%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP034^2 : 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 : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:01 AM UTC 2026
% Result : Theorem 2.73s 1.61s
% Output : Refutation 2.73s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP034^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n011.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:24:03 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.46 ---- Original DTF file ---
% 0.32/0.46 thf(spec,logic,$$dhol).
% 0.32/0.46 %------------------------------------------------------------------------------
% 0.32/0.46 % File : ITP034^2 : TPTP v9.2.1. Released v7.5.0.
% 0.32/0.46 % Domain : Interactive Theorem Proving
% 0.32/0.46 % Problem : Sledgehammer BinaryTree problem prob_562__3255854_1
% 0.32/0.46 % Version : Especial.
% 0.32/0.46 % English :
% 0.32/0.46
% 0.32/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.32/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.32/0.46 % Source : [Des21]
% 0.32/0.46 % Names : BinaryTree/prob_562__3255854_1 [Des21]
% 0.32/0.46
% 0.32/0.46 % Status : Theorem
% 0.32/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.32/0.46 % Syntax : Number of formulae : 329 ( 113 unt; 40 typ; 0 def)
% 0.32/0.46 % Number of atoms : 801 ( 273 equ; 0 cnn)
% 0.32/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.32/0.46 % Number of connectives : 3400 ( 77 ~; 20 |; 71 &;2879 @)
% 0.32/0.46 % ( 0 <=>; 353 =>; 0 <=; 0 <~>)
% 0.32/0.46 % Maximal formula depth : 21 ( 7 avg)
% 0.32/0.46 % Number of types : 3 ( 2 usr)
% 0.32/0.47 % Number of type conns : 142 ( 142 >; 0 *; 0 +; 0 <<)
% 0.32/0.47 % Number of symbols : 40 ( 38 usr; 5 con; 0-5 aty)
% 0.32/0.47 % Number of variables : 1009 ( 88 ^; 856 !; 34 ?;1009 :)
% 0.32/0.47 % ( 31 !>; 0 ?*; 0 @-; 0 @+)
% 0.32/0.47 % SPC : TH1_THM_EQU_NAR
% 0.32/0.47
% 0.32/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:17:10.404
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % Could-be-implicit typings (4)
% 0.32/0.47 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
% 0.32/0.47 binary1291135688e_Tree: $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Set_Oset,type,
% 0.32/0.47 set: $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Int_Oint,type,
% 0.32/0.47 int: $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_tf_a,type,
% 0.32/0.47 a: $tType ).
% 0.32/0.47
% 0.32/0.47 % Explicit typings (36)
% 0.32/0.47 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.32/0.47 bounded_lattice:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_HOL_Otype,type,
% 0.32/0.47 type:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Obot,type,
% 0.32/0.47 bot:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.32/0.47 ord:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.32/0.47 order:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Lattices_Olattice,type,
% 0.32/0.47 lattice:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/0.47 linorder:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/0.47 preorder:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.32/0.47 linordered_idom:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.32/0.47 semilattice_sup:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.32/0.47 bounde1808546759up_bot:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.32/0.47 condit1656338222tinuum:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.32/0.47 condit1037483654norder:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
% 0.32/0.47 binary210054475elle_T:
% 0.32/0.47 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
% 0.32/0.47 binary1746293266le_Tip:
% 0.32/0.47 !>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert,type,
% 0.32/0.47 binary1830089824insert:
% 0.32/0.47 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
% 0.32/0.47 binary827270440e_memb:
% 0.32/0.47 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oremove,type,
% 0.32/0.47 binary997842527remove:
% 0.32/0.47 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Orm,type,
% 0.32/0.47 binary576689334lle_rm:
% 0.32/0.47 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
% 0.32/0.47 binary1653327646_setOf:
% 0.32/0.47 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
% 0.32/0.47 binary1610619414edTree:
% 0.32/0.47 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
% 0.32/0.47 binary231205461t_pred:
% 0.32/0.47 !>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrm,type,
% 0.32/0.47 binary213313527le_wrm:
% 0.32/0.47 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.32/0.47 sup_sup:
% 0.32/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.32/0.47 bot_bot:
% 0.32/0.47 !>[A: $tType] : A ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.32/0.47 ord_less:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/0.47 ord_less_eq:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_OCollect,type,
% 0.32/0.47 collect:
% 0.32/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_Oinsert,type,
% 0.32/0.47 insert:
% 0.32/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_Othe__elem,type,
% 0.32/0.47 the_elem:
% 0.32/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_member,type,
% 0.32/0.47 member:
% 0.32/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_h,type,
% 0.32/0.47 h: a > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_l____,type,
% 0.32/0.47 l: a ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_t1____,type,
% 0.32/0.47 t1: binary1291135688e_Tree @ a ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_t2____,type,
% 0.32/0.47 t2: binary1291135688e_Tree @ a ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_x____,type,
% 0.32/0.47 x: a ).
% 0.32/0.47
% 0.32/0.47 % Relevant facts (255)
% 0.32/0.47 thf(fact_0_hLess,axiom,
% 0.32/0.47 ord_less @ int @ ( h @ l ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % hLess
% 0.32/0.47 thf(fact_1_rm__res,axiom,
% 0.32/0.47 ( ( binary576689334lle_rm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
% 0.32/0.47 = ( binary576689334lle_rm @ a @ h @ t2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % rm_res
% 0.32/0.47 thf(fact_2_t2nTip,axiom,
% 0.32/0.47 ( t2
% 0.32/0.47 != ( binary1746293266le_Tip @ a ) ) ).
% 0.32/0.47
% 0.32/0.47 % t2nTip
% 0.32/0.47 thf(fact_3_Tree_Oinject,axiom,
% 0.32/0.47 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
% 0.32/0.47 = ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
% 0.32/0.47 = ( ( X21 = Y21 )
% 0.32/0.47 & ( X22 = Y22 )
% 0.32/0.47 & ( X23 = Y23 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Tree.inject
% 0.32/0.47 thf(fact_4_s,axiom,
% 0.32/0.47 binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% 0.32/0.47
% 0.32/0.47 % s
% 0.32/0.47 thf(fact_5_wrm__res,axiom,
% 0.32/0.47 ( ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
% 0.32/0.47 = ( binary210054475elle_T @ a @ t1 @ x @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % wrm_res
% 0.32/0.47 thf(fact_6_ldef,axiom,
% 0.32/0.47 member @ a @ l @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ldef
% 0.32/0.47 thf(fact_7_s1,axiom,
% 0.32/0.47 binary1610619414edTree @ a @ h @ t1 ).
% 0.32/0.47
% 0.32/0.47 % s1
% 0.32/0.47 thf(fact_8_s2,axiom,
% 0.32/0.47 binary1610619414edTree @ a @ h @ t2 ).
% 0.32/0.47
% 0.32/0.47 % s2
% 0.32/0.47 thf(fact_9_binsert_Osimps_I2_J,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.32/0.47 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
% 0.32/0.47 & ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.32/0.47 => ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.32/0.47 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
% 0.32/0.47 & ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.32/0.47 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % binsert.simps(2)
% 0.32/0.47 thf(fact_10_rm_Osimps,axiom,
% 0.32/0.47 ! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
% 0.32/0.47 ( ( ( T2
% 0.32/0.47 = ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 => ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = X ) )
% 0.32/0.47 & ( ( T2
% 0.32/0.47 != ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 => ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = ( binary576689334lle_rm @ A @ H @ T2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rm.simps
% 0.32/0.47 thf(fact_11_minf_I11_J,axiom,
% 0.32/0.47 ! [C: $tType,D: $tType] :
% 0.32/0.47 ( ( ord @ C )
% 0.32/0.47 => ! [F: D] :
% 0.32/0.47 ? [Z: C] :
% 0.32/0.47 ! [X2: C] :
% 0.32/0.47 ( ( ord_less @ C @ X2 @ Z )
% 0.32/0.47 => ( F = F ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(11)
% 0.32/0.47 thf(fact_12_minf_I7_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z )
% 0.32/0.47 => ~ ( ord_less @ A @ T @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(7)
% 0.32/0.47 thf(fact_13_minf_I5_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z )
% 0.32/0.47 => ( ord_less @ A @ X2 @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(5)
% 0.32/0.47 thf(fact_14_minf_I4_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z )
% 0.32/0.47 => ( X2 != T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(4)
% 0.32/0.47 thf(fact_15_minf_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z )
% 0.32/0.47 => ( X2 != T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(3)
% 0.32/0.47 thf(fact_16_minf_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/0.47 ( ? [Z2: A] :
% 0.32/0.47 ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Z2 )
% 0.32/0.47 => ( ( P @ X3 )
% 0.32/0.47 = ( P2 @ X3 ) ) )
% 0.32/0.47 => ( ? [Z2: A] :
% 0.32/0.47 ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Z2 )
% 0.32/0.47 => ( ( Q @ X3 )
% 0.32/0.47 = ( Q2 @ X3 ) ) )
% 0.32/0.47 => ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z )
% 0.32/0.47 => ( ( ( P @ X2 )
% 0.32/0.47 | ( Q @ X2 ) )
% 0.32/0.47 = ( ( P2 @ X2 )
% 0.32/0.47 | ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(2)
% 0.32/0.47 thf(fact_17_sortedTree_Osimps_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % sortedTree.simps(1)
% 0.32/0.47 thf(fact_18_wrm__sort,axiom,
% 0.32/0.47 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.32/0.47 ( ( ( T
% 0.32/0.47 != ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.32/0.47 => ( binary1610619414edTree @ A @ H @ ( binary213313527le_wrm @ A @ H @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % wrm_sort
% 0.32/0.47 thf(fact_19_binsert__sorted,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
% 0.32/0.47 ( ( binary1610619414edTree @ A @ H @ T )
% 0.32/0.47 => ( binary1610619414edTree @ A @ H @ ( binary1830089824insert @ A @ H @ X @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % binsert_sorted
% 0.32/0.47 thf(fact_20_rm__set,axiom,
% 0.32/0.47 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.32/0.47 ( ( ( T
% 0.32/0.47 != ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.32/0.47 => ( member @ A @ ( binary576689334lle_rm @ A @ H @ T ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rm_set
% 0.32/0.47 thf(fact_21_binsert_Osimps_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,E: A] :
% 0.32/0.47 ( ( binary1830089824insert @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 = ( binary210054475elle_T @ A @ ( binary1746293266le_Tip @ A ) @ E @ ( binary1746293266le_Tip @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % binsert.simps(1)
% 0.32/0.47 thf(fact_22_wrm_Osimps,axiom,
% 0.32/0.47 ! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
% 0.32/0.47 ( ( ( T2
% 0.32/0.47 = ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 => ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = T1 ) )
% 0.32/0.47 & ( ( T2
% 0.32/0.47 != ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 => ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary213313527le_wrm @ A @ H @ T2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % wrm.simps
% 0.32/0.47 thf(fact_23_sortedTree_Osimps_I2_J,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = ( ( binary1610619414edTree @ A @ H @ T1 )
% 0.32/0.47 & ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T1 ) )
% 0.32/0.47 => ( ord_less @ int @ ( H @ X4 ) @ ( H @ X ) ) )
% 0.32/0.47 & ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T2 ) )
% 0.32/0.47 => ( ord_less @ int @ ( H @ X ) @ ( H @ X4 ) ) )
% 0.32/0.47 & ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sortedTree.simps(2)
% 0.32/0.47 thf(fact_24_Tree_Oexhaust,axiom,
% 0.32/0.47 ! [A: $tType,Y: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( Y
% 0.32/0.47 != ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 => ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( Y
% 0.32/0.47 != ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Tree.exhaust
% 0.32/0.47 thf(fact_25_Tree_Oinduct,axiom,
% 0.32/0.47 ! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( P @ ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 => ( ! [X1: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( P @ X1 )
% 0.32/0.47 => ( ( P @ X32 )
% 0.32/0.47 => ( P @ ( binary210054475elle_T @ A @ X1 @ X24 @ X32 ) ) ) )
% 0.32/0.47 => ( P @ Tree ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Tree.induct
% 0.32/0.47 thf(fact_26_Tree_Odistinct_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( binary1746293266le_Tip @ A )
% 0.32/0.47 != ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Tree.distinct(1)
% 0.32/0.47 thf(fact_27_sortLemmaR,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 => ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % sortLemmaR
% 0.32/0.47 thf(fact_28_sortLemmaL,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 => ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% 0.32/0.47
% 0.32/0.47 % sortLemmaL
% 0.32/0.47 thf(fact_29_pinf_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/0.47 ( ? [Z2: A] :
% 0.32/0.47 ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/0.47 => ( ( P @ X3 )
% 0.32/0.47 = ( P2 @ X3 ) ) )
% 0.32/0.47 => ( ? [Z2: A] :
% 0.32/0.47 ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/0.47 => ( ( Q @ X3 )
% 0.32/0.47 = ( Q2 @ X3 ) ) )
% 0.32/0.47 => ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.32/0.47 => ( ( ( P @ X2 )
% 0.32/0.47 & ( Q @ X2 ) )
% 0.32/0.47 = ( ( P2 @ X2 )
% 0.32/0.47 & ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(1)
% 0.32/0.47 thf(fact_30_pinf_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/0.47 ( ? [Z2: A] :
% 0.32/0.47 ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/0.47 => ( ( P @ X3 )
% 0.32/0.47 = ( P2 @ X3 ) ) )
% 0.32/0.47 => ( ? [Z2: A] :
% 0.32/0.47 ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/0.47 => ( ( Q @ X3 )
% 0.32/0.47 = ( Q2 @ X3 ) ) )
% 0.32/0.47 => ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.32/0.47 => ( ( ( P @ X2 )
% 0.32/0.47 | ( Q @ X2 ) )
% 0.32/0.47 = ( ( P2 @ X2 )
% 0.32/0.47 | ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(2)
% 0.32/0.47 thf(fact_31_pinf_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.32/0.47 => ( X2 != T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(3)
% 0.32/0.47 thf(fact_32_pinf_I4_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.32/0.47 => ( X2 != T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(4)
% 0.32/0.47 thf(fact_33_pinf_I5_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.32/0.47 => ~ ( ord_less @ A @ X2 @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(5)
% 0.32/0.47 thf(fact_34_pinf_I7_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.32/0.47 => ( ord_less @ A @ T @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(7)
% 0.32/0.47 thf(fact_35_pinf_I11_J,axiom,
% 0.32/0.47 ! [C: $tType,D: $tType] :
% 0.32/0.47 ( ( ord @ C )
% 0.32/0.47 => ! [F: D] :
% 0.32/0.47 ? [Z: C] :
% 0.32/0.47 ! [X2: C] :
% 0.32/0.47 ( ( ord_less @ C @ Z @ X2 )
% 0.32/0.47 => ( F = F ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(11)
% 0.32/0.47 thf(fact_36_minf_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/0.47 ( ? [Z2: A] :
% 0.32/0.47 ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Z2 )
% 0.32/0.47 => ( ( P @ X3 )
% 0.32/0.47 = ( P2 @ X3 ) ) )
% 0.32/0.47 => ( ? [Z2: A] :
% 0.32/0.47 ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Z2 )
% 0.32/0.47 => ( ( Q @ X3 )
% 0.32/0.47 = ( Q2 @ X3 ) ) )
% 0.32/0.47 => ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z )
% 0.32/0.47 => ( ( ( P @ X2 )
% 0.32/0.47 & ( Q @ X2 ) )
% 0.32/0.47 = ( ( P2 @ X2 )
% 0.32/0.47 & ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(1)
% 0.32/0.47 thf(fact_37_h1,axiom,
% 0.32/0.47 ( ( ( t1
% 0.32/0.47 != ( binary1746293266le_Tip @ a ) )
% 0.32/0.47 & ( binary1610619414edTree @ a @ h @ t1 ) )
% 0.32/0.47 => ! [X2: a] :
% 0.32/0.47 ( ( member @ a @ X2 @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t1 ) ) )
% 0.32/0.47 => ( ord_less @ int @ ( h @ X2 ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t1 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % h1
% 0.32/0.47 thf(fact_38_h2,axiom,
% 0.32/0.47 ( ( ( t2
% 0.32/0.47 != ( binary1746293266le_Tip @ a ) )
% 0.32/0.47 & ( binary1610619414edTree @ a @ h @ t2 ) )
% 0.32/0.47 => ! [X2: a] :
% 0.32/0.47 ( ( member @ a @ X2 @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t2 ) ) )
% 0.32/0.47 => ( ord_less @ int @ ( h @ X2 ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % h2
% 0.32/0.47 thf(fact_39_memb__spec,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
% 0.32/0.47 ( ( binary1610619414edTree @ A @ H @ T )
% 0.32/0.47 => ( ( binary827270440e_memb @ A @ H @ X @ T )
% 0.32/0.47 = ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % memb_spec
% 0.32/0.47 thf(fact_40_sorted__distinct__pred__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( binary231205461t_pred @ A )
% 0.32/0.47 = ( ^ [H2: A > int,A2: A,B: A,T3: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( ( binary1610619414edTree @ A @ H2 @ T3 )
% 0.32/0.47 & ( member @ A @ A2 @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.32/0.47 & ( member @ A @ B @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.32/0.47 & ( ( H2 @ A2 )
% 0.32/0.47 = ( H2 @ B ) ) )
% 0.32/0.47 => ( A2 = B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sorted_distinct_pred_def
% 0.32/0.47 thf(fact_41_l__scope,axiom,
% 0.32/0.47 member @ a @ l @ ( sup_sup @ ( set @ a ) @ ( sup_sup @ ( set @ a ) @ ( insert @ a @ x @ ( bot_bot @ ( set @ a ) ) ) @ ( binary1653327646_setOf @ a @ t1 ) ) @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % l_scope
% 0.32/0.47 thf(fact_42_wrm__set1,axiom,
% 0.32/0.47 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.32/0.47 ( ( ( T
% 0.32/0.47 != ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( binary1653327646_setOf @ A @ ( binary213313527le_wrm @ A @ H @ T ) ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % wrm_set1
% 0.32/0.47 thf(fact_43_remove_Osimps_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,E: A] :
% 0.32/0.47 ( ( binary997842527remove @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 = ( binary1746293266le_Tip @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % remove.simps(1)
% 0.32/0.47 thf(fact_44_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,P: A > $o] :
% 0.32/0.47 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.32/0.47 = ( P @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_45_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.32/0.47 = A4 ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_46_Collect__cong,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 = ( Q @ X3 ) )
% 0.32/0.47 => ( ( collect @ A @ P )
% 0.32/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_47_ext,axiom,
% 0.32/0.47 ! [B2: $tType,A: $tType,F2: A > B2,G: A > B2] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( F2 @ X3 )
% 0.32/0.47 = ( G @ X3 ) )
% 0.32/0.47 => ( F2 = G ) ) ).
