
% 2.90/1.65 % (29187)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.65 % (29187)Termination reason: Unknown
% 2.90/1.65 % (29187)Termination phase: shuffling
% 2.90/1.65
% 2.90/1.65 % (29187)Memory used [KB]: 1918
% 2.90/1.65 % (29187)Time elapsed: 0.010 s
% 2.90/1.65 % (29187)Instructions burned: 23 (million)
% 2.90/1.65 % (29187)------------------------------
% 2.90/1.65 % (29187)------------------------------
% 2.90/1.65 % (29188)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/5Mi)
% 2.90/1.65 % (29188)Instruction limit reached!
% 2.90/1.65 % (29188)------------------------------
% 2.90/1.65 % (29188)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.65 % (29188)Termination reason: Unknown
% 2.90/1.65 % (29188)Termination phase: shuffling
% 2.90/1.65
% 2.90/1.65 % (29188)Memory used [KB]: 1535
% 2.90/1.65 % (29188)Time elapsed: 0.004 s
% 2.90/1.65 % (29188)Instructions burned: 7 (million)
% 2.90/1.65 % (29188)------------------------------
% 2.90/1.65 % (29188)------------------------------
% 2.90/1.65 % (29189)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_28889 for (2999ds/6Mi)
% 2.90/1.65 % (29189)Instruction limit reached!
% 2.90/1.65 % (29189)------------------------------
% 2.90/1.65 % (29189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.65 % (29189)Termination reason: Unknown
% 2.90/1.65 % (29189)Termination phase: shuffling
% 2.90/1.65
% 2.90/1.65 % (29189)Memory used [KB]: 1407
% 2.90/1.65 % (29189)Time elapsed: 0.003 s
% 2.90/1.65 % (29189)Instructions burned: 6 (million)
% 2.90/1.65 % (29189)------------------------------
% 2.90/1.65 % (29189)------------------------------
% 2.90/1.66 % (29190)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_28889 for (2999ds/377Mi)
% 2.90/1.66 % (29191)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_28889 for (2999ds/779Mi)
% 2.90/1.66 % (29192)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_28889 for (2999ds/19Mi)
% 2.90/1.67 % (29192)Instruction limit reached!
% 2.90/1.67 % (29192)------------------------------
% 2.90/1.67 % (29192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (29192)Termination reason: Unknown
% 2.90/1.67 % (29192)Termination phase: shuffling
% 2.90/1.67
% 2.90/1.67 % (29192)Memory used [KB]: 1791
% 2.90/1.67 % (29192)Time elapsed: 0.007 s
% 2.90/1.67 % (29192)Instructions burned: 19 (million)
% 2.90/1.67 % (29192)------------------------------
% 2.90/1.67 % (29192)------------------------------
% 2.90/1.67 % (29186)Refutation not found, incomplete strategy
% 2.90/1.67 % (29186)------------------------------
% 2.90/1.67 % (29186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (29186)Termination reason: Refutation not found, incomplete strategy
% 2.90/1.67
% 2.90/1.67
% 2.90/1.67 % (29186)Memory used [KB]: 7164
% 2.90/1.67 % (29186)Time elapsed: 0.036 s
% 2.90/1.67 % (29186)Instructions burned: 115 (million)
% 2.90/1.67 % (29186)------------------------------
% 2.90/1.67 % (29186)------------------------------
% 2.90/1.68 % (29193)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_28889 for (2998ds/879Mi)
% 2.90/1.68 % (29194)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_28889 for (2998ds/17Mi)
% 2.90/1.69 % (29194)Instruction limit reached!
% 2.90/1.69 % (29194)------------------------------
% 2.90/1.69 % (29194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (29194)Termination reason: Unknown
% 2.90/1.69 % (29194)Termination phase: shuffling
% 2.90/1.69
% 2.90/1.69 % (29194)Memory used [KB]: 1791
% 2.90/1.69 % (29194)Time elapsed: 0.007 s
% 2.90/1.69 % (29194)Instructions burned: 17 (million)
% 2.90/1.69 % (29194)------------------------------
% 2.90/1.69 % (29194)------------------------------
% 2.90/1.69 % (29164)Instruction limit reached!