% 0.32/0.47
% 0.32/0.47 % ext
% 0.32/0.47 thf(fact_48_ex__gt__or__lt,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( condit1656338222tinuum @ A )
% 0.32/0.47 => ! [A3: A] :
% 0.32/0.47 ? [B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/0.47 | ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ex_gt_or_lt
% 0.32/0.47 thf(fact_49_linorder__neqE__linordered__idom,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_idom @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( X != Y )
% 0.32/0.47 => ( ~ ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_neqE_linordered_idom
% 0.32/0.47 thf(fact_50_complete__interval,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( condit1037483654norder @ A )
% 0.32/0.47 => ! [A3: A,B4: A,P: A > $o] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B4 )
% 0.32/0.47 => ( ( P @ A3 )
% 0.32/0.47 => ( ~ ( P @ B4 )
% 0.32/0.47 => ? [C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.32/0.47 & ( ord_less_eq @ A @ C2 @ B4 )
% 0.32/0.47 & ! [X2: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ A3 @ X2 )
% 0.32/0.47 & ( ord_less @ A @ X2 @ C2 ) )
% 0.32/0.47 => ( P @ X2 ) )
% 0.32/0.47 & ! [D2: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ A3 @ X3 )
% 0.32/0.47 & ( ord_less @ A @ X3 @ D2 ) )
% 0.32/0.47 => ( P @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ D2 @ C2 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % complete_interval
% 0.32/0.47 thf(fact_51_setOf_Osimps_I2_J,axiom,
% 0.32/0.47 ! [A: $tType,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.32/0.47 ( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/0.47 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T2 ) ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % setOf.simps(2)
% 0.32/0.47 thf(fact_52_minf_I8_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(8)
% 0.32/0.47 thf(fact_53_minf_I6_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Z )
% 0.32/0.47 => ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minf(6)
% 0.32/0.47 thf(fact_54_pinf_I8_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.32/0.47 => ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(8)
% 0.32/0.47 thf(fact_55_pinf_I6_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [T: A] :
% 0.32/0.47 ? [Z: A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pinf(6)
% 0.32/0.47 thf(fact_56_sorted__distinct,axiom,
% 0.32/0.47 ! [A: $tType,H: A > int,A3: A,B4: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A3 @ B4 @ T ) ).
% 0.32/0.47
% 0.32/0.47 % sorted_distinct
% 0.32/0.47 thf(fact_57_setOf_Osimps_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % setOf.simps(1)
% 0.32/0.47 thf(fact_58_singleton__insert__inj__eq_H,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,A4: set @ A,B4: A] :
% 0.32/0.47 ( ( ( insert @ A @ A3 @ A4 )
% 0.32/0.47 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 = ( ( A3 = B4 )
% 0.32/0.47 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % singleton_insert_inj_eq'
% 0.32/0.47 thf(fact_59_singleton__insert__inj__eq,axiom,
% 0.32/0.47 ! [A: $tType,B4: A,A3: A,A4: set @ A] :
% 0.32/0.47 ( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 = ( insert @ A @ A3 @ A4 ) )
% 0.32/0.47 = ( ( A3 = B4 )
% 0.32/0.47 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % singleton_insert_inj_eq
% 0.32/0.47 thf(fact_60_singleton__conv2,axiom,
% 0.32/0.47 ! [A: $tType,A3: A] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ( ^ [Y2: A,Z3: A] : ( Y2 = Z3 )
% 0.32/0.47 @ A3 ) )
% 0.32/0.47 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % singleton_conv2
% 0.32/0.47 thf(fact_61_singleton__conv,axiom,
% 0.32/0.47 ! [A: $tType,A3: A] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ^ [X4: A] : ( X4 = A3 ) )
% 0.32/0.47 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % singleton_conv
% 0.32/0.47 thf(fact_62_Un__insert__right,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,A3: A,B5: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B5 ) )
% 0.32/0.47 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_insert_right
% 0.32/0.47 thf(fact_63_Un__insert__left,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,B5: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.32/0.47 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_insert_left
% 0.32/0.47 thf(fact_64_Un__subset__iff,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
% 0.32/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.32/0.47 & ( ord_less_eq @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_subset_iff
% 0.32/0.47 thf(fact_65_Un__empty,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 = ( ( A4
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 & ( B5
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_empty
% 0.32/0.47 thf(fact_66_empty__iff,axiom,
% 0.32/0.47 ! [A: $tType,C4: A] :
% 0.32/0.47 ~ ( member @ A @ C4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % empty_iff
% 0.32/0.47 thf(fact_67_all__not__in__conv,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ( ( ! [X4: A] :
% 0.32/0.47 ~ ( member @ A @ X4 @ A4 ) )
% 0.32/0.47 = ( A4
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % all_not_in_conv
% 0.32/0.47 thf(fact_68_Collect__empty__eq,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o] :
% 0.32/0.47 ( ( ( collect @ A @ P )
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 = ( ! [X4: A] :
% 0.32/0.47 ~ ( P @ X4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_empty_eq
% 0.32/0.47 thf(fact_69_empty__Collect__eq,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o] :
% 0.32/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.32/0.47 = ( collect @ A @ P ) )
% 0.32/0.47 = ( ! [X4: A] :
% 0.32/0.47 ~ ( P @ X4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % empty_Collect_eq
% 0.32/0.47 thf(fact_70_subsetI,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A4 )
% 0.32/0.47 => ( member @ A @ X3 @ B5 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetI
% 0.32/0.47 thf(fact_71_subset__antisym,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
% 0.32/0.47 => ( A4 = B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_antisym
% 0.32/0.47 thf(fact_72_psubsetI,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( A4 != B5 )
% 0.32/0.47 => ( ord_less @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubsetI
% 0.32/0.47 thf(fact_73_insertCI,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,B5: set @ A,B4: A] :
% 0.32/0.47 ( ( ~ ( member @ A @ A3 @ B5 )
% 0.32/0.47 => ( A3 = B4 ) )
% 0.32/0.47 => ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insertCI
% 0.32/0.47 thf(fact_74_insert__iff,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,B4: A,A4: set @ A] :
% 0.32/0.47 ( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
% 0.32/0.47 = ( ( A3 = B4 )
% 0.32/0.47 | ( member @ A @ A3 @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_iff
% 0.32/0.47 thf(fact_75_insert__absorb2,axiom,
% 0.32/0.47 ! [A: $tType,X: A,A4: set @ A] :
% 0.32/0.47 ( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
% 0.32/0.47 = ( insert @ A @ X @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_absorb2
% 0.32/0.47 thf(fact_76_UnCI,axiom,
% 0.32/0.47 ! [A: $tType,C4: A,B5: set @ A,A4: set @ A] :
% 0.32/0.47 ( ( ~ ( member @ A @ C4 @ B5 )
% 0.32/0.47 => ( member @ A @ C4 @ A4 ) )
% 0.32/0.47 => ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UnCI
% 0.32/0.47 thf(fact_77_Un__iff,axiom,
% 0.32/0.47 ! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/0.47 = ( ( member @ A @ C4 @ A4 )
% 0.32/0.47 | ( member @ A @ C4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_iff
% 0.32/0.47 thf(fact_78_subset__empty,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 = ( A4
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_empty
% 0.32/0.47 thf(fact_79_empty__subsetI,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.32/0.47
% 0.32/0.47 % empty_subsetI
% 0.32/0.47 thf(fact_80_singletonI,axiom,
% 0.32/0.47 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % singletonI
% 0.32/0.47 thf(fact_81_insert__subset,axiom,
% 0.32/0.47 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.32/0.47 = ( ( member @ A @ X @ B5 )
% 0.32/0.47 & ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_subset
% 0.32/0.47 thf(fact_82_less__eq__set__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( ord_less_eq @ ( A > $o )
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A5 )
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_eq_set_def
% 0.32/0.47 thf(fact_83_emptyE,axiom,
% 0.32/0.47 ! [A: $tType,A3: A] :
% 0.32/0.47 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % emptyE
% 0.32/0.47 thf(fact_84_equals0D,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,A3: A] :
% 0.32/0.47 ( ( A4
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 => ~ ( member @ A @ A3 @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % equals0D
% 0.32/0.47 thf(fact_85_equals0I,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ( ! [Y3: A] :
% 0.32/0.47 ~ ( member @ A @ Y3 @ A4 )
% 0.32/0.47 => ( A4
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % equals0I
% 0.32/0.47 thf(fact_86_ex__in__conv,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.32/0.47 = ( A4
% 0.32/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ex_in_conv
% 0.32/0.47 thf(fact_87_not__psubset__empty,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_psubset_empty
% 0.32/0.47 thf(fact_88_in__mono,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( member @ A @ X @ A4 )
% 0.32/0.47 => ( member @ A @ X @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_mono
% 0.32/0.47 thf(fact_89_subsetD,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C4: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( member @ A @ C4 @ A4 )
% 0.32/0.47 => ( member @ A @ C4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetD
% 0.32/0.47 thf(fact_90_equalityE,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( A4 = B5 )
% 0.32/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityE
% 0.32/0.47 thf(fact_91_subset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A5 )
% 0.32/0.47 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_eq
% 0.32/0.47 thf(fact_92_equalityD1,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( A4 = B5 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityD1
% 0.32/0.47 thf(fact_93_equalityD2,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( A4 = B5 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityD2
% 0.32/0.47 thf(fact_94_subset__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ! [T3: A] :
% 0.32/0.47 ( ( member @ A @ T3 @ A5 )
% 0.32/0.47 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_iff
% 0.32/0.47 thf(fact_95_subset__refl,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.32/0.47
% 0.32/0.47 % subset_refl
% 0.32/0.47 thf(fact_96_Collect__mono,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( Q @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mono
% 0.32/0.47 thf(fact_97_subset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_trans
% 0.32/0.47 thf(fact_98_set__eq__subset,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ^ [Y2: set @ A,Z3: set @ A] : ( Y2 = Z3 ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.32/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % set_eq_subset
% 0.32/0.47 thf(fact_99_Collect__mono__iff,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.32/0.47 = ( ! [X4: A] :
% 0.32/0.47 ( ( P @ X4 )
% 0.32/0.47 => ( Q @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mono_iff
% 0.32/0.47 thf(fact_100_subset__iff__psubset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 0.32/0.47 | ( A5 = B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_iff_psubset_eq
% 0.32/0.47 thf(fact_101_subset__psubset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
% 0.32/0.47 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_psubset_trans
% 0.32/0.47 thf(fact_102_subset__not__subset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.32/0.47 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_not_subset_eq
% 0.32/0.47 thf(fact_103_psubset__subset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
% 0.32/0.47 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubset_subset_trans
% 0.32/0.47 thf(fact_104_psubset__imp__subset,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubset_imp_subset
% 0.32/0.47 thf(fact_105_psubset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.32/0.47 & ( A5 != B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubset_eq
% 0.32/0.47 thf(fact_106_psubsetE,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubsetE
% 0.32/0.47 thf(fact_107_insertE,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,B4: A,A4: set @ A] :
% 0.32/0.47 ( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
% 0.32/0.47 => ( ( A3 != B4 )
% 0.32/0.47 => ( member @ A @ A3 @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insertE
% 0.32/0.47 thf(fact_108_insertI1,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % insertI1
% 0.32/0.47 thf(fact_109_insertI2,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,B5: set @ A,B4: A] :
% 0.32/0.47 ( ( member @ A @ A3 @ B5 )
% 0.32/0.47 => ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insertI2
% 0.32/0.47 thf(fact_110_Set_Oset__insert,axiom,
% 0.32/0.47 ! [A: $tType,X: A,A4: set @ A] :
% 0.32/0.47 ( ( member @ A @ X @ A4 )
% 0.32/0.47 => ~ ! [B7: set @ A] :
% 0.32/0.47 ( ( A4
% 0.32/0.47 = ( insert @ A @ X @ B7 ) )
% 0.32/0.47 => ( member @ A @ X @ B7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Set.set_insert
% 0.32/0.47 thf(fact_111_insert__ident,axiom,
% 0.32/0.47 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ~ ( member @ A @ X @ A4 )
% 0.32/0.47 => ( ~ ( member @ A @ X @ B5 )
% 0.32/0.47 => ( ( ( insert @ A @ X @ A4 )
% 0.32/0.47 = ( insert @ A @ X @ B5 ) )
% 0.32/0.47 = ( A4 = B5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_ident
% 0.32/0.47 thf(fact_112_insert__absorb,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,A4: set @ A] :
% 0.32/0.47 ( ( member @ A @ A3 @ A4 )
% 0.32/0.47 => ( ( insert @ A @ A3 @ A4 )
% 0.32/0.47 = A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_absorb
% 0.32/0.47 thf(fact_113_insert__eq__iff,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,A4: set @ A,B4: A,B5: set @ A] :
% 0.32/0.47 ( ~ ( member @ A @ A3 @ A4 )
% 0.32/0.47 => ( ~ ( member @ A @ B4 @ B5 )
% 0.32/0.47 => ( ( ( insert @ A @ A3 @ A4 )
% 0.32/0.47 = ( insert @ A @ B4 @ B5 ) )
% 0.32/0.47 = ( ( ( A3 = B4 )
% 0.32/0.47 => ( A4 = B5 ) )
% 0.32/0.47 & ( ( A3 != B4 )
% 0.32/0.47 => ? [C5: set @ A] :
% 0.32/0.47 ( ( A4
% 0.32/0.47 = ( insert @ A @ B4 @ C5 ) )
% 0.32/0.47 & ~ ( member @ A @ B4 @ C5 )
% 0.32/0.47 & ( B5
% 0.32/0.47 = ( insert @ A @ A3 @ C5 ) )
% 0.32/0.47 & ~ ( member @ A @ A3 @ C5 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_eq_iff
% 0.32/0.47 thf(fact_114_insert__commute,axiom,
% 0.32/0.47 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 0.32/0.47 ( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
% 0.32/0.47 = ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_commute
% 0.32/0.47 thf(fact_115_mk__disjoint__insert,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,A4: set @ A] :
% 0.32/0.47 ( ( member @ A @ A3 @ A4 )
% 0.32/0.47 => ? [B7: set @ A] :
% 0.32/0.47 ( ( A4
% 0.32/0.47 = ( insert @ A @ A3 @ B7 ) )
% 0.32/0.47 & ~ ( member @ A @ A3 @ B7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mk_disjoint_insert
% 0.32/0.47 thf(fact_116_UnE,axiom,
% 0.32/0.47 ! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/0.47 => ( ~ ( member @ A @ C4 @ A4 )
% 0.32/0.47 => ( member @ A @ C4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UnE
% 0.32/0.47 thf(fact_117_UnI1,axiom,
% 0.32/0.47 ! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( member @ A @ C4 @ A4 )
% 0.32/0.47 => ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UnI1
% 0.32/0.47 thf(fact_118_UnI2,axiom,
% 0.32/0.47 ! [A: $tType,C4: A,B5: set @ A,A4: set @ A] :
% 0.32/0.47 ( ( member @ A @ C4 @ B5 )
% 0.32/0.47 => ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UnI2
% 0.32/0.47 thf(fact_119_bex__Un,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
% 0.32/0.47 ( ( ? [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/0.47 & ( P @ X4 ) ) )
% 0.32/0.47 = ( ? [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A4 )
% 0.32/0.47 & ( P @ X4 ) )
% 0.32/0.47 | ? [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ B5 )
% 0.32/0.47 & ( P @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bex_Un
% 0.32/0.47 thf(fact_120_ball__Un,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
% 0.32/0.47 ( ( ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/0.47 => ( P @ X4 ) ) )
% 0.32/0.47 = ( ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A4 )
% 0.32/0.47 => ( P @ X4 ) )
% 0.32/0.47 & ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ B5 )
% 0.32/0.47 => ( P @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ball_Un
% 0.32/0.47 thf(fact_121_Un__assoc,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
% 0.32/0.47 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_assoc
% 0.32/0.47 thf(fact_122_Un__absorb,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 0.32/0.47 = A4 ) ).
% 0.32/0.47
% 0.32/0.47 % Un_absorb
% 0.32/0.47 thf(fact_123_Un__commute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_commute
% 0.32/0.47 thf(fact_124_Un__left__absorb,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/0.47 = ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_left_absorb
% 0.32/0.47 thf(fact_125_Un__left__commute,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
% 0.32/0.47 = ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_left_commute
% 0.32/0.47 thf(fact_126_empty__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.32/0.47 = ( collect @ A
% 0.32/0.47 @ ^ [X4: A] : $false ) ) ).
% 0.32/0.47
% 0.32/0.47 % empty_def
% 0.32/0.47 thf(fact_127_Collect__subset,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,P: A > $o] :
% 0.32/0.47 ( ord_less_eq @ ( set @ A )
% 0.32/0.47 @ ( collect @ A
% 0.32/0.47 @ ^ [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A4 )
% 0.32/0.47 & ( P @ X4 ) ) )
% 0.32/0.47 @ A4 ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_subset
% 0.32/0.47 thf(fact_128_insert__compr,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( insert @ A )
% 0.32/0.47 = ( ^ [A2: A,B6: set @ A] :
% 0.32/0.47 ( collect @ A
% 0.32/0.47 @ ^ [X4: A] :
% 0.32/0.47 ( ( X4 = A2 )
% 0.32/0.47 | ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_compr
% 0.32/0.47 thf(fact_129_insert__Collect,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,P: A > $o] :
% 0.32/0.47 ( ( insert @ A @ A3 @ ( collect @ A @ P ) )
% 0.32/0.47 = ( collect @ A
% 0.32/0.47 @ ^ [U: A] :
% 0.32/0.47 ( ( U != A3 )
% 0.32/0.47 => ( P @ U ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_Collect
% 0.32/0.47 thf(fact_130_Un__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( collect @ A
% 0.32/0.47 @ ^ [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A5 )
% 0.32/0.47 | ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_def
% 0.32/0.47 thf(fact_131_Collect__disj__eq,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ^ [X4: A] :
% 0.32/0.47 ( ( P @ X4 )
% 0.32/0.47 | ( Q @ X4 ) ) )
% 0.32/0.47 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_disj_eq
% 0.32/0.47 thf(fact_132_singletonD,axiom,
% 0.32/0.47 ! [A: $tType,B4: A,A3: A] :
% 0.32/0.47 ( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 => ( B4 = A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % singletonD
% 0.32/0.47 thf(fact_133_singleton__iff,axiom,
% 0.32/0.47 ! [A: $tType,B4: A,A3: A] :
% 0.32/0.47 ( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 = ( B4 = A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % singleton_iff
% 0.32/0.47 thf(fact_134_doubleton__eq__iff,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,B4: A,C4: A,D3: A] :
% 0.32/0.47 ( ( ( insert @ A @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 = ( insert @ A @ C4 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/0.47 = ( ( ( A3 = C4 )
% 0.32/0.47 & ( B4 = D3 ) )
% 0.32/0.47 | ( ( A3 = D3 )
% 0.32/0.47 & ( B4 = C4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % doubleton_eq_iff
% 0.32/0.47 thf(fact_135_insert__not__empty,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,A4: set @ A] :
% 0.32/0.47 ( ( insert @ A @ A3 @ A4 )
% 0.32/0.47 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_not_empty
% 0.32/0.47 thf(fact_136_singleton__inject,axiom,
% 0.32/0.47 ! [A: $tType,A3: A,B4: A] :
% 0.32/0.47 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 => ( A3 = B4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % singleton_inject
% 0.32/0.47 thf(fact_137_insert__mono,axiom,
% 0.32/0.47 ! [A: $tType,C3: set @ A,D4: set @ A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ C3 @ D4 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C3 ) @ ( insert @ A @ A3 @ D4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_mono
% 0.32/0.47 thf(fact_138_subset__insert,axiom,
% 0.32/0.47 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ~ ( member @ A @ X @ A4 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
% 0.32/0.47 = ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_insert
% 0.32/0.47 thf(fact_139_subset__insertI,axiom,
% 0.32/0.47 ! [A: $tType,B5: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A3 @ B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_insertI
% 0.32/0.47 thf(fact_140_subset__insertI2,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_insertI2
% 0.32/0.47 thf(fact_141_Un__empty__left,axiom,
% 0.32/0.47 ! [A: $tType,B5: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
% 0.32/0.47 = B5 ) ).
% 0.32/0.47
% 0.32/0.47 % Un_empty_left
% 0.32/0.47 thf(fact_142_Un__empty__right,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 = A4 ) ).
% 0.32/0.47
% 0.32/0.47 % Un_empty_right
% 0.32/0.47 thf(fact_143_Un__mono,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,C3: set @ A,B5: set @ A,D4: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_mono
% 0.32/0.47 thf(fact_144_Un__least,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,C3: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_least
% 0.32/0.47 thf(fact_145_Un__upper1,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_upper1
% 0.32/0.47 thf(fact_146_Un__upper2,axiom,
% 0.32/0.47 ! [A: $tType,B5: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_upper2
% 0.32/0.47 thf(fact_147_Un__absorb1,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 = B5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_absorb1
% 0.32/0.47 thf(fact_148_Un__absorb2,axiom,
% 0.32/0.47 ! [A: $tType,B5: set @ A,A4: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
% 0.32/0.47 => ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 = A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_absorb2
% 0.32/0.47 thf(fact_149_subset__UnE,axiom,
% 0.32/0.47 ! [A: $tType,C3: set @ A,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/0.47 => ~ ! [A6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ A4 )
% 0.32/0.47 => ! [B8: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ B8 @ B5 )
% 0.32/0.47 => ( C3
% 0.32/0.47 != ( sup_sup @ ( set @ A ) @ A6 @ B8 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_UnE
% 0.32/0.47 thf(fact_150_subset__Un__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
% 0.32/0.47 = B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_Un_eq
% 0.32/0.47 thf(fact_151_Collect__conv__if,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,A3: A] :
% 0.32/0.47 ( ( ( P @ A3 )
% 0.32/0.47 => ( ( collect @ A
% 0.32/0.47 @ ^ [X4: A] :
% 0.32/0.47 ( ( X4 = A3 )
% 0.32/0.47 & ( P @ X4 ) ) )
% 0.32/0.47 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/0.47 & ( ~ ( P @ A3 )
% 0.32/0.47 => ( ( collect @ A
% 0.32/0.47 @ ^ [X4: A] :
% 0.32/0.47 ( ( X4 = A3 )
% 0.32/0.47 & ( P @ X4 ) ) )
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_conv_if
% 0.32/0.47 thf(fact_152_Collect__conv__if2,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,A3: A] :
% 0.32/0.47 ( ( ( P @ A3 )
% 0.32/0.47 => ( ( collect @ A
% 0.32/0.47 @ ^ [X4: A] :
% 0.32/0.47 ( ( A3 = X4 )
% 0.32/0.47 & ( P @ X4 ) ) )
% 0.32/0.47 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/0.47 & ( ~ ( P @ A3 )
% 0.32/0.47 => ( ( collect @ A
% 0.32/0.47 @ ^ [X4: A] :
% 0.32/0.47 ( ( A3 = X4 )
% 0.32/0.47 & ( P @ X4 ) ) )
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_conv_if2
% 0.32/0.47 thf(fact_153_insert__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( insert @ A )
% 0.32/0.47 = ( ^ [A2: A] :
% 0.32/0.47 ( sup_sup @ ( set @ A )
% 0.32/0.47 @ ( collect @ A
% 0.32/0.47 @ ^ [X4: A] : ( X4 = A2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_def
% 0.32/0.47 thf(fact_154_subset__singletonD,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 => ( ( A4
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 | ( A4
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_singletonD
% 0.32/0.47 thf(fact_155_subset__singleton__iff,axiom,
% 0.32/0.47 ! [A: $tType,X5: set @ A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 = ( ( X5
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 | ( X5
% 0.32/0.47 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_singleton_iff
% 0.32/0.47 thf(fact_156_insert__is__Un,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( insert @ A )
% 0.32/0.47 = ( ^ [A2: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % insert_is_Un
% 0.32/0.47 thf(fact_157_Un__singleton__iff,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
% 0.32/0.47 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 = ( ( ( A4
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 & ( B5
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/0.47 | ( ( A4
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 & ( B5
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 | ( ( A4
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 & ( B5
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Un_singleton_iff
% 0.32/0.47 thf(fact_158_singleton__Un__iff,axiom,
% 0.32/0.47 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.32/0.47 ( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 = ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/0.47 = ( ( ( A4
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.32/0.47 & ( B5
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/0.47 | ( ( A4
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 & ( B5
% 0.32/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 | ( ( A4
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 & ( B5
% 0.32/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % singleton_Un_iff
% 0.32/0.47 thf(fact_159_sup__bot__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bounde1808546759up_bot @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 0.32/0.47 = X ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_bot_left
% 0.32/0.47 thf(fact_160_sup__bot__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bounde1808546759up_bot @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 0.32/0.47 = X ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_bot_right
% 0.32/0.47 thf(fact_161_bot__eq__sup__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bounde1808546759up_bot @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ( bot_bot @ A )
% 0.32/0.47 = ( sup_sup @ A @ X @ Y ) )
% 0.32/0.47 = ( ( X
% 0.32/0.47 = ( bot_bot @ A ) )
% 0.32/0.47 & ( Y
% 0.32/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_eq_sup_iff
% 0.32/0.47 thf(fact_162_sup__eq__bot__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bounde1808546759up_bot @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ( sup_sup @ A @ X @ Y )
% 0.32/0.47 = ( bot_bot @ A ) )
% 0.32/0.47 = ( ( X
% 0.32/0.47 = ( bot_bot @ A ) )
% 0.32/0.47 & ( Y
% 0.32/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_eq_bot_iff
% 0.32/0.47 thf(fact_163_sup__bot_Oeq__neutr__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bounde1808546759up_bot @ A )
% 0.32/0.47 => ! [A3: A,B4: A] :
% 0.32/0.47 ( ( ( sup_sup @ A @ A3 @ B4 )
% 0.32/0.47 = ( bot_bot @ A ) )
% 0.32/0.47 = ( ( A3
% 0.32/0.47 = ( bot_bot @ A ) )
% 0.32/0.47 & ( B4
% 0.32/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_bot.eq_neutr_iff
% 0.32/0.47 thf(fact_164_sup__bot_Oleft__neutral,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bounde1808546759up_bot @ A )
% 0.32/0.47 => ! [A3: A] :
% 0.32/0.47 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
% 0.32/0.47 = A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_bot.left_neutral
% 0.32/0.47 thf(fact_165_sup__bot_Oneutr__eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bounde1808546759up_bot @ A )
% 0.32/0.47 => ! [A3: A,B4: A] :
% 0.32/0.47 ( ( ( bot_bot @ A )
% 0.32/0.47 = ( sup_sup @ A @ A3 @ B4 ) )
% 0.32/0.47 = ( ( A3
% 0.32/0.47 = ( bot_bot @ A ) )
% 0.32/0.47 & ( B4
% 0.32/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_bot.neutr_eq_iff
% 0.32/0.47 thf(fact_166_sup_Oright__idem,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,B4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B4 ) @ B4 )
% 0.32/0.47 = ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.right_idem
% 0.32/0.47 thf(fact_167_sup__left__idem,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.32/0.47 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_left_idem
% 0.32/0.47 thf(fact_168_sup_Oleft__idem,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,B4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B4 ) )
% 0.32/0.47 = ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.left_idem
% 0.32/0.47 thf(fact_169_sup__idem,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ( ( sup_sup @ A @ X @ X )
% 0.32/0.47 = X ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_idem
% 0.32/0.47 thf(fact_170_sup_Oidem,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A] :
% 0.32/0.47 ( ( sup_sup @ A @ A3 @ A3 )
% 0.32/0.47 = A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.idem
% 0.32/0.47 thf(fact_171_sup__apply,axiom,
% 0.32/0.47 ! [B2: $tType,A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ B2 )
% 0.32/0.47 => ( ( sup_sup @ ( A > B2 ) )
% 0.32/0.47 = ( ^ [F3: A > B2,G2: A > B2,X4: A] : ( sup_sup @ B2 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_apply
% 0.32/0.47 thf(fact_172_sup_Obounded__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B4: A,C4: A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
% 0.32/0.47 = ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/0.47 & ( ord_less_eq @ A @ C4 @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.bounded_iff
% 0.32/0.47 thf(fact_173_le__sup__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
% 0.32/0.47 = ( ( ord_less_eq @ A @ X @ Z4 )
% 0.32/0.47 & ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_sup_iff
% 0.32/0.47 thf(fact_174_sup__bot_Oright__neutral,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bounde1808546759up_bot @ A )
% 0.32/0.47 => ! [A3: A] :
% 0.32/0.47 ( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
% 0.32/0.47 = A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_bot.right_neutral
% 0.32/0.47 thf(fact_175_less__set__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( ord_less @ ( A > $o )
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A5 )
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_set_def
% 0.32/0.47 thf(fact_176_psubset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
% 0.32/0.47 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubset_trans
% 0.32/0.47 thf(fact_177_psubsetD,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C4: A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/0.47 => ( ( member @ A @ C4 @ A4 )
% 0.32/0.47 => ( member @ A @ C4 @ B5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubsetD
% 0.32/0.47 thf(fact_178_bot__set__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.32/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_set_def
% 0.32/0.47 thf(fact_179_sup__set__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( sup_sup @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/0.47 ( collect @ A
% 0.32/0.47 @ ( sup_sup @ ( A > $o )
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A5 )
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_set_def
% 0.32/0.47 thf(fact_180_sup__left__commute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) )
% 0.32/0.47 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_left_commute
% 0.32/0.47 thf(fact_181_sup_Oleft__commute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B4: A,A3: A,C4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ B4 @ ( sup_sup @ A @ A3 @ C4 ) )
% 0.32/0.47 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B4 @ C4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.left_commute
% 0.32/0.47 thf(fact_182_sup__commute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( sup_sup @ A )
% 0.32/0.47 = ( ^ [X4: A,Y4: A] : ( sup_sup @ A @ Y4 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_commute
% 0.32/0.47 thf(fact_183_sup_Ocommute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( sup_sup @ A )
% 0.32/0.47 = ( ^ [A2: A,B: A] : ( sup_sup @ A @ B @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.commute
% 0.32/0.47 thf(fact_184_sup__assoc,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
% 0.32/0.47 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_assoc
% 0.32/0.47 thf(fact_185_sup_Oassoc,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,B4: A,C4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B4 ) @ C4 )
% 0.32/0.47 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B4 @ C4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.assoc
% 0.32/0.47 thf(fact_186_boolean__algebra__cancel_Osup2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B5: A,K: A,B4: A,A3: A] :
% 0.32/0.47 ( ( B5
% 0.32/0.47 = ( sup_sup @ A @ K @ B4 ) )
% 0.32/0.47 => ( ( sup_sup @ A @ A3 @ B5 )
% 0.32/0.47 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % boolean_algebra_cancel.sup2
% 0.32/0.47 thf(fact_187_boolean__algebra__cancel_Osup1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A4: A,K: A,A3: A,B4: A] :
% 0.32/0.47 ( ( A4
% 0.32/0.47 = ( sup_sup @ A @ K @ A3 ) )
% 0.32/0.47 => ( ( sup_sup @ A @ A4 @ B4 )
% 0.32/0.47 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % boolean_algebra_cancel.sup1
% 0.32/0.47 thf(fact_188_sup__fun__def,axiom,
% 0.32/0.47 ! [B2: $tType,A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ B2 )
% 0.32/0.47 => ( ( sup_sup @ ( A > B2 ) )
% 0.32/0.47 = ( ^ [F3: A > B2,G2: A > B2,X4: A] : ( sup_sup @ B2 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_fun_def
% 0.32/0.47 thf(fact_189_inf__sup__aci_I5_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ( ( sup_sup @ A )
% 0.32/0.47 = ( ^ [X4: A,Y4: A] : ( sup_sup @ A @ Y4 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_aci(5)
% 0.32/0.47 thf(fact_190_inf__sup__aci_I6_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
% 0.32/0.47 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_aci(6)
% 0.32/0.47 thf(fact_191_inf__sup__aci_I7_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) )
% 0.32/0.47 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_aci(7)
% 0.32/0.47 thf(fact_192_inf__sup__aci_I8_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.32/0.47 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_aci(8)
% 0.32/0.47 thf(fact_193_sup_OcoboundedI2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [C4: A,B4: A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ C4 @ B4 )
% 0.32/0.47 => ( ord_less_eq @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.coboundedI2
% 0.32/0.47 thf(fact_194_sup_OcoboundedI1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [C4: A,A3: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ C4 @ A3 )
% 0.32/0.47 => ( ord_less_eq @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.coboundedI1
% 0.32/0.47 thf(fact_195_sup_Oabsorb__iff2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A2: A,B: A] :
% 0.32/0.47 ( ( sup_sup @ A @ A2 @ B )
% 0.32/0.47 = B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.absorb_iff2
% 0.32/0.47 thf(fact_196_sup_Oabsorb__iff1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [B: A,A2: A] :
% 0.32/0.47 ( ( sup_sup @ A @ A2 @ B )
% 0.32/0.47 = A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.absorb_iff1
% 0.32/0.47 thf(fact_197_sup_Ocobounded2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B4: A,A3: A] : ( ord_less_eq @ A @ B4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.cobounded2
% 0.32/0.47 thf(fact_198_sup_Ocobounded1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,B4: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.cobounded1
% 0.32/0.47 thf(fact_199_sup_Oorder__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [B: A,A2: A] :
% 0.32/0.47 ( A2
% 0.32/0.47 = ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.order_iff
% 0.32/0.47 thf(fact_200_sup_OboundedI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B4: A,A3: A,C4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C4 @ A3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.boundedI
% 0.32/0.47 thf(fact_201_sup_OboundedE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B4: A,C4: A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ C4 @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.boundedE
% 0.32/0.47 thf(fact_202_sup__absorb2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( sup_sup @ A @ X @ Y )
% 0.32/0.47 = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_absorb2
% 0.32/0.47 thf(fact_203_sup__absorb1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [Y: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( ( sup_sup @ A @ X @ Y )
% 0.32/0.47 = X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_absorb1
% 0.32/0.47 thf(fact_204_sup_Oabsorb2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/0.47 => ( ( sup_sup @ A @ A3 @ B4 )
% 0.32/0.47 = B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.absorb2
% 0.32/0.47 thf(fact_205_sup_Oabsorb1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B4: A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/0.47 => ( ( sup_sup @ A @ A3 @ B4 )
% 0.32/0.47 = A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.absorb1
% 0.32/0.47 thf(fact_206_sup__unique,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [F2: A > A > A,X: A,Y: A] :
% 0.32/0.47 ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y3 ) )
% 0.32/0.47 => ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F2 @ X3 @ Y3 ) )
% 0.32/0.47 => ( ! [X3: A,Y3: A,Z: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Z @ X3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
% 0.32/0.47 => ( ( sup_sup @ A @ X @ Y )
% 0.32/0.47 = ( F2 @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_unique
% 0.32/0.47 thf(fact_207_sup_OorderI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,B4: A] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( sup_sup @ A @ A3 @ B4 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ B4 @ A3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.orderI
% 0.32/0.47 thf(fact_208_sup_OorderE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B4: A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/0.47 => ( A3
% 0.32/0.47 = ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.orderE
% 0.32/0.47 thf(fact_209_le__iff__sup,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [X4: A,Y4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ X4 @ Y4 )
% 0.32/0.47 = Y4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_iff_sup
% 0.32/0.47 thf(fact_210_sup__least,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [Y: A,X: A,Z4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Z4 @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z4 ) @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_least
% 0.32/0.47 thf(fact_211_sup__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,C4: A,B4: A,D3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ C4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B4 @ D3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ ( sup_sup @ A @ C4 @ D3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_mono
% 0.32/0.47 thf(fact_212_sup_Omono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [C4: A,A3: A,D3: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ C4 @ A3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ D3 @ B4 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ C4 @ D3 ) @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.mono
% 0.32/0.47 thf(fact_213_le__supI2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,B4: A,A3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ B4 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_supI2
% 0.32/0.47 thf(fact_214_le__supI1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,A3: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ A3 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_supI1
% 0.32/0.47 thf(fact_215_sup__ge2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_ge2
% 0.32/0.47 thf(fact_216_sup__ge1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_ge1
% 0.32/0.47 thf(fact_217_le__supI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,X: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B4 @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_supI
% 0.32/0.47 thf(fact_218_le__supE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A3: A,B4: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ X )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ A3 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ B4 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_supE
% 0.32/0.47 thf(fact_219_inf__sup__ord_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_ord(3)
% 0.32/0.47 thf(fact_220_inf__sup__ord_I4_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_ord(4)
% 0.32/0.47 thf(fact_221_less__supI1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,A3: A,B4: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ A3 )
% 0.32/0.47 => ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_supI1
% 0.32/0.47 thf(fact_222_less__supI2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,B4: A,A3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ B4 )
% 0.32/0.47 => ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_supI2
% 0.32/0.47 thf(fact_223_sup_Ostrict__boundedE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B4: A,C4: A,A3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
% 0.32/0.47 => ~ ( ( ord_less @ A @ B4 @ A3 )
% 0.32/0.47 => ~ ( ord_less @ A @ C4 @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.strict_boundedE
% 0.32/0.47 thf(fact_224_sup_Ostrict__order__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [B: A,A2: A] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 = ( sup_sup @ A @ A2 @ B ) )
% 0.32/0.47 & ( A2 != B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.strict_order_iff
% 0.32/0.47 thf(fact_225_sup_Ostrict__coboundedI1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [C4: A,A3: A,B4: A] :
% 0.32/0.47 ( ( ord_less @ A @ C4 @ A3 )
% 0.32/0.47 => ( ord_less @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.strict_coboundedI1
% 0.32/0.47 thf(fact_226_sup_Ostrict__coboundedI2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [C4: A,B4: A,A3: A] :
% 0.32/0.47 ( ( ord_less @ A @ C4 @ B4 )
% 0.32/0.47 => ( ord_less @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.strict_coboundedI2
% 0.32/0.47 thf(fact_227_predicate1I,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( Q @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.32/0.47
% 0.32/0.47 % predicate1I
% 0.32/0.47 thf(fact_228_pred__subset__eq,axiom,
% 0.32/0.47 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > $o )
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ R )
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ S ) )
% 0.32/0.47 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.32/0.47
% 0.32/0.47 % pred_subset_eq
% 0.32/0.47 thf(fact_229_the__elem__eq,axiom,
% 0.32/0.47 ! [A: $tType,X: A] :
% 0.32/0.47 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/0.47 = X ) ).