% 2.90/1.69 % (29164)------------------------------
% 2.90/1.69 % (29164)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (29164)Termination reason: Unknown
% 2.90/1.69 % (29164)Termination phase: Saturation
% 2.90/1.69
% 2.90/1.69 % (29164)Memory used [KB]: 7036
% 2.90/1.69 % (29164)Time elapsed: 0.105 s
% 2.90/1.69 % (29164)Instructions burned: 183 (million)
% 2.90/1.69 % (29164)------------------------------
% 2.90/1.69 % (29164)------------------------------
% 2.90/1.70 % (29195)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_28889 for (2998ds/3Mi)
% 2.90/1.70 % (29195)Instruction limit reached!
% 2.90/1.70 % (29195)------------------------------
% 2.90/1.70 % (29195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.70 % (29195)Termination reason: Unknown
% 2.90/1.70 % (29195)Termination phase: shuffling
% 2.90/1.70
% 2.90/1.70 % (29195)Memory used [KB]: 1407
% 2.90/1.70 % (29195)Time elapsed: 0.002 s
% 2.90/1.70 % (29195)Instructions burned: 3 (million)
% 2.90/1.70 % (29195)------------------------------
% 2.90/1.70 % (29195)------------------------------
% 2.90/1.70 % (29184)First to succeed.
% 2.90/1.70 % (29184)Refutation found. Thanks to Tanya!
% 2.90/1.70 % SZS status Theorem for DTF2THF_28889
% 2.90/1.70 % SZS output start Proof for DTF2THF_28889
% 2.90/1.70 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.90/1.70 thf(type_def_6, type, set: ($tType) > $tType).
% 2.90/1.70 thf(type_def_8, type, binary1291135688e_Tree: ($tType) > $tType).
% 2.90/1.70 thf(type_def_9, type, int: $tType).
% 2.90/1.70 thf(type_def_10, type, a: $tType).
% 2.90/1.70 thf(func_def_0, type, binary1291135688e_Tree: ($tType) > $tType).
% 2.90/1.70 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 2.90/1.70 thf(func_def_2, type, set: ($tType) > $tType).
% 2.90/1.70 thf(func_def_3, type, int: $tType).
% 2.90/1.70 thf(func_def_4, type, a: $tType).
% 2.90/1.70 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_6, type, bot: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_7, type, ord: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_8, type, order: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_9, type, no_bot: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_10, type, no_top: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_13, type, order_bot: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_14, type, wellorder: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_15, type, dense_order: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_16, type, dense_linorder: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_17, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_18, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.90/1.70 thf(func_def_19, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 2.90/1.70 thf(func_def_20, type, binary210054475elle_T: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.90/1.70 thf(func_def_21, type, binary1746293266le_Tip: !>[X0: $tType]:(binary1291135688e_Tree @ X0)).
% 2.90/1.70 thf(func_def_22, type, binary536355927e_Tree: !>[X0: $tType, X1: $tType]:(X0 > (binary1291135688e_Tree @ X1 > X1 > binary1291135688e_Tree @ X1 > X0) > binary1291135688e_Tree @ X1 > X0)).
% 2.90/1.70 thf(func_def_23, type, binary1929596613c_Tree: !>[X0: $tType, X1: $tType]:(X0 > (binary1291135688e_Tree @ X1 > X1 > binary1291135688e_Tree @ X1 > X0 > X0 > X0) > binary1291135688e_Tree @ X1 > X0)).
% 2.90/1.70 thf(func_def_24, type, binary1830089824insert: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.90/1.70 thf(func_def_25, type, binary827270440e_memb: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > $o)).
% 2.90/1.70 thf(func_def_26, type, binary997842527remove: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.90/1.70 thf(func_def_27, type, binary576689334lle_rm: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > X0)).
% 2.90/1.70 thf(func_def_28, type, binary1653327646_setOf: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > set @ X0)).
% 2.90/1.70 thf(func_def_29, type, binary1610619414edTree: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > $o)).
% 2.90/1.70 thf(func_def_30, type, binary231205461t_pred: !>[X0: $tType]:((X0 > int) > X0 > X0 > binary1291135688e_Tree @ X0 > $o)).
% 2.90/1.70 thf(func_def_31, type, binary213313527le_wrm: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.90/1.70 thf(func_def_32, type, binary1271298290_wrmrm: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > product_prod @ binary1291135688e_Tree @ X0 @ X0)).