% 0.32/0.47
% 0.32/0.47 % the_elem_eq
% 0.32/0.47 thf(fact_230_order__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_refl
% 0.32/0.47 thf(fact_231_sup1CI,axiom,
% 0.32/0.47 ! [A: $tType,B5: A > $o,X: A,A4: A > $o] :
% 0.32/0.47 ( ( ~ ( B5 @ X )
% 0.32/0.47 => ( A4 @ X ) )
% 0.32/0.47 => ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup1CI
% 0.32/0.47 thf(fact_232_bot__apply,axiom,
% 0.32/0.47 ! [C: $tType,D: $tType] :
% 0.32/0.47 ( ( bot @ C )
% 0.32/0.47 => ( ( bot_bot @ ( D > C ) )
% 0.32/0.47 = ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_apply
% 0.32/0.47 thf(fact_233_sup1I2,axiom,
% 0.32/0.47 ! [A: $tType,B5: A > $o,X: A,A4: A > $o] :
% 0.32/0.47 ( ( B5 @ X )
% 0.32/0.47 => ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup1I2
% 0.32/0.47 thf(fact_234_sup1I1,axiom,
% 0.32/0.47 ! [A: $tType,A4: A > $o,X: A,B5: A > $o] :
% 0.32/0.47 ( ( A4 @ X )
% 0.32/0.47 => ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup1I1
% 0.32/0.47 thf(fact_235_sup1E,axiom,
% 0.32/0.47 ! [A: $tType,A4: A > $o,B5: A > $o,X: A] :
% 0.32/0.47 ( ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X )
% 0.32/0.47 => ( ~ ( A4 @ X )
% 0.32/0.47 => ( B5 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup1E
% 0.32/0.47 thf(fact_236_order__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B2: $tType] :
% 0.32/0.47 ( ( ( order @ B2 )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,F2: B2 > A,B4: B2,C4: B2] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ ( F2 @ B4 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B2 @ B4 @ C4 )
% 0.32/0.47 => ( ! [X3: B2,Y3: B2] :
% 0.32/0.47 ( ( ord_less_eq @ B2 @ X3 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ ( F2 @ C4 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_237_order__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType] :
% 0.32/0.47 ( ( ( order @ C )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A3: A,B4: A,F2: A > C,C4: C] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/0.47 => ( ( ord_less_eq @ C @ ( F2 @ B4 ) @ C4 )
% 0.32/0.47 => ( ! [X3: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ C @ ( F2 @ A3 ) @ C4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_238_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: $tType,B2: $tType] :
% 0.32/0.47 ( ( ( ord @ B2 )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A3: A,F2: B2 > A,B4: B2,C4: B2] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( F2 @ B4 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B2 @ B4 @ C4 )
% 0.32/0.47 => ( ! [X3: B2,Y3: B2] :
% 0.32/0.47 ( ( ord_less_eq @ B2 @ X3 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ ( F2 @ C4 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_239_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: $tType,B2: $tType] :
% 0.32/0.47 ( ( ( ord @ B2 )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A3: A,B4: A,F2: A > B2,C4: B2] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/0.47 => ( ( ( F2 @ B4 )
% 0.32/0.47 = C4 )
% 0.32/0.47 => ( ! [X3: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ B2 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ B2 @ ( F2 @ A3 ) @ C4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_240_eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y2: A,Z3: A] : ( Y2 = Z3 ) )
% 0.32/0.47 = ( ^ [X4: A,Y4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.32/0.47 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_241_antisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_242_linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_243_eq__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( X = Y )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_244_le__cases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_245_order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B4: A,C4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B4 @ C4 )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_246_le__cases3,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y @ Z4 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ Z4 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ X @ Z4 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z4 @ Y ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Z4 @ Y )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z4 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z4 @ X ) )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ Z4 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_247_antisym__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [Y: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 = ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y2: A,Z3: A] : ( Y2 = Z3 ) )
% 0.32/0.47 = ( ^ [A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 & ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_249_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A3: A,B4: A,C4: A] :
% 0.32/0.47 ( ( A3 = B4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B4 @ C4 )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_250_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A3: A,B4: A,C4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/0.47 => ( ( B4 = C4 )
% 0.32/0.47 => ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_251_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/0.47 => ( A3 = B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_252_order__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y @ Z4 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_253_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_254_linorder__wlog,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > A > $o,A3: A,B4: A] :
% 0.32/0.47 ( ! [A7: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A7 @ B3 )
% 0.32/0.47 => ( P @ A7 @ B3 ) )
% 0.32/0.47 => ( ! [A7: A,B3: A] :
% 0.32/0.47 ( ( P @ B3 @ A7 )
% 0.32/0.47 => ( P @ A7 @ B3 ) )
% 0.32/0.47 => ( P @ A3 @ B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47
% 0.32/0.47 % Type constructors (33)
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.32/0.47 bounded_lattice @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.32/0.47 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( bounded_lattice @ A9 )
% 0.32/0.47 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( bounded_lattice @ A9 )
% 0.32/0.47 => ( bounde1808546759up_bot @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A9 )
% 0.32/0.47 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( preorder @ A9 )
% 0.32/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( lattice @ A9 )
% 0.32/0.47 => ( lattice @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( order @ A9 )
% 0.32/0.47 => ( order @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( ord @ A9 )
% 0.32/0.47 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( bot @ A9 )
% 0.32/0.47 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.32/0.47 condit1037483654norder @ int ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
% 0.32/0.47 semilattice_sup @ int ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.32/0.47 linordered_idom @ int ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Int_Oint___Orderings_Opreorder_4,axiom,
% 0.32/0.47 preorder @ int ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.32/0.47 linorder @ int ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Int_Oint___Lattices_Olattice_5,axiom,
% 0.32/0.47 lattice @ int ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Int_Oint___Orderings_Oorder_6,axiom,
% 0.32/0.47 order @ int ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Int_Oint___Orderings_Oord_7,axiom,
% 0.32/0.47 ord @ int ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_8,axiom,
% 0.32/0.47 ! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_9,axiom,
% 0.32/0.47 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_10,axiom,
% 0.32/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Olattice_11,axiom,
% 0.32/0.47 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oorder_12,axiom,
% 0.32/0.47 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oord_13,axiom,
% 0.32/0.47 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Obot_14,axiom,
% 0.32/0.47 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_15,axiom,
% 0.32/0.47 bounde1808546759up_bot @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_16,axiom,
% 0.32/0.47 semilattice_sup @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_17,axiom,
% 0.32/0.47 preorder @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_18,axiom,
% 0.32/0.47 linorder @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Olattice_19,axiom,
% 0.32/0.47 lattice @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_20,axiom,
% 0.32/0.47 order @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oord_21,axiom,
% 0.32/0.47 ord @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Obot_22,axiom,
% 0.32/0.47 bot @ $o ).
% 0.32/0.47
% 0.32/0.47 % Conjectures (1)
% 0.32/0.47 thf(conj_0,conjecture,
% 0.32/0.47 ord_less @ int @ ( h @ l ) @ ( h @ ( binary576689334lle_rm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 0.32/1.50 ---- Embedded in THF ---
% 0.32/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.32/1.50 %%% Generated on Tue Feb 24 08:24:04 EST 2026
% 0.32/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.32/1.50 %%% Logic specification used:
% 0.32/1.50 %%% thf(spec, logic, $$dhol).
% 0.32/1.50
% 0.32/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.7sjdMpQ3z4/DTF2DTF18142.p
% 0.32/1.50 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree, type, binary1291135688e_Tree: ($tType > $tType)).
% 0.32/1.50 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.32/1.50 thf(ty_t_Int_Oint, type, int: $tType).
% 0.32/1.50 thf(ty_tf_a, type, a: $tType).
% 0.32/1.50 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde1808546759up_bot: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT, type, binary210054475elle_T: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip, type, binary1746293266le_Tip: (!> [A:$tType]: ((binary1291135688e_Tree @ A)))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert, type, binary1830089824insert: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb, type, binary827270440e_memb: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > $o)))))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oremove, type, binary997842527remove: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Orm, type, binary576689334lle_rm: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > A))))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf, type, binary1653327646_setOf: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (set @ A))))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree, type, binary1610619414edTree: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > $o))))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred, type, binary231205461t_pred: (!> [A:$tType]: (((A > int) > (A > (A > ((binary1291135688e_Tree @ A) > $o))))))).
% 0.32/1.50 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrm, type, binary213313527le_wrm: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A)))))).
% 0.32/1.50 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 0.32/1.50 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 0.32/1.50 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.50 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.32/1.50 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 0.32/1.50 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 0.32/1.50 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.32/1.50 thf(sy_v_h, type, h: (a > int)).
% 0.32/1.50 thf(sy_v_l____, type, l: a).
% 0.32/1.50 thf(sy_v_t1____, type, t1: (binary1291135688e_Tree @ a)).
% 0.32/1.50 thf(sy_v_t2____, type, t2: (binary1291135688e_Tree @ a)).
% 0.32/1.50 thf(sy_v_x____, type, x: a).
% 0.32/1.50 thf(fact_0_hLess, axiom, (((ord_less @ int) @ (h @ l)) @ (h @ (((binary576689334lle_rm @ a) @ h) @ t2)))).
% 0.32/1.50 thf(fact_1_rm__res, axiom, ((((binary576689334lle_rm @ a) @ h) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2)) = (((binary576689334lle_rm @ a) @ h) @ t2))).
% 0.32/1.50 thf(fact_2_t2nTip, axiom, (t2 != (binary1746293266le_Tip @ a))).
% 0.32/1.50 thf(fact_3_Tree_Oinject, axiom, (! [A:$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))))))).
% 0.32/1.50 thf(fact_4_s, axiom, (((binary1610619414edTree @ a) @ h) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2))).
% 0.32/1.50 thf(fact_5_wrm__res, axiom, ((((binary213313527le_wrm @ a) @ h) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2)) = ((((binary210054475elle_T @ a) @ t1) @ x) @ (((binary213313527le_wrm @ a) @ h) @ t2)))).
% 0.32/1.50 thf(fact_6_ldef, axiom, (((member @ a) @ l) @ ((binary1653327646_setOf @ a) @ (((binary213313527le_wrm @ a) @ h) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2))))).
% 0.32/1.50 thf(fact_7_s1, axiom, (((binary1610619414edTree @ a) @ h) @ t1)).
% 0.32/1.50 thf(fact_8_s2, axiom, (((binary1610619414edTree @ a) @ h) @ t2)).
% 0.32/1.50 thf(fact_9_binsert_Osimps_I2_J, axiom, (! [A:$tType,H:(A > int),E:A,X:A,T1:(binary1291135688e_Tree @ A),T2:(binary1291135688e_Tree @ A)]: ((((((ord_less @ int) @ (H @ E)) @ (H @ X)) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ ((((binary1830089824insert @ A) @ H) @ E) @ T1)) @ X) @ T2))) & ((~ ((((ord_less @ int) @ (H @ E)) @ (H @ X)))) => (((((ord_less @ int) @ (H @ X)) @ (H @ E)) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ X) @ ((((binary1830089824insert @ A) @ H) @ E) @ T2)))) & ((~ ((((ord_less @ int) @ (H @ X)) @ (H @ E)))) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ E) @ T2))))))))).
% 0.32/1.50 thf(fact_10_rm_Osimps, axiom, (! [A:$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))))))).
% 0.32/1.50 thf(fact_11_minf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F:D]: ((? [Z:C]: ((! [X2:C]: (((((ord_less @ C) @ X2) @ Z) => (F = F)))))))))))).
% 0.32/1.50 thf(fact_12_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (~ ((((ord_less @ A) @ T) @ X2)))))))))))))).
% 0.32/1.50 thf(fact_13_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (((ord_less @ A) @ X2) @ T)))))))))))).
% 0.32/1.50 thf(fact_14_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (X2 != T)))))))))))).
% 0.32/1.50 thf(fact_15_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (X2 != T)))))))))))).
% 0.32/1.50 thf(fact_16_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z2) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z2) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (((P @ X2) | (Q @ X2)) = ((P2 @ X2) | (Q2 @ X2)))))))))))))))).
% 0.32/1.50 thf(fact_17_sortedTree_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int)]: ((((binary1610619414edTree @ A) @ H) @ (binary1746293266le_Tip @ A))))).
% 0.32/1.50 thf(fact_18_wrm__sort, axiom, (! [A:$tType,T:(binary1291135688e_Tree @ A),H:(A > int)]: ((((T != (binary1746293266le_Tip @ A)) & (((binary1610619414edTree @ A) @ H) @ T)) => (((binary1610619414edTree @ A) @ H) @ (((binary213313527le_wrm @ A) @ H) @ T)))))).
% 0.32/1.50 thf(fact_19_binsert__sorted, axiom, (! [A:$tType,H:(A > int),T:(binary1291135688e_Tree @ A),X:A]: (((((binary1610619414edTree @ A) @ H) @ T) => (((binary1610619414edTree @ A) @ H) @ ((((binary1830089824insert @ A) @ H) @ X) @ T)))))).
% 0.32/1.50 thf(fact_20_rm__set, axiom, (! [A:$tType,T:(binary1291135688e_Tree @ A),H:(A > int)]: ((((T != (binary1746293266le_Tip @ A)) & (((binary1610619414edTree @ A) @ H) @ T)) => (((member @ A) @ (((binary576689334lle_rm @ A) @ H) @ T)) @ ((binary1653327646_setOf @ A) @ T)))))).
% 0.32/1.50 thf(fact_21_binsert_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int),E:A]: ((((((binary1830089824insert @ A) @ H) @ E) @ (binary1746293266le_Tip @ A)) = ((((binary210054475elle_T @ A) @ (binary1746293266le_Tip @ A)) @ E) @ (binary1746293266le_Tip @ A)))))).
% 0.32/1.50 thf(fact_22_wrm_Osimps, axiom, (! [A:$tType,T2:(binary1291135688e_Tree @ A),H:(A > int),T1:(binary1291135688e_Tree @ A),X:A]: ((((T2 = (binary1746293266le_Tip @ A)) => ((((binary213313527le_wrm @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = T1)) & ((T2 != (binary1746293266le_Tip @ A)) => ((((binary213313527le_wrm @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ X) @ (((binary213313527le_wrm @ A) @ H) @ T2)))))))).
% 0.32/1.50 thf(fact_23_sortedTree_Osimps_I2_J, axiom, (! [A:$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)))))))).
% 0.32/1.50 thf(fact_24_Tree_Oexhaust, axiom, (! [A:$tType,Y:(binary1291135688e_Tree @ A)]: (((Y != (binary1746293266le_Tip @ A)) => (~ ((! [X212:(binary1291135688e_Tree @ A),X222:A,X232:(binary1291135688e_Tree @ A)]: ((Y != ((((binary210054475elle_T @ A) @ X212) @ X222) @ X232)))))))))).
% 0.32/1.50 thf(fact_25_Tree_Oinduct, axiom, (! [A:$tType,P:((binary1291135688e_Tree @ A) > $o),Tree:(binary1291135688e_Tree @ A)]: (((P @ (binary1746293266le_Tip @ A)) => ((! [X1:(binary1291135688e_Tree @ A),X24:A,X32:(binary1291135688e_Tree @ A)]: (((P @ X1) => ((P @ X32) => (P @ ((((binary210054475elle_T @ A) @ X1) @ X24) @ X32)))))) => (P @ Tree)))))).
% 0.32/1.50 thf(fact_26_Tree_Odistinct_I1_J, axiom, (! [A:$tType,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: (((binary1746293266le_Tip @ A) != ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23))))).
% 0.32/1.50 thf(fact_27_sortLemmaR, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X:A,T2:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) => (((binary1610619414edTree @ A) @ H) @ T2))))).
% 0.32/1.50 thf(fact_28_sortLemmaL, axiom, (! [A:$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))))).
% 0.32/1.50 thf(fact_29_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((P @ X2) & (Q @ X2)) = ((P2 @ X2) & (Q2 @ X2)))))))))))))))).
% 0.32/1.50 thf(fact_30_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((P @ X2) | (Q @ X2)) = ((P2 @ X2) | (Q2 @ X2)))))))))))))))).
% 0.32/1.50 thf(fact_31_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (X2 != T)))))))))))).
% 0.32/1.50 thf(fact_32_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (X2 != T)))))))))))).
% 0.32/1.50 thf(fact_33_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (~ ((((ord_less @ A) @ X2) @ T)))))))))))))).
% 0.32/1.50 thf(fact_34_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((ord_less @ A) @ T) @ X2)))))))))))).
% 0.32/1.50 thf(fact_35_pinf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F:D]: ((? [Z:C]: ((! [X2:C]: (((((ord_less @ C) @ Z) @ X2) => (F = F)))))))))))).
% 0.32/1.50 thf(fact_36_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z2) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z2) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (((P @ X2) & (Q @ X2)) = ((P2 @ X2) & (Q2 @ X2)))))))))))))))).
% 0.32/1.50 thf(fact_37_h1, axiom, (((t1 != (binary1746293266le_Tip @ a)) & (((binary1610619414edTree @ a) @ h) @ t1)) => (! [X2:a]: (((((member @ a) @ X2) @ ((binary1653327646_setOf @ a) @ (((binary213313527le_wrm @ a) @ h) @ t1))) => (((ord_less @ int) @ (h @ X2)) @ (h @ (((binary576689334lle_rm @ a) @ h) @ t1)))))))).
% 0.32/1.50 thf(fact_38_h2, axiom, (((t2 != (binary1746293266le_Tip @ a)) & (((binary1610619414edTree @ a) @ h) @ t2)) => (! [X2:a]: (((((member @ a) @ X2) @ ((binary1653327646_setOf @ a) @ (((binary213313527le_wrm @ a) @ h) @ t2))) => (((ord_less @ int) @ (h @ X2)) @ (h @ (((binary576689334lle_rm @ a) @ h) @ t2)))))))).
% 0.32/1.50 thf(fact_39_memb__spec, axiom, (! [A:$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))))))).
% 0.32/1.50 thf(fact_40_sorted__distinct__pred__def, axiom, (! [A:$tType]: (((binary231205461t_pred @ A) = (^ [H2:(A > int),A2:A,B:A,T3:(binary1291135688e_Tree @ A)]: ((((((binary1610619414edTree @ A) @ H2) @ T3) & ((((member @ A) @ A2) @ ((binary1653327646_setOf @ A) @ T3)) & ((((member @ A) @ B) @ ((binary1653327646_setOf @ A) @ T3)) & ((H2 @ A2) = (H2 @ B))))) => (A2 = B)))))))).
% 0.32/1.50 thf(fact_41_l__scope, axiom, (((member @ a) @ l) @ (((sup_sup @ (set @ a)) @ (((sup_sup @ (set @ a)) @ (((insert @ a) @ x) @ (bot_bot @ (set @ a)))) @ ((binary1653327646_setOf @ a) @ t1))) @ ((binary1653327646_setOf @ a) @ (((binary213313527le_wrm @ a) @ h) @ t2))))).
% 0.32/1.50 thf(fact_42_wrm__set1, axiom, (! [A:$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)))))).
% 0.32/1.50 thf(fact_43_remove_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int),E:A]: ((((((binary997842527remove @ A) @ H) @ E) @ (binary1746293266le_Tip @ A)) = (binary1746293266le_Tip @ A))))).
% 0.32/1.50 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 0.32/1.50 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A4)))) = A4)))).
% 0.32/1.50 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.32/1.50 thf(fact_47_ext, axiom, (! [B2:$tType,A:$tType,F2:(A > B2),G:(A > B2)]: (((! [X3:A]: (((F2 @ X3) = (G @ X3)))) => (F2 = G))))).
% 0.32/1.50 thf(fact_48_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A3:A]: ((? [B3:A]: (((((ord_less @ A) @ A3) @ B3) | (((ord_less @ A) @ B3) @ A3)))))))))).
% 0.32/1.50 thf(fact_49_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 0.32/1.50 thf(fact_50_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A3:A,B4:A,P:(A > $o)]: (((((ord_less @ A) @ A3) @ B4) => ((P @ A3) => ((~ ((P @ B4))) => (? [C2:A]: (((((ord_less_eq @ A) @ A3) @ C2) & ((((ord_less_eq @ A) @ C2) @ B4) & ((! [X2:A]: ((((((ord_less_eq @ A) @ A3) @ X2) & (((ord_less @ A) @ X2) @ C2)) => (P @ X2)))) & (! [D2:A]: (((! [X3:A]: ((((((ord_less_eq @ A) @ A3) @ X3) & (((ord_less @ A) @ X3) @ D2)) => (P @ X3)))) => (((ord_less_eq @ A) @ D2) @ C2)))))))))))))))))).
% 0.32/1.50 thf(fact_51_setOf_Osimps_I2_J, axiom, (! [A:$tType,T1:(binary1291135688e_Tree @ A),X:A,T2:(binary1291135688e_Tree @ A)]: ((((binary1653327646_setOf @ A) @ ((((binary210054475elle_T @ A) @ T1) @ X) @ T2)) = (((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ ((binary1653327646_setOf @ A) @ T1)) @ ((binary1653327646_setOf @ A) @ T2))) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))).
% 0.32/1.50 thf(fact_52_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (~ ((((ord_less_eq @ A) @ T) @ X2)))))))))))))).
% 0.32/1.50 thf(fact_53_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => (((ord_less_eq @ A) @ X2) @ T)))))))))))).
% 0.32/1.50 thf(fact_54_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((ord_less_eq @ A) @ T) @ X2)))))))))))).
% 0.32/1.50 thf(fact_55_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ T)))))))))))))).
% 0.32/1.50 thf(fact_56_sorted__distinct, axiom, (! [A:$tType,H:(A > int),A3:A,B4:A,T:(binary1291135688e_Tree @ A)]: ((((((binary231205461t_pred @ A) @ H) @ A3) @ B4) @ T)))).
% 0.32/1.50 thf(fact_57_setOf_Osimps_I1_J, axiom, (! [A:$tType]: ((((binary1653327646_setOf @ A) @ (binary1746293266le_Tip @ A)) = (bot_bot @ (set @ A)))))).
% 0.32/1.50 thf(fact_58_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A3:A,A4:(set @ A),B4:A]: ((((((insert @ A) @ A3) @ A4) = (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) = ((A3 = B4) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))))))).
% 0.32/1.50 thf(fact_59_singleton__insert__inj__eq, axiom, (! [A:$tType,B4:A,A3:A,A4:(set @ A)]: ((((((insert @ A) @ B4) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A3) @ A4)) = ((A3 = B4) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))))))).
% 0.32/1.50 thf(fact_60_singleton__conv2, axiom, (! [A:$tType,A3:A]: ((((collect @ A) @ ((^ [Y2:A,Z3:A]: ((Y2 = Z3))) @ A3)) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 0.32/1.50 thf(fact_61_singleton__conv, axiom, (! [A:$tType,A3:A]: ((((collect @ A) @ (^ [X4:A]: ((X4 = A3)))) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 0.32/1.50 thf(fact_62_Un__insert__right, axiom, (! [A:$tType,A4:(set @ A),A3:A,B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((insert @ A) @ A3) @ B5)) = (((insert @ A) @ A3) @ (((sup_sup @ (set @ A)) @ A4) @ B5)))))).
% 0.32/1.50 thf(fact_63_Un__insert__left, axiom, (! [A:$tType,A3:A,B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A3) @ B5)) @ C3) = (((insert @ A) @ A3) @ (((sup_sup @ (set @ A)) @ B5) @ C3)))))).
% 0.32/1.50 thf(fact_64_Un__subset__iff, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) @ C3) = ((((ord_less_eq @ (set @ A)) @ A4) @ C3) & (((ord_less_eq @ (set @ A)) @ B5) @ C3)))))).
% 0.32/1.50 thf(fact_65_Un__empty, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A4) @ B5) = (bot_bot @ (set @ A))) = ((A4 = (bot_bot @ (set @ A))) & (B5 = (bot_bot @ (set @ A)))))))).
% 0.32/1.50 thf(fact_66_empty__iff, axiom, (! [A:$tType,C4:A]: ((~ ((((member @ A) @ C4) @ (bot_bot @ (set @ A)))))))).
% 0.32/1.50 thf(fact_67_all__not__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A4))))) = (A4 = (bot_bot @ (set @ A))))))).
% 0.32/1.50 thf(fact_68_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 0.32/1.50 thf(fact_69_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 0.32/1.50 thf(fact_70_subsetI, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A4) => (((member @ A) @ X3) @ B5)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B5))))).
% 0.32/1.50 thf(fact_71_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A4) => (A4 = B5)))))).
% 0.32/1.50 thf(fact_72_psubsetI, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((A4 != B5) => (((ord_less @ (set @ A)) @ A4) @ B5)))))).
% 0.32/1.50 thf(fact_73_insertCI, axiom, (! [A:$tType,A3:A,B5:(set @ A),B4:A]: ((((~ ((((member @ A) @ A3) @ B5))) => (A3 = B4)) => (((member @ A) @ A3) @ (((insert @ A) @ B4) @ B5)))))).
% 0.32/1.50 thf(fact_74_insert__iff, axiom, (! [A:$tType,A3:A,B4:A,A4:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B4) @ A4)) = ((A3 = B4) | (((member @ A) @ A3) @ A4)))))).
% 0.32/1.50 thf(fact_75_insert__absorb2, axiom, (! [A:$tType,X:A,A4:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A4)) = (((insert @ A) @ X) @ A4))))).
% 0.32/1.50 thf(fact_76_UnCI, axiom, (! [A:$tType,C4:A,B5:(set @ A),A4:(set @ A)]: ((((~ ((((member @ A) @ C4) @ B5))) => (((member @ A) @ C4) @ A4)) => (((member @ A) @ C4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)))))).
% 0.32/1.50 thf(fact_77_Un__iff, axiom, (! [A:$tType,C4:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) = ((((member @ A) @ C4) @ A4) | (((member @ A) @ C4) @ B5)))))).
% 0.32/1.50 thf(fact_78_subset__empty, axiom, (! [A:$tType,A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = (A4 = (bot_bot @ (set @ A))))))).
% 0.32/1.50 thf(fact_79_empty__subsetI, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A4)))).
% 0.32/1.50 thf(fact_80_singletonI, axiom, (! [A:$tType,A3:A]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 0.32/1.50 thf(fact_81_insert__subset, axiom, (! [A:$tType,X:A,A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A4)) @ B5) = ((((member @ A) @ X) @ B5) & (((ord_less_eq @ (set @ A)) @ A4) @ B5)))))).
% 0.32/1.50 thf(fact_82_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A5)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ B6)))))))))).
% 0.32/1.50 thf(fact_83_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 0.32/1.50 thf(fact_84_equals0D, axiom, (! [A:$tType,A4:(set @ A),A3:A]: (((A4 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A4))))))).
% 0.32/1.50 thf(fact_85_equals0I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A4))))) => (A4 = (bot_bot @ (set @ A))))))).
% 0.32/1.50 thf(fact_86_ex__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ A4))) = (A4 != (bot_bot @ (set @ A))))))).
% 0.32/1.50 thf(fact_87_not__psubset__empty, axiom, (! [A:$tType,A4:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A4) @ (bot_bot @ (set @ A)))))))).
% 0.32/1.50 thf(fact_88_in__mono, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B5)))))).
% 0.32/1.50 thf(fact_89_subsetD, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C4:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((member @ A) @ C4) @ A4) => (((member @ A) @ C4) @ B5)))))).
% 0.32/1.50 thf(fact_90_equalityE, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((A4 = B5) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A4)))))))))).
% 0.32/1.50 thf(fact_91_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B6)))))))))).
% 0.32/1.50 thf(fact_92_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((A4 = B5) => (((ord_less_eq @ (set @ A)) @ A4) @ B5))))).
% 0.32/1.50 thf(fact_93_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((A4 = B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A4))))).
% 0.32/1.50 thf(fact_94_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A5) => (((member @ A) @ T3) @ B6)))))))))).
% 0.32/1.50 thf(fact_95_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 0.32/1.50 thf(fact_96_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 0.32/1.50 thf(fact_97_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C3) => (((ord_less_eq @ (set @ A)) @ A4) @ C3)))))).
% 0.32/1.50 thf(fact_98_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z3:(set @ A)]: ((Y2 = Z3))) = (^ [A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A5)))))))).
% 0.32/1.50 thf(fact_99_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 0.32/1.50 thf(fact_100_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B6) | (A5 = B6)))))))).
% 0.32/1.50 thf(fact_101_subset__psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((ord_less @ (set @ A)) @ B5) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 0.32/1.50 thf(fact_102_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) & (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A5)))))))))).
% 0.32/1.50 thf(fact_103_psubset__subset__trans, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 0.32/1.50 thf(fact_104_psubset__imp__subset, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B5) => (((ord_less_eq @ (set @ A)) @ A4) @ B5))))).
% 0.32/1.50 thf(fact_105_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) & (A5 != B6)))))))).
% 0.32/1.50 thf(fact_106_psubsetE, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B5) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A4)))))))).
% 0.32/1.50 thf(fact_107_insertE, axiom, (! [A:$tType,A3:A,B4:A,A4:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B4) @ A4)) => ((A3 != B4) => (((member @ A) @ A3) @ A4)))))).
% 0.32/1.50 thf(fact_108_insertI1, axiom, (! [A:$tType,A3:A,B5:(set @ A)]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ B5))))).
% 0.32/1.50 thf(fact_109_insertI2, axiom, (! [A:$tType,A3:A,B5:(set @ A),B4:A]: (((((member @ A) @ A3) @ B5) => (((member @ A) @ A3) @ (((insert @ A) @ B4) @ B5)))))).
% 0.32/1.50 thf(fact_110_Set_Oset__insert, axiom, (! [A:$tType,X:A,A4:(set @ A)]: (((((member @ A) @ X) @ A4) => (~ ((! [B7:(set @ A)]: (((A4 = (((insert @ A) @ X) @ B7)) => (((member @ A) @ X) @ B7)))))))))).
% 0.32/1.50 thf(fact_111_insert__ident, axiom, (! [A:$tType,X:A,A4:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X) @ A4))) => ((~ ((((member @ A) @ X) @ B5))) => (((((insert @ A) @ X) @ A4) = (((insert @ A) @ X) @ B5)) = (A4 = B5))))))).
% 0.32/1.50 thf(fact_112_insert__absorb, axiom, (! [A:$tType,A3:A,A4:(set @ A)]: (((((member @ A) @ A3) @ A4) => ((((insert @ A) @ A3) @ A4) = A4))))).
% 0.32/1.50 thf(fact_113_insert__eq__iff, axiom, (! [A:$tType,A3:A,A4:(set @ A),B4:A,B5:(set @ A)]: (((~ ((((member @ A) @ A3) @ A4))) => ((~ ((((member @ A) @ B4) @ B5))) => (((((insert @ A) @ A3) @ A4) = (((insert @ A) @ B4) @ B5)) = (((A3 = B4) => (A4 = B5)) & ((A3 != B4) => (? [C5:(set @ A)]: (((A4 = (((insert @ A) @ B4) @ C5)) & ((~ ((((member @ A) @ B4) @ C5))) & ((B5 = (((insert @ A) @ A3) @ C5)) & (~ ((((member @ A) @ A3) @ C5)))))))))))))))).
% 0.32/1.50 thf(fact_114_insert__commute, axiom, (! [A:$tType,X:A,Y:A,A4:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ Y) @ A4)) = (((insert @ A) @ Y) @ (((insert @ A) @ X) @ A4)))))).
% 0.32/1.50 thf(fact_115_mk__disjoint__insert, axiom, (! [A:$tType,A3:A,A4:(set @ A)]: (((((member @ A) @ A3) @ A4) => (? [B7:(set @ A)]: (((A4 = (((insert @ A) @ A3) @ B7)) & (~ ((((member @ A) @ A3) @ B7)))))))))).
% 0.32/1.50 thf(fact_116_UnE, axiom, (! [A:$tType,C4:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) => ((~ ((((member @ A) @ C4) @ A4))) => (((member @ A) @ C4) @ B5)))))).