% 2.90/1.70 thf(func_def_33, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.90/1.70 thf(func_def_34, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.90/1.70 thf(func_def_35, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.90/1.70 thf(func_def_36, type, ordering_top: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0 > $o)).
% 2.90/1.70 thf(func_def_37, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.90/1.70 thf(func_def_38, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.90/1.70 thf(func_def_39, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 2.90/1.70 thf(func_def_40, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.90/1.70 thf(func_def_41, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.90/1.70 thf(func_def_42, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.90/1.70 thf(func_def_43, type, h: a > int).
% 2.90/1.70 thf(func_def_44, type, t1: binary1291135688e_Tree @ a).
% 2.90/1.70 thf(func_def_45, type, t2: binary1291135688e_Tree @ a).
% 2.90/1.70 thf(func_def_46, type, x: a).
% 2.90/1.70 thf(func_def_63, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.90/1.70 thf(func_def_65, type, ph2: !>[X0: $tType]:(X0)).
% 2.90/1.70 thf(f1185,plain,(
% 2.90/1.70 $false),
% 2.90/1.70 inference(avatar_sat_refutation,[status(thm)],[f840,f1184])).
% 2.90/1.70 thf(f1184,plain,(
% 2.90/1.70 spl1_3),
% 2.90/1.70 inference(avatar_contradiction_clause,[status(thm)],[f1183])).
% 2.90/1.70 thf(f1183,plain,(
% 2.90/1.70 $false | spl1_3),
% 2.90/1.70 inference(subsumption_resolution,[status(thm)],[f1171,f838])).
% 2.90/1.70 thf(f838,plain,(
% 2.90/1.70 ((binary1610619414edTree @ a @ h @ t1) != $true) | spl1_3),
% 2.90/1.70 inference(avatar_component_clause,[status(thm)],[f836])).
% 2.90/1.70 thf(f836,definition,(
% 2.90/1.70 spl1_3 <=> ((binary1610619414edTree @ a @ h @ t1) = $true)),
% 2.90/1.70 introduced(definition,[new_symbols(naming,[spl1_3])],[avatar_definition])).
% 2.90/1.70 thf(f1171,plain,(
% 2.90/1.70 ((binary1610619414edTree @ a @ h @ t1) = $true)),
% 2.90/1.70 inference(trivial_inequality_removal,[status(thm)],[f1166])).
% 2.90/1.70 thf(f1166,plain,(
% 2.90/1.70 ($true != $true) | ((binary1610619414edTree @ a @ h @ t1) = $true)),
% 2.90/1.70 inference(superposition,[status(thm)],[f823,f820])).
% 2.90/1.70 thf(f820,plain,(
% 2.90/1.70 ((binary1610619414edTree @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2)) = $true)),
% 2.90/1.70 inference(cnf_transformation,[status(thm)],[f766])).
% 2.90/1.70 thf(f766,plain,(
% 2.90/1.70 ((binary1610619414edTree @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2)) = $true)),
% 2.90/1.70 inference(fool_elimination,[status(thm)],[f765])).
% 2.90/1.70 thf(f765,plain,(
% 2.90/1.70 (binary1610619414edTree @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2))),
% 2.90/1.70 inference(rectify,[status(thm)],[f2])).
% 2.90/1.70 thf(f2,axiom,(
% 2.90/1.70 (binary1610619414edTree @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2))),
% 2.90/1.70 file('/export/starexec/sandbox/tmp/tmp.yadyxb1T5r/DTF2THF_28889.p',fact_1_s)).
% 2.90/1.70 thf(f823,plain,(
% 2.90/1.70 ( ! [X0 : $tType,X2 : binary1291135688e_Tree @ X0,X3 : X0,X1 : X0 > int,X4 : binary1291135688e_Tree @ X0] : (((binary1610619414edTree @ X0 @ X1 @ (binary210054475elle_T @ X0 @ X2 @ X3 @ X4)) != $true) | ((binary1610619414edTree @ X0 @ X1 @ X2) = $true)) )),
% 2.90/1.70 inference(cnf_transformation,[status(thm)],[f809])).
% 2.90/1.70 thf(f809,plain,(
% 2.90/1.70 ! [X0 : $tType,X1 : X0 > int,X2 : binary1291135688e_Tree @ X0,X3 : X0,X4 : binary1291135688e_Tree @ X0] : (((binary1610619414edTree @ X0 @ X1 @ (binary210054475elle_T @ X0 @ X2 @ X3 @ X4)) != $true) | ((binary1610619414edTree @ X0 @ X1 @ X2) = $true))),
% 2.90/1.70 inference(ennf_transformation,[status(thm)],[f534])).
% 2.90/1.70 thf(f534,plain,(
% 2.90/1.70 ! [X0 : $tType,X1 : X0 > int,X2 : binary1291135688e_Tree @ X0,X3 : X0,X4 : binary1291135688e_Tree @ X0] : (((binary1610619414edTree @ X0 @ X1 @ (binary210054475elle_T @ X0 @ X2 @ X3 @ X4)) = $true) => ((binary1610619414edTree @ X0 @ X1 @ X2) = $true))),
% 2.90/1.70 inference(fool_elimination,[status(thm)],[f533])).
% 2.90/1.70 thf(f533,plain,(
% 2.90/1.70 ! [X0 : $tType,X1 : X0 > int,X2 : binary1291135688e_Tree @ X0,X3 : X0,X4 : binary1291135688e_Tree @ X0] : ((binary1610619414edTree @ X0 @ X1 @ (binary210054475elle_T @ X0 @ X2 @ X3 @ X4)) => (binary1610619414edTree @ X0 @ X1 @ X2))),
% 2.90/1.70 inference(rectify,[status(thm)],[f4])).
% 2.90/1.70 thf(f4,axiom,(
% 2.90/1.70 ! [X0 : $tType,X3 : X0 > int,X4 : binary1291135688e_Tree @ X0,X5 : X0,X6 : binary1291135688e_Tree @ X0] : ((binary1610619414edTree @ X0 @ X3 @ (binary210054475elle_T @ X0 @ X4 @ X5 @ X6)) => (binary1610619414edTree @ X0 @ X3 @ X4))),
% 2.90/1.70 file('/export/starexec/sandbox/tmp/tmp.yadyxb1T5r/DTF2THF_28889.p',fact_3_sortLemmaL)).
% 2.90/1.70 thf(f840,plain,(
% 2.90/1.70 ~spl1_3),
% 2.90/1.70 inference(avatar_split_clause,[status(thm)],[f817,f836])).
% 2.90/1.70 thf(f817,plain,(
% 2.90/1.70 ((binary1610619414edTree @ a @ h @ t1) != $true)),
% 2.90/1.70 inference(cnf_transformation,[status(thm)],[f801])).
% 2.90/1.70 thf(f801,plain,(
% 2.90/1.70 ((binary1610619414edTree @ a @ h @ t1) != $true)),
% 2.90/1.70 inference(flattening,[status(thm)],[f800])).
% 2.90/1.70 thf(f800,plain,(
% 2.90/1.70 ~((binary1610619414edTree @ a @ h @ t1) = $true)),
% 2.90/1.70 inference(fool_elimination,[status(thm)],[f799])).
% 2.90/1.70 thf(f799,plain,(
% 2.90/1.70 ~(binary1610619414edTree @ a @ h @ t1)),
% 2.90/1.70 inference(rectify,[status(thm)],[f281])).
% 2.90/1.70 thf(f281,negated_conjecture,(
% 2.90/1.70 ~(binary1610619414edTree @ a @ h @ t1)),
% 2.90/1.70 inference(negated_conjecture,[status(cth)],[f280])).
% 2.90/1.70 thf(f280,conjecture,(
% 2.90/1.70 (binary1610619414edTree @ a @ h @ t1)),
% 2.90/1.70 file('/export/starexec/sandbox/tmp/tmp.yadyxb1T5r/DTF2THF_28889.p',conj_0)).
% 2.90/1.70 % SZS output end Proof for DTF2THF_28889
% 2.90/1.70 % (29184)------------------------------
% 2.90/1.70 % (29184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.70 % (29184)Termination reason: Refutation
% 2.90/1.70
% 2.90/1.70 % (29184)Memory used [KB]: 7164
% 2.90/1.70 % (29184)Time elapsed: 0.069 s
% 2.90/1.70 % (29184)Instructions burned: 205 (million)
% 2.90/1.70 % (29184)------------------------------
% 2.90/1.70 % (29184)------------------------------
% 2.90/1.70 % (29163)Success in time 0.136 s
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