% 0.32/1.50 thf(fact_117_UnI1, axiom, (! [A:$tType,C4:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C4) @ A4) => (((member @ A) @ C4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)))))).
% 0.32/1.50 thf(fact_118_UnI2, axiom, (! [A:$tType,C4:A,B5:(set @ A),A4:(set @ A)]: (((((member @ A) @ C4) @ B5) => (((member @ A) @ C4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)))))).
% 0.32/1.50 thf(fact_119_bex__Un, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) & (P @ X4)))) = ((? [X4:A]: (((((member @ A) @ X4) @ A4) & (P @ X4)))) | (? [X4:A]: (((((member @ A) @ X4) @ B5) & (P @ X4))))))))).
% 0.32/1.50 thf(fact_120_ball__Un, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),P:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) => (P @ X4)))) = ((! [X4:A]: (((((member @ A) @ X4) @ A4) => (P @ X4)))) & (! [X4:A]: (((((member @ A) @ X4) @ B5) => (P @ X4))))))))).
% 0.32/1.50 thf(fact_121_Un__assoc, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) @ C3) = (((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B5) @ C3)))))).
% 0.32/1.50 thf(fact_122_Un__absorb, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ A4) = A4)))).
% 0.32/1.50 thf(fact_123_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((((sup_sup @ (set @ A)) @ B6) @ A5))))))).
% 0.32/1.50 thf(fact_124_Un__left__absorb, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) = (((sup_sup @ (set @ A)) @ A4) @ B5))))).
% 0.32/1.50 thf(fact_125_Un__left__commute, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B5) @ C3)) = (((sup_sup @ (set @ A)) @ B5) @ (((sup_sup @ (set @ A)) @ A4) @ C3)))))).
% 0.32/1.50 thf(fact_126_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X4:A]: ($false))))))).
% 0.32/1.50 thf(fact_127_Collect__subset, axiom, (! [A:$tType,A4:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A4) & (P @ X4)))))) @ A4)))).
% 0.32/1.50 thf(fact_128_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A2:A,B6:(set @ A)]: (((collect @ A) @ (^ [X4:A]: (((X4 = A2) | (((member @ A) @ X4) @ B6))))))))))).
% 0.32/1.50 thf(fact_129_insert__Collect, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((insert @ A) @ A3) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U:A]: (((U != A3) => (P @ U))))))))).
% 0.32/1.50 thf(fact_130_Un__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A5) | (((member @ A) @ X4) @ B6))))))))))).
% 0.32/1.50 thf(fact_131_Collect__disj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X4:A]: (((P @ X4) | (Q @ X4))))) = (((sup_sup @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 0.32/1.50 thf(fact_132_singletonD, axiom, (! [A:$tType,B4:A,A3:A]: (((((member @ A) @ B4) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) => (B4 = A3))))).
% 0.32/1.50 thf(fact_133_singleton__iff, axiom, (! [A:$tType,B4:A,A3:A]: (((((member @ A) @ B4) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = (B4 = A3))))).
% 0.32/1.50 thf(fact_134_doubleton__eq__iff, axiom, (! [A:$tType,A3:A,B4:A,C4:A,D3:A]: ((((((insert @ A) @ A3) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C4) @ (((insert @ A) @ D3) @ (bot_bot @ (set @ A))))) = (((A3 = C4) & (B4 = D3)) | ((A3 = D3) & (B4 = C4))))))).
% 0.32/1.50 thf(fact_135_insert__not__empty, axiom, (! [A:$tType,A3:A,A4:(set @ A)]: (((((insert @ A) @ A3) @ A4) != (bot_bot @ (set @ A)))))).
% 0.32/1.50 thf(fact_136_singleton__inject, axiom, (! [A:$tType,A3:A,B4:A]: ((((((insert @ A) @ A3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) => (A3 = B4))))).
% 0.32/1.50 thf(fact_137_insert__mono, axiom, (! [A:$tType,C3:(set @ A),D4:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ C3) @ D4) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A3) @ C3)) @ (((insert @ A) @ A3) @ D4)))))).
% 0.32/1.50 thf(fact_138_subset__insert, axiom, (! [A:$tType,X:A,A4:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X) @ A4))) => ((((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ B5)) = (((ord_less_eq @ (set @ A)) @ A4) @ B5)))))).
% 0.32/1.50 thf(fact_139_subset__insertI, axiom, (! [A:$tType,B5:(set @ A),A3:A]: ((((ord_less_eq @ (set @ A)) @ B5) @ (((insert @ A) @ A3) @ B5))))).
% 0.32/1.50 thf(fact_140_subset__insertI2, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),B4:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B4) @ B5)))))).
% 0.32/1.50 thf(fact_141_Un__empty__left, axiom, (! [A:$tType,B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B5) = B5)))).
% 0.32/1.50 thf(fact_142_Un__empty__right, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = A4)))).
% 0.32/1.50 thf(fact_143_Un__mono, axiom, (! [A:$tType,A4:(set @ A),C3:(set @ A),B5:(set @ A),D4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ C3) => ((((ord_less_eq @ (set @ A)) @ B5) @ D4) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) @ (((sup_sup @ (set @ A)) @ C3) @ D4))))))).
% 0.32/1.50 thf(fact_144_Un__least, axiom, (! [A:$tType,A4:(set @ A),C3:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ C3) => ((((ord_less_eq @ (set @ A)) @ B5) @ C3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) @ C3)))))).
% 0.32/1.50 thf(fact_145_Un__upper1, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ A4) @ B5))))).
% 0.32/1.50 thf(fact_146_Un__upper2, axiom, (! [A:$tType,B5:(set @ A),A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B5) @ (((sup_sup @ (set @ A)) @ A4) @ B5))))).
% 0.32/1.50 thf(fact_147_Un__absorb1, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((sup_sup @ (set @ A)) @ A4) @ B5) = B5))))).
% 0.32/1.50 thf(fact_148_Un__absorb2, axiom, (! [A:$tType,B5:(set @ A),A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B5) @ A4) => ((((sup_sup @ (set @ A)) @ A4) @ B5) = A4))))).
% 0.32/1.50 thf(fact_149_subset__UnE, axiom, (! [A:$tType,C3:(set @ A),A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C3) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) => (~ ((! [A6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ A4) => (! [B8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B8) @ B5) => (C3 != (((sup_sup @ (set @ A)) @ A6) @ B8)))))))))))))).
% 0.32/1.50 thf(fact_150_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A5) @ B6) = B6))))))).
% 0.32/1.50 thf(fact_151_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A3:A]: ((((P @ A3) => (((collect @ A) @ (^ [X4:A]: (((X4 = A3) & (P @ X4))))) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A3))) => (((collect @ A) @ (^ [X4:A]: (((X4 = A3) & (P @ X4))))) = (bot_bot @ (set @ A)))))))).
% 0.32/1.50 thf(fact_152_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A3:A]: ((((P @ A3) => (((collect @ A) @ (^ [X4:A]: (((A3 = X4) & (P @ X4))))) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A3))) => (((collect @ A) @ (^ [X4:A]: (((A3 = X4) & (P @ X4))))) = (bot_bot @ (set @ A)))))))).
% 0.32/1.50 thf(fact_153_insert__def, axiom, (! [A:$tType]: (((insert @ A) = (^ [A2:A]: (((sup_sup @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: ((X4 = A2))))))))))).
% 0.32/1.50 thf(fact_154_subset__singletonD, axiom, (! [A:$tType,A4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) => ((A4 = (bot_bot @ (set @ A))) | (A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 0.32/1.50 thf(fact_155_subset__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ X5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((X5 = (bot_bot @ (set @ A))) | (X5 = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))).
% 0.32/1.50 thf(fact_156_insert__is__Un, axiom, (! [A:$tType]: (((insert @ A) = (^ [A2:A]: (((sup_sup @ (set @ A)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))))).
% 0.32/1.50 thf(fact_157_Un__singleton__iff, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),X:A]: ((((((sup_sup @ (set @ A)) @ A4) @ B5) = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (((A4 = (bot_bot @ (set @ A))) & (B5 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) | (((A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) & (B5 = (bot_bot @ (set @ A)))) | ((A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) & (B5 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))).
% 0.32/1.50 thf(fact_158_singleton__Un__iff, axiom, (! [A:$tType,X:A,A4:(set @ A),B5:(set @ A)]: ((((((insert @ A) @ X) @ (bot_bot @ (set @ A))) = (((sup_sup @ (set @ A)) @ A4) @ B5)) = (((A4 = (bot_bot @ (set @ A))) & (B5 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) | (((A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) & (B5 = (bot_bot @ (set @ A)))) | ((A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) & (B5 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))).
% 0.32/1.50 thf(fact_159_sup__bot__left, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X) = X))))))).
% 0.32/1.50 thf(fact_160_sup__bot__right, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ (bot_bot @ A)) = X))))))).
% 0.32/1.50 thf(fact_161_bot__eq__sup__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X:A,Y:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X) @ Y)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 0.32/1.50 thf(fact_162_sup__eq__bot__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X:A,Y:A]: ((((((sup_sup @ A) @ X) @ Y) = (bot_bot @ A)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 0.32/1.50 thf(fact_163_sup__bot_Oeq__neutr__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A3:A,B4:A]: ((((((sup_sup @ A) @ A3) @ B4) = (bot_bot @ A)) = ((A3 = (bot_bot @ A)) & (B4 = (bot_bot @ A)))))))))).
% 0.32/1.50 thf(fact_164_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A3:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A3) = A3))))))).
% 0.32/1.50 thf(fact_165_sup__bot_Oneutr__eq__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A3:A,B4:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ A3) @ B4)) = ((A3 = (bot_bot @ A)) & (B4 = (bot_bot @ A)))))))))).
% 0.32/1.50 thf(fact_166_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B4:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B4)) @ B4) = (((sup_sup @ A) @ A3) @ B4)))))))).
% 0.32/1.50 thf(fact_167_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 0.32/1.50 thf(fact_168_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B4:A]: (((((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B4)) = (((sup_sup @ A) @ A3) @ B4)))))))).
% 0.32/1.50 thf(fact_169_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 0.32/1.50 thf(fact_170_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ A3) = A3))))))).
% 0.32/1.50 thf(fact_171_sup__apply, axiom, (! [B2:$tType,A:$tType]: (((semilattice_sup @ B2) => ((sup_sup @ (A > B2)) = (^ [F3:(A > B2),G2:(A > B2),X4:A]: ((((sup_sup @ B2) @ (F3 @ X4)) @ (G2 @ X4))))))))).
% 0.32/1.50 thf(fact_172_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,C4:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B4) @ C4)) @ A3) = ((((ord_less_eq @ A) @ B4) @ A3) & (((ord_less_eq @ A) @ C4) @ A3))))))))).
% 0.32/1.50 thf(fact_173_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A,Z4:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z4) = ((((ord_less_eq @ A) @ X) @ Z4) & (((ord_less_eq @ A) @ Y) @ Z4))))))))).
% 0.32/1.50 thf(fact_174_sup__bot_Oright__neutral, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ (bot_bot @ A)) = A3))))))).
% 0.32/1.50 thf(fact_175_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A5)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ B6)))))))))).
% 0.32/1.50 thf(fact_176_psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B5) => ((((ord_less @ (set @ A)) @ B5) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 0.32/1.50 thf(fact_177_psubsetD, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C4:A]: (((((ord_less @ (set @ A)) @ A4) @ B5) => ((((member @ A) @ C4) @ A4) => (((member @ A) @ C4) @ B5)))))).
% 0.32/1.50 thf(fact_178_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 0.32/1.50 thf(fact_179_sup__set__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((sup_sup @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A5)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ B6))))))))))).
% 0.32/1.50 thf(fact_180_sup__left__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A,Z4:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z4)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z4))))))))).
% 0.32/1.50 thf(fact_181_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,A3:A,C4:A]: (((((sup_sup @ A) @ B4) @ (((sup_sup @ A) @ A3) @ C4)) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B4) @ C4))))))))).
% 0.32/1.50 thf(fact_182_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X4:A,Y4:A]: ((((sup_sup @ A) @ Y4) @ X4)))))))).
% 0.32/1.50 thf(fact_183_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A2:A,B:A]: ((((sup_sup @ A) @ B) @ A2)))))))).
% 0.32/1.50 thf(fact_184_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A,Z4:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z4) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z4))))))))).
% 0.32/1.50 thf(fact_185_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B4:A,C4:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B4)) @ C4) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B4) @ C4))))))))).
% 0.32/1.50 thf(fact_186_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B5:A,K:A,B4:A,A3:A]: (((B5 = (((sup_sup @ A) @ K) @ B4)) => ((((sup_sup @ A) @ A3) @ B5) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A3) @ B4)))))))))).
% 0.32/1.50 thf(fact_187_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,K:A,A3:A,B4:A]: (((A4 = (((sup_sup @ A) @ K) @ A3)) => ((((sup_sup @ A) @ A4) @ B4) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A3) @ B4)))))))))).
% 0.32/1.50 thf(fact_188_sup__fun__def, axiom, (! [B2:$tType,A:$tType]: (((semilattice_sup @ B2) => ((sup_sup @ (A > B2)) = (^ [F3:(A > B2),G2:(A > B2),X4:A]: ((((sup_sup @ B2) @ (F3 @ X4)) @ (G2 @ X4))))))))).
% 0.32/1.50 thf(fact_189_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X4:A,Y4:A]: ((((sup_sup @ A) @ Y4) @ X4)))))))).
% 0.32/1.50 thf(fact_190_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z4:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z4) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z4))))))))).
% 0.32/1.50 thf(fact_191_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z4:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z4)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z4))))))))).
% 0.32/1.50 thf(fact_192_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 0.32/1.50 thf(fact_193_sup_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C4:A,B4:A,A3:A]: (((((ord_less_eq @ A) @ C4) @ B4) => (((ord_less_eq @ A) @ C4) @ (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_194_sup_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C4:A,A3:A,B4:A]: (((((ord_less_eq @ A) @ C4) @ A3) => (((ord_less_eq @ A) @ C4) @ (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_195_sup_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [A2:A,B:A]: (((((sup_sup @ A) @ A2) @ B) = B)))))))).
% 0.32/1.50 thf(fact_196_sup_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B:A,A2:A]: (((((sup_sup @ A) @ A2) @ B) = A2)))))))).
% 0.32/1.50 thf(fact_197_sup_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,A3:A]: ((((ord_less_eq @ A) @ B4) @ (((sup_sup @ A) @ A3) @ B4)))))))).
% 0.32/1.50 thf(fact_198_sup_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B4:A]: ((((ord_less_eq @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B4)))))))).
% 0.32/1.50 thf(fact_199_sup_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B:A,A2:A]: ((A2 = (((sup_sup @ A) @ A2) @ B))))))))).
% 0.32/1.50 thf(fact_200_sup_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,A3:A,C4:A]: (((((ord_less_eq @ A) @ B4) @ A3) => ((((ord_less_eq @ A) @ C4) @ A3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B4) @ C4)) @ A3))))))))).
% 0.32/1.50 thf(fact_201_sup_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,C4:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B4) @ C4)) @ A3) => (~ (((((ord_less_eq @ A) @ B4) @ A3) => (~ ((((ord_less_eq @ A) @ C4) @ A3))))))))))))).
% 0.32/1.50 thf(fact_202_sup__absorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((sup_sup @ A) @ X) @ Y) = Y)))))))).
% 0.32/1.50 thf(fact_203_sup__absorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((sup_sup @ A) @ X) @ Y) = X)))))))).
% 0.32/1.50 thf(fact_204_sup_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B4:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((((sup_sup @ A) @ A3) @ B4) = B4)))))))).
% 0.32/1.50 thf(fact_205_sup_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,A3:A]: (((((ord_less_eq @ A) @ B4) @ A3) => ((((sup_sup @ A) @ A3) @ B4) = A3)))))))).
% 0.32/1.50 thf(fact_206_sup__unique, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [F2:(A > (A > A)),X:A,Y:A]: (((! [X3:A,Y3:A]: ((((ord_less_eq @ A) @ X3) @ ((F2 @ X3) @ Y3)))) => ((! [X3:A,Y3:A]: ((((ord_less_eq @ A) @ Y3) @ ((F2 @ X3) @ Y3)))) => ((! [X3:A,Y3:A,Z:A]: (((((ord_less_eq @ A) @ Y3) @ X3) => ((((ord_less_eq @ A) @ Z) @ X3) => (((ord_less_eq @ A) @ ((F2 @ Y3) @ Z)) @ X3))))) => ((((sup_sup @ A) @ X) @ Y) = ((F2 @ X) @ Y))))))))))).
% 0.32/1.50 thf(fact_207_sup_OorderI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B4:A]: (((A3 = (((sup_sup @ A) @ A3) @ B4)) => (((ord_less_eq @ A) @ B4) @ A3)))))))).
% 0.32/1.50 thf(fact_208_sup_OorderE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,A3:A]: (((((ord_less_eq @ A) @ B4) @ A3) => (A3 = (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_209_le__iff__sup, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [X4:A,Y4:A]: (((((sup_sup @ A) @ X4) @ Y4) = Y4)))))))).
% 0.32/1.50 thf(fact_210_sup__least, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X:A,Z4:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ Z4) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y) @ Z4)) @ X))))))))).
% 0.32/1.50 thf(fact_211_sup__mono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,C4:A,B4:A,D3:A]: (((((ord_less_eq @ A) @ A3) @ C4) => ((((ord_less_eq @ A) @ B4) @ D3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B4)) @ (((sup_sup @ A) @ C4) @ D3)))))))))).
% 0.32/1.50 thf(fact_212_sup_Omono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C4:A,A3:A,D3:A,B4:A]: (((((ord_less_eq @ A) @ C4) @ A3) => ((((ord_less_eq @ A) @ D3) @ B4) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C4) @ D3)) @ (((sup_sup @ A) @ A3) @ B4)))))))))).
% 0.32/1.50 thf(fact_213_le__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,B4:A,A3:A]: (((((ord_less_eq @ A) @ X) @ B4) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_214_le__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,A3:A,B4:A]: (((((ord_less_eq @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_215_sup__ge2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 0.32/1.50 thf(fact_216_sup__ge1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 0.32/1.50 thf(fact_217_le__supI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,X:A,B4:A]: (((((ord_less_eq @ A) @ A3) @ X) => ((((ord_less_eq @ A) @ B4) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B4)) @ X))))))))).
% 0.32/1.50 thf(fact_218_le__supE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B4:A,X:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B4)) @ X) => (~ (((((ord_less_eq @ A) @ A3) @ X) => (~ ((((ord_less_eq @ A) @ B4) @ X))))))))))))).
% 0.32/1.50 thf(fact_219_inf__sup__ord_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 0.32/1.50 thf(fact_220_inf__sup__ord_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 0.32/1.50 thf(fact_221_less__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,A3:A,B4:A]: (((((ord_less @ A) @ X) @ A3) => (((ord_less @ A) @ X) @ (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_222_less__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,B4:A,A3:A]: (((((ord_less @ A) @ X) @ B4) => (((ord_less @ A) @ X) @ (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_223_sup_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,C4:A,A3:A]: (((((ord_less @ A) @ (((sup_sup @ A) @ B4) @ C4)) @ A3) => (~ (((((ord_less @ A) @ B4) @ A3) => (~ ((((ord_less @ A) @ C4) @ A3))))))))))))).
% 0.32/1.50 thf(fact_224_sup_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less @ A) = (^ [B:A,A2:A]: (((A2 = (((sup_sup @ A) @ A2) @ B)) & (A2 != B))))))))).
% 0.32/1.50 thf(fact_225_sup_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C4:A,A3:A,B4:A]: (((((ord_less @ A) @ C4) @ A3) => (((ord_less @ A) @ C4) @ (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_226_sup_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C4:A,B4:A,A3:A]: (((((ord_less @ A) @ C4) @ B4) => (((ord_less @ A) @ C4) @ (((sup_sup @ A) @ A3) @ B4))))))))).
% 0.32/1.50 thf(fact_227_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 0.32/1.50 thf(fact_228_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ R)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 0.32/1.50 thf(fact_229_the__elem__eq, axiom, (! [A:$tType,X:A]: ((((the_elem @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X)))).
% 0.32/1.50 thf(fact_230_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 0.32/1.50 thf(fact_231_sup1CI, axiom, (! [A:$tType,B5:(A > $o),X:A,A4:(A > $o)]: ((((~ ((B5 @ X))) => (A4 @ X)) => ((((sup_sup @ (A > $o)) @ A4) @ B5) @ X))))).
% 0.32/1.50 thf(fact_232_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X4:D]: ((bot_bot @ C)))))))).
% 0.32/1.50 thf(fact_233_sup1I2, axiom, (! [A:$tType,B5:(A > $o),X:A,A4:(A > $o)]: (((B5 @ X) => ((((sup_sup @ (A > $o)) @ A4) @ B5) @ X))))).
% 0.32/1.50 thf(fact_234_sup1I1, axiom, (! [A:$tType,A4:(A > $o),X:A,B5:(A > $o)]: (((A4 @ X) => ((((sup_sup @ (A > $o)) @ A4) @ B5) @ X))))).
% 0.32/1.50 thf(fact_235_sup1E, axiom, (! [A:$tType,A4:(A > $o),B5:(A > $o),X:A]: ((((((sup_sup @ (A > $o)) @ A4) @ B5) @ X) => ((~ ((A4 @ X))) => (B5 @ X)))))).
% 0.32/1.50 thf(fact_236_order__subst1, axiom, (! [A:$tType,B2:$tType]: ((((order @ B2) & (order @ A)) => (! [A3:A,F2:(B2 > A),B4:B2,C4:B2]: (((((ord_less_eq @ A) @ A3) @ (F2 @ B4)) => ((((ord_less_eq @ B2) @ B4) @ C4) => ((! [X3:B2,Y3:B2]: (((((ord_less_eq @ B2) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F2 @ C4))))))))))).
% 0.32/1.50 thf(fact_237_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B4:A,F2:(A > C),C4:C]: (((((ord_less_eq @ A) @ A3) @ B4) => ((((ord_less_eq @ C) @ (F2 @ B4)) @ C4) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ C) @ (F2 @ X3)) @ (F2 @ Y3))))) => (((ord_less_eq @ C) @ (F2 @ A3)) @ C4)))))))))).
% 0.32/1.50 thf(fact_238_ord__eq__le__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A3:A,F2:(B2 > A),B4:B2,C4:B2]: (((A3 = (F2 @ B4)) => ((((ord_less_eq @ B2) @ B4) @ C4) => ((! [X3:B2,Y3:B2]: (((((ord_less_eq @ B2) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F2 @ C4))))))))))).
% 0.32/1.50 thf(fact_239_ord__le__eq__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A3:A,B4:A,F2:(A > B2),C4:B2]: (((((ord_less_eq @ A) @ A3) @ B4) => (((F2 @ B4) = C4) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B2) @ (F2 @ X3)) @ (F2 @ Y3))))) => (((ord_less_eq @ B2) @ (F2 @ A3)) @ C4)))))))))).
% 0.32/1.50 thf(fact_240_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z3:A]: ((Y2 = Z3))) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X4))))))))).
% 0.32/1.50 thf(fact_241_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 0.32/1.50 thf(fact_242_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.32/1.50 thf(fact_243_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 0.32/1.50 thf(fact_244_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.32/1.50 thf(fact_245_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B4:A,C4:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((((ord_less_eq @ A) @ B4) @ C4) => (((ord_less_eq @ A) @ A3) @ C4))))))))).
% 0.32/1.50 thf(fact_246_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z4:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z4)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z4)))) => (((((ord_less_eq @ A) @ X) @ Z4) => (~ ((((ord_less_eq @ A) @ Z4) @ Y)))) => (((((ord_less_eq @ A) @ Z4) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z4) => (~ ((((ord_less_eq @ A) @ Z4) @ X)))) => (~ (((((ord_less_eq @ A) @ Z4) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 0.32/1.50 thf(fact_247_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 0.32/1.50 thf(fact_248_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z3:A]: ((Y2 = Z3))) = (^ [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) & (((ord_less_eq @ A) @ B) @ A2))))))))).
% 0.32/1.50 thf(fact_249_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B4:A,C4:A]: (((A3 = B4) => ((((ord_less_eq @ A) @ B4) @ C4) => (((ord_less_eq @ A) @ A3) @ C4))))))))).
% 0.32/1.50 thf(fact_250_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B4:A,C4:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((B4 = C4) => (((ord_less_eq @ A) @ A3) @ C4))))))))).
% 0.32/1.50 thf(fact_251_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B4:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((((ord_less_eq @ A) @ B4) @ A3) => (A3 = B4))))))))).
% 0.32/1.50 thf(fact_252_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z4:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z4) => (((ord_less_eq @ A) @ X) @ Z4))))))))).
% 0.32/1.50 thf(fact_253_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 0.32/1.50 thf(fact_254_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B4:A]: (((! [A7:A,B3:A]: (((((ord_less_eq @ A) @ A7) @ B3) => ((P @ A7) @ B3)))) => ((! [A7:A,B3:A]: ((((P @ B3) @ A7) => ((P @ A7) @ B3)))) => ((P @ A3) @ B4))))))))).
% 0.32/1.50 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, (bounded_lattice @ $o)).
% 0.32/1.50 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A8:$tType]: ((bounded_lattice @ (set @ A8))))).
% 0.32/1.50 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounded_lattice @ (A8 > A9)))))).
% 0.32/1.50 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounde1808546759up_bot @ (A8 > A9)))))).
% 0.32/1.50 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A8:$tType,A9:$tType]: (((semilattice_sup @ A9) => (semilattice_sup @ (A8 > A9)))))).
% 0.32/1.50 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 0.32/1.50 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: (((lattice @ A9) => (lattice @ (A8 > A9)))))).
% 0.32/1.50 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 0.32/1.50 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 0.32/1.50 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 0.32/1.50 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ int)).
% 0.32/1.50 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3, axiom, (semilattice_sup @ int)).
% 0.32/1.50 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 0.32/1.50 thf(tcon_Int_Oint___Orderings_Opreorder_4, axiom, (preorder @ int)).
% 0.32/1.50 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 0.32/1.50 thf(tcon_Int_Oint___Lattices_Olattice_5, axiom, (lattice @ int)).
% 0.32/1.50 thf(tcon_Int_Oint___Orderings_Oorder_6, axiom, (order @ int)).
% 0.32/1.50 thf(tcon_Int_Oint___Orderings_Oord_7, axiom, (ord @ int)).
% 0.32/1.50 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_8, axiom, (! [A8:$tType]: ((bounde1808546759up_bot @ (set @ A8))))).
% 0.32/1.50 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_9, axiom, (! [A8:$tType]: ((semilattice_sup @ (set @ A8))))).
% 0.32/1.50 thf(tcon_Set_Oset___Orderings_Opreorder_10, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 0.32/1.50 thf(tcon_Set_Oset___Lattices_Olattice_11, axiom, (! [A8:$tType]: ((lattice @ (set @ A8))))).
% 0.32/1.50 thf(tcon_Set_Oset___Orderings_Oorder_12, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 0.32/1.50 thf(tcon_Set_Oset___Orderings_Oord_13, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 0.32/1.50 thf(tcon_Set_Oset___Orderings_Obot_14, axiom, (! [A8:$tType]: ((bot @ (set @ A8))))).
% 0.32/1.50 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_15, axiom, (bounde1808546759up_bot @ $o)).
% 0.32/1.50 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_16, axiom, (semilattice_sup @ $o)).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Opreorder_17, axiom, (preorder @ $o)).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Olinorder_18, axiom, (linorder @ $o)).
% 0.32/1.50 thf(tcon_HOL_Obool___Lattices_Olattice_19, axiom, (lattice @ $o)).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Oorder_20, axiom, (order @ $o)).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Oord_21, axiom, (ord @ $o)).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Obot_22, axiom, (bot @ $o)).
% 0.32/1.50 thf(conj_0, conjecture, (((ord_less @ int) @ (h @ l)) @ (h @ (((binary576689334lle_rm @ a) @ h) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2))))).
% 0.32/1.50 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.7sjdMpQ3z4/DTF2DTF18142.p
% 0.32/1.50 ------------------------
% 0.32/1.53 ---- Cleaned THF ---
% 0.32/1.53 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
% 0.32/1.53 binary1291135688e_Tree: $tType > $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_t_Set_Oset,type,
% 0.32/1.53 set: $tType > $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_t_Int_Oint,type,
% 0.32/1.53 int: $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_tf_a,type,
% 0.32/1.53 a: $tType ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.32/1.53 bounded_lattice:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_HOL_Otype,type,
% 0.32/1.53 type:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Obot,type,
% 0.32/1.53 bot:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Oord,type,
% 0.32/1.53 ord:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Oorder,type,
% 0.32/1.53 order:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Lattices_Olattice,type,
% 0.32/1.53 lattice:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/1.53 linorder:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/1.53 preorder:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.32/1.53 linordered_idom:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.32/1.53 semilattice_sup:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.32/1.53 bounde1808546759up_bot:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.32/1.53 condit1656338222tinuum:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.32/1.53 condit1037483654norder:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
% 0.32/1.53 binary210054475elle_T:
% 0.32/1.53 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
% 0.32/1.53 binary1746293266le_Tip:
% 0.32/1.53 !>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert,type,
% 0.32/1.53 binary1830089824insert:
% 0.32/1.53 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
% 0.32/1.53 binary827270440e_memb:
% 0.32/1.53 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oremove,type,
% 0.32/1.53 binary997842527remove:
% 0.32/1.53 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Orm,type,
% 0.32/1.53 binary576689334lle_rm:
% 0.32/1.53 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
% 0.32/1.53 binary1653327646_setOf:
% 0.32/1.53 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
% 0.32/1.53 binary1610619414edTree:
% 0.32/1.53 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
% 0.32/1.53 binary231205461t_pred:
% 0.32/1.53 !>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrm,type,
% 0.32/1.53 binary213313527le_wrm:
% 0.32/1.53 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.32/1.53 sup_sup:
% 0.32/1.53 !>[A: $tType] : ( A > A > A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.32/1.53 bot_bot:
% 0.32/1.53 !>[A: $tType] : A ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.32/1.53 ord_less:
% 0.32/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/1.53 ord_less_eq:
% 0.32/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Set_OCollect,type,
% 0.32/1.53 collect:
% 0.32/1.53 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Set_Oinsert,type,
% 0.32/1.53 insert:
% 0.32/1.53 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Set_Othe__elem,type,
% 0.32/1.53 the_elem:
% 0.32/1.53 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_member,type,
% 0.32/1.53 member:
% 0.32/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_h,type,
% 0.32/1.53 h: a > int ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_l____,type,
% 0.32/1.53 l: a ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_t1____,type,
% 0.32/1.53 t1: binary1291135688e_Tree @ a ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_t2____,type,
% 0.32/1.53 t2: binary1291135688e_Tree @ a ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_x____,type,
% 0.32/1.53 x: a ).
% 0.32/1.53
% 0.32/1.53 thf(fact_0_hLess,axiom,
% 0.32/1.53 ord_less @ int @ ( h @ l ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_1_rm__res,axiom,
% 0.32/1.53 ( ( binary576689334lle_rm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
% 0.32/1.53 = ( binary576689334lle_rm @ a @ h @ t2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_2_t2nTip,axiom,
% 0.32/1.53 ( t2
% 0.32/1.53 != ( binary1746293266le_Tip @ a ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_3_Tree_Oinject,axiom,
% 0.32/1.53 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
% 0.32/1.53 = ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
% 0.32/1.53 = ( ( X21 = Y21 )
% 0.32/1.53 & ( X22 = Y22 )
% 0.32/1.53 & ( X23 = Y23 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_4_s,axiom,
% 0.32/1.53 binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_5_wrm__res,axiom,
% 0.32/1.53 ( ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
% 0.32/1.53 = ( binary210054475elle_T @ a @ t1 @ x @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_6_ldef,axiom,
% 0.32/1.53 member @ a @ l @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_7_s1,axiom,
% 0.32/1.53 binary1610619414edTree @ a @ h @ t1 ).
% 0.32/1.53
% 0.32/1.53 thf(fact_8_s2,axiom,
% 0.32/1.53 binary1610619414edTree @ a @ h @ t2 ).
% 0.32/1.53
% 0.32/1.53 thf(fact_9_binsert_Osimps_I2_J,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.32/1.53 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
% 0.32/1.53 & ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
% 0.32/1.53 => ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.32/1.53 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
% 0.32/1.53 & ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
% 0.32/1.53 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_10_rm_Osimps,axiom,
% 0.32/1.53 ! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
% 0.32/1.53 ( ( ( T2
% 0.32/1.53 = ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 => ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = X ) )
% 0.32/1.53 & ( ( T2
% 0.32/1.53 != ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 => ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = ( binary576689334lle_rm @ A @ H @ T2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_11_minf_I11_J,axiom,
% 0.32/1.53 ! [C: $tType,D: $tType] :
% 0.32/1.53 ( ( ord @ C )
% 0.32/1.53 => ! [F: D] :
% 0.32/1.53 ? [Z: C] :
% 0.32/1.53 ! [X2: C] :
% 0.32/1.53 ( ( ord_less @ C @ X2 @ Z )
% 0.32/1.53 => ( F = F ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_12_minf_I7_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ X2 @ Z )
% 0.32/1.53 => ~ ( ord_less @ A @ T @ X2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_13_minf_I5_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ X2 @ Z )
% 0.32/1.53 => ( ord_less @ A @ X2 @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_14_minf_I4_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ X2 @ Z )
% 0.32/1.53 => ( X2 != T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_15_minf_I3_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ X2 @ Z )
% 0.32/1.53 => ( X2 != T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_16_minf_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/1.53 ( ? [Z2: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Z2 )
% 0.32/1.53 => ( ( P @ X3 )
% 0.32/1.53 = ( P2 @ X3 ) ) )
% 0.32/1.53 => ( ? [Z2: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Z2 )
% 0.32/1.53 => ( ( Q @ X3 )
% 0.32/1.53 = ( Q2 @ X3 ) ) )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ X2 @ Z )
% 0.32/1.53 => ( ( ( P @ X2 )
% 0.32/1.53 | ( Q @ X2 ) )
% 0.32/1.53 = ( ( P2 @ X2 )
% 0.32/1.53 | ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_17_sortedTree_Osimps_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_18_wrm__sort,axiom,
% 0.32/1.53 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.32/1.53 ( ( ( T
% 0.32/1.53 != ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.32/1.53 => ( binary1610619414edTree @ A @ H @ ( binary213313527le_wrm @ A @ H @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_19_binsert__sorted,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
% 0.32/1.53 ( ( binary1610619414edTree @ A @ H @ T )
% 0.32/1.53 => ( binary1610619414edTree @ A @ H @ ( binary1830089824insert @ A @ H @ X @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_20_rm__set,axiom,
% 0.32/1.53 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.32/1.53 ( ( ( T
% 0.32/1.53 != ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.32/1.53 => ( member @ A @ ( binary576689334lle_rm @ A @ H @ T ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_21_binsert_Osimps_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,E: A] :
% 0.32/1.53 ( ( binary1830089824insert @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 = ( binary210054475elle_T @ A @ ( binary1746293266le_Tip @ A ) @ E @ ( binary1746293266le_Tip @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_22_wrm_Osimps,axiom,
% 0.32/1.53 ! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
% 0.32/1.53 ( ( ( T2
% 0.32/1.53 = ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 => ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = T1 ) )
% 0.32/1.53 & ( ( T2
% 0.32/1.53 != ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 => ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = ( binary210054475elle_T @ A @ T1 @ X @ ( binary213313527le_wrm @ A @ H @ T2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_23_sortedTree_Osimps_I2_J,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = ( ( binary1610619414edTree @ A @ H @ T1 )
% 0.32/1.53 & ! [X4: A] :
% 0.32/1.53 ( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T1 ) )
% 0.32/1.53 => ( ord_less @ int @ ( H @ X4 ) @ ( H @ X ) ) )
% 0.32/1.53 & ! [X4: A] :
% 0.32/1.53 ( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T2 ) )
% 0.32/1.53 => ( ord_less @ int @ ( H @ X ) @ ( H @ X4 ) ) )
% 0.32/1.53 & ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_24_Tree_Oexhaust,axiom,
% 0.32/1.53 ! [A: $tType,Y: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( Y
% 0.32/1.53 != ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 => ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( Y
% 0.32/1.53 != ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_25_Tree_Oinduct,axiom,
% 0.32/1.53 ! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( P @ ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 => ( ! [X1: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( P @ X1 )
% 0.32/1.53 => ( ( P @ X32 )
% 0.32/1.53 => ( P @ ( binary210054475elle_T @ A @ X1 @ X24 @ X32 ) ) ) )
% 0.32/1.53 => ( P @ Tree ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_26_Tree_Odistinct_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( binary1746293266le_Tip @ A )
% 0.32/1.53 != ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_27_sortLemmaR,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 => ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_28_sortLemmaL,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 => ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_29_pinf_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/1.53 ( ? [Z2: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/1.53 => ( ( P @ X3 )
% 0.32/1.53 = ( P2 @ X3 ) ) )
% 0.32/1.53 => ( ? [Z2: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/1.53 => ( ( Q @ X3 )
% 0.32/1.53 = ( Q2 @ X3 ) ) )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X2 )
% 0.32/1.53 => ( ( ( P @ X2 )
% 0.32/1.53 & ( Q @ X2 ) )
% 0.32/1.53 = ( ( P2 @ X2 )
% 0.32/1.53 & ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_30_pinf_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/1.53 ( ? [Z2: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/1.53 => ( ( P @ X3 )
% 0.32/1.53 = ( P2 @ X3 ) ) )
% 0.32/1.53 => ( ? [Z2: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z2 @ X3 )
% 0.32/1.53 => ( ( Q @ X3 )
% 0.32/1.53 = ( Q2 @ X3 ) ) )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X2 )
% 0.32/1.53 => ( ( ( P @ X2 )
% 0.32/1.53 | ( Q @ X2 ) )
% 0.32/1.53 = ( ( P2 @ X2 )
% 0.32/1.53 | ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_31_pinf_I3_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X2 )
% 0.32/1.53 => ( X2 != T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_32_pinf_I4_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X2 )
% 0.32/1.53 => ( X2 != T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_33_pinf_I5_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X2 )
% 0.32/1.53 => ~ ( ord_less @ A @ X2 @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_34_pinf_I7_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X2 )
% 0.32/1.53 => ( ord_less @ A @ T @ X2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_35_pinf_I11_J,axiom,
% 0.32/1.53 ! [C: $tType,D: $tType] :
% 0.32/1.53 ( ( ord @ C )
% 0.32/1.53 => ! [F: D] :
% 0.32/1.53 ? [Z: C] :
% 0.32/1.53 ! [X2: C] :
% 0.32/1.53 ( ( ord_less @ C @ Z @ X2 )
% 0.32/1.53 => ( F = F ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_36_minf_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.32/1.53 ( ? [Z2: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Z2 )
% 0.32/1.53 => ( ( P @ X3 )
% 0.32/1.53 = ( P2 @ X3 ) ) )
% 0.32/1.53 => ( ? [Z2: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Z2 )
% 0.32/1.53 => ( ( Q @ X3 )
% 0.32/1.53 = ( Q2 @ X3 ) ) )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ X2 @ Z )
% 0.32/1.53 => ( ( ( P @ X2 )
% 0.32/1.53 & ( Q @ X2 ) )
% 0.32/1.53 = ( ( P2 @ X2 )
% 0.32/1.53 & ( Q2 @ X2 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_37_h1,axiom,
% 0.32/1.53 ( ( ( t1
% 0.32/1.53 != ( binary1746293266le_Tip @ a ) )
% 0.32/1.53 & ( binary1610619414edTree @ a @ h @ t1 ) )
% 0.32/1.53 => ! [X2: a] :
% 0.32/1.53 ( ( member @ a @ X2 @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t1 ) ) )
% 0.32/1.53 => ( ord_less @ int @ ( h @ X2 ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t1 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_38_h2,axiom,
% 0.32/1.53 ( ( ( t2
% 0.32/1.53 != ( binary1746293266le_Tip @ a ) )
% 0.32/1.53 & ( binary1610619414edTree @ a @ h @ t2 ) )
% 0.32/1.53 => ! [X2: a] :
% 0.32/1.53 ( ( member @ a @ X2 @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t2 ) ) )
% 0.32/1.53 => ( ord_less @ int @ ( h @ X2 ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_39_memb__spec,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
% 0.32/1.53 ( ( binary1610619414edTree @ A @ H @ T )
% 0.32/1.53 => ( ( binary827270440e_memb @ A @ H @ X @ T )
% 0.32/1.53 = ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_40_sorted__distinct__pred__def,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( binary231205461t_pred @ A )
% 0.32/1.53 = ( ^ [H2: A > int,A2: A,B: A,T3: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( ( binary1610619414edTree @ A @ H2 @ T3 )
% 0.32/1.53 & ( member @ A @ A2 @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.32/1.53 & ( member @ A @ B @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.32/1.53 & ( ( H2 @ A2 )
% 0.32/1.53 = ( H2 @ B ) ) )
% 0.32/1.53 => ( A2 = B ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_41_l__scope,axiom,
% 0.32/1.53 member @ a @ l @ ( sup_sup @ ( set @ a ) @ ( sup_sup @ ( set @ a ) @ ( insert @ a @ x @ ( bot_bot @ ( set @ a ) ) ) @ ( binary1653327646_setOf @ a @ t1 ) ) @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_42_wrm__set1,axiom,
% 0.32/1.53 ! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
% 0.32/1.53 ( ( ( T
% 0.32/1.53 != ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 & ( binary1610619414edTree @ A @ H @ T ) )
% 0.32/1.53 => ( ord_less_eq @ ( set @ A ) @ ( binary1653327646_setOf @ A @ ( binary213313527le_wrm @ A @ H @ T ) ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_43_remove_Osimps_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,E: A] :
% 0.32/1.53 ( ( binary997842527remove @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 = ( binary1746293266le_Tip @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_44_mem__Collect__eq,axiom,
% 0.32/1.53 ! [A: $tType,A3: A,P: A > $o] :
% 0.32/1.53 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.32/1.53 = ( P @ A3 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_45_Collect__mem__eq,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A] :
% 0.32/1.53 ( ( collect @ A
% 0.32/1.53 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.32/1.53 = A4 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_46_Collect__cong,axiom,
% 0.32/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ( P @ X3 )
% 0.32/1.53 = ( Q @ X3 ) )
% 0.32/1.53 => ( ( collect @ A @ P )
% 0.32/1.53 = ( collect @ A @ Q ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_47_ext,axiom,
% 0.32/1.53 ! [B2: $tType,A: $tType,F2: A > B2,G: A > B2] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ( F2 @ X3 )
% 0.32/1.53 = ( G @ X3 ) )
% 0.32/1.53 => ( F2 = G ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_48_ex__gt__or__lt,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( condit1656338222tinuum @ A )
% 0.32/1.53 => ! [A3: A] :
% 0.32/1.53 ? [B3: A] :
% 0.32/1.53 ( ( ord_less @ A @ A3 @ B3 )
% 0.32/1.53 | ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_49_linorder__neqE__linordered__idom,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_idom @ A )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( X != Y )
% 0.32/1.53 => ( ~ ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_50_complete__interval,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( condit1037483654norder @ A )
% 0.32/1.53 => ! [A3: A,B4: A,P: A > $o] :
% 0.32/1.53 ( ( ord_less @ A @ A3 @ B4 )
% 0.32/1.53 => ( ( P @ A3 )
% 0.32/1.53 => ( ~ ( P @ B4 )
% 0.32/1.53 => ? [C2: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.32/1.53 & ( ord_less_eq @ A @ C2 @ B4 )
% 0.32/1.53 & ! [X2: A] :
% 0.32/1.53 ( ( ( ord_less_eq @ A @ A3 @ X2 )
% 0.32/1.53 & ( ord_less @ A @ X2 @ C2 ) )
% 0.32/1.53 => ( P @ X2 ) )
% 0.32/1.53 & ! [D2: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ( ( ord_less_eq @ A @ A3 @ X3 )
% 0.32/1.53 & ( ord_less @ A @ X3 @ D2 ) )
% 0.32/1.53 => ( P @ X3 ) )
% 0.32/1.53 => ( ord_less_eq @ A @ D2 @ C2 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_51_setOf_Osimps_I2_J,axiom,
% 0.32/1.53 ! [A: $tType,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
% 0.32/1.53 ( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
% 0.32/1.53 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T2 ) ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_52_minf_I8_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ X2 @ Z )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_53_minf_I6_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ X2 @ Z )
% 0.32/1.53 => ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_54_pinf_I8_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X2 )
% 0.32/1.53 => ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_55_pinf_I6_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X2 )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_56_sorted__distinct,axiom,
% 0.32/1.53 ! [A: $tType,H: A > int,A3: A,B4: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A3 @ B4 @ T ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_57_setOf_Osimps_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
% 0.32/1.53 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_58_singleton__insert__inj__eq_H,axiom,
% 0.32/1.53 ! [A: $tType,A3: A,A4: set @ A,B4: A] :
% 0.32/1.53 ( ( ( insert @ A @ A3 @ A4 )
% 0.32/1.53 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.53 = ( ( A3 = B4 )
% 0.32/1.53 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_59_singleton__insert__inj__eq,axiom,
% 0.32/1.53 ! [A: $tType,B4: A,A3: A,A4: set @ A] :
% 0.32/1.53 ( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
% 0.32/1.53 = ( insert @ A @ A3 @ A4 ) )
% 0.32/1.53 = ( ( A3 = B4 )
% 0.32/1.53 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_60_singleton__conv2,axiom,
% 0.32/1.53 ! [A: $tType,A3: A] :
% 0.32/1.53 ( ( collect @ A
% 0.32/1.53 @ ( ^ [Y2: A,Z3: A] : ( Y2 = Z3 )
% 0.32/1.53 @ A3 ) )
% 0.32/1.53 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_61_singleton__conv,axiom,
% 0.32/1.53 ! [A: $tType,A3: A] :
% 0.32/1.53 ( ( collect @ A
% 0.32/1.53 @ ^ [X4: A] : ( X4 = A3 ) )
% 0.32/1.53 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_62_Un__insert__right,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A,A3: A,B5: set @ A] :
% 0.32/1.53 ( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B5 ) )
% 0.32/1.53 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_63_Un__insert__left,axiom,
% 0.32/1.53 ! [A: $tType,A3: A,B5: set @ A,C3: set @ A] :
% 0.32/1.53 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.32/1.53 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_64_Un__subset__iff,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/1.53 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
% 0.32/1.53 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.32/1.53 & ( ord_less_eq @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_65_Un__empty,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.53 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.53 = ( bot_bot @ ( set @ A ) ) )
% 0.32/1.53 = ( ( A4
% 0.32/1.53 = ( bot_bot @ ( set @ A ) ) )
% 0.32/1.53 & ( B5
% 0.32/1.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_66_empty__iff,axiom,
% 0.32/1.53 ! [A: $tType,C4: A] :
% 0.32/1.53 ~ ( member @ A @ C4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_67_all__not__in__conv,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A] :
% 0.32/1.53 ( ( ! [X4: A] :
% 0.32/1.53 ~ ( member @ A @ X4 @ A4 ) )
% 0.32/1.53 = ( A4
% 0.32/1.53 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_68_Collect__empty__eq,axiom,
% 0.32/1.53 ! [A: $tType,P: A > $o] :
% 0.32/1.53 ( ( ( collect @ A @ P )
% 0.32/1.53 = ( bot_bot @ ( set @ A ) ) )
% 0.32/1.53 = ( ! [X4: A] :
% 0.32/1.53 ~ ( P @ X4 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_69_empty__Collect__eq,axiom,
% 0.32/1.53 ! [A: $tType,P: A > $o] :
% 0.32/1.53 ( ( ( bot_bot @ ( set @ A ) )
% 0.32/1.53 = ( collect @ A @ P ) )
% 0.32/1.53 = ( ! [X4: A] :
% 0.32/1.53 ~ ( P @ X4 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_70_subsetI,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ( member @ A @ X3 @ A4 )
% 0.32/1.53 => ( member @ A @ X3 @ B5 ) )
% 0.32/1.53 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_71_subset__antisym,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.53 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.53 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
% 0.32/1.53 => ( A4 = B5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_72_psubsetI,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.53 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.53 => ( ( A4 != B5 )
% 0.32/1.53 => ( ord_less @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_73_insertCI,axiom,
% 0.32/1.53 ! [A: $tType,A3: A,B5: set @ A,B4: A] :
% 0.32/1.53 ( ( ~ ( member @ A @ A3 @ B5 )
% 0.32/1.53 => ( A3 = B4 ) )
% 0.32/1.53 => ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_74_insert__iff,axiom,
% 0.32/1.53 ! [A: $tType,A3: A,B4: A,A4: set @ A] :
% 0.32/1.53 ( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
% 0.32/1.53 = ( ( A3 = B4 )
% 0.32/1.53 | ( member @ A @ A3 @ A4 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_75_insert__absorb2,axiom,
% 0.32/1.53 ! [A: $tType,X: A,A4: set @ A] :
% 0.32/1.53 ( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
% 0.32/1.53 = ( insert @ A @ X @ A4 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_76_UnCI,axiom,
% 0.32/1.53 ! [A: $tType,C4: A,B5: set @ A,A4: set @ A] :
% 0.32/1.53 ( ( ~ ( member @ A @ C4 @ B5 )
% 0.32/1.53 => ( member @ A @ C4 @ A4 ) )
% 0.32/1.53 => ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_77_Un__iff,axiom,
% 0.32/1.53 ! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
% 0.32/1.53 ( ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/1.53 = ( ( member @ A @ C4 @ A4 )
% 0.32/1.53 | ( member @ A @ C4 @ B5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_78_subset__empty,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A] :
% 0.32/1.53 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.32/1.53 = ( A4
% 0.32/1.53 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_79_empty__subsetI,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_80_singletonI,axiom,
% 0.32/1.53 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_81_insert__subset,axiom,
% 0.32/1.53 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.32/1.53 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.32/1.53 = ( ( member @ A @ X @ B5 )
% 0.32/1.53 & ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_82_less__eq__set__def,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( ord_less_eq @ ( A > $o )
% 0.32/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ A5 )
% 0.32/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_83_emptyE,axiom,
% 0.32/1.54 ! [A: $tType,A3: A] :
% 0.32/1.54 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_84_equals0D,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,A3: A] :
% 0.32/1.54 ( ( A4
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) )
% 0.32/1.54 => ~ ( member @ A @ A3 @ A4 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_85_equals0I,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A] :
% 0.32/1.54 ( ! [Y3: A] :
% 0.32/1.54 ~ ( member @ A @ Y3 @ A4 )
% 0.32/1.54 => ( A4
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_86_ex__in__conv,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A] :
% 0.32/1.54 ( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.32/1.54 = ( A4
% 0.32/1.54 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_87_not__psubset__empty,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A] :
% 0.32/1.54 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_88_in__mono,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ( member @ A @ X @ A4 )
% 0.32/1.54 => ( member @ A @ X @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_89_subsetD,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,C4: A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ( member @ A @ C4 @ A4 )
% 0.32/1.54 => ( member @ A @ C4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_90_equalityE,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( A4 = B5 )
% 0.32/1.54 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_91_subset__eq,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ! [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ A5 )
% 0.32/1.54 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_92_equalityD1,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( A4 = B5 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_93_equalityD2,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( A4 = B5 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_94_subset__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ! [T3: A] :
% 0.32/1.54 ( ( member @ A @ T3 @ A5 )
% 0.32/1.54 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_95_subset__refl,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_96_Collect__mono,axiom,
% 0.32/1.54 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.54 ( ! [X3: A] :
% 0.32/1.54 ( ( P @ X3 )
% 0.32/1.54 => ( Q @ X3 ) )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_97_subset__trans,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_98_set__eq__subset,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ^ [Y2: set @ A,Z3: set @ A] : ( Y2 = Z3 ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.32/1.54 & ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_99_Collect__mono__iff,axiom,
% 0.32/1.54 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.32/1.54 = ( ! [X4: A] :
% 0.32/1.54 ( ( P @ X4 )
% 0.32/1.54 => ( Q @ X4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_100_subset__iff__psubset__eq,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 0.32/1.54 | ( A5 = B6 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_101_subset__psubset__trans,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
% 0.32/1.54 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_102_subset__not__subset__eq,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.32/1.54 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_103_psubset__subset__trans,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
% 0.32/1.54 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_104_psubset__imp__subset,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_105_psubset__eq,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.32/1.54 & ( A5 != B6 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_106_psubsetE,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_107_insertE,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,B4: A,A4: set @ A] :
% 0.32/1.54 ( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
% 0.32/1.54 => ( ( A3 != B4 )
% 0.32/1.54 => ( member @ A @ A3 @ A4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_108_insertI1,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_109_insertI2,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,B5: set @ A,B4: A] :
% 0.32/1.54 ( ( member @ A @ A3 @ B5 )
% 0.32/1.54 => ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_110_Set_Oset__insert,axiom,
% 0.32/1.54 ! [A: $tType,X: A,A4: set @ A] :
% 0.32/1.54 ( ( member @ A @ X @ A4 )
% 0.32/1.54 => ~ ! [B7: set @ A] :
% 0.32/1.54 ( ( A4
% 0.32/1.54 = ( insert @ A @ X @ B7 ) )
% 0.32/1.54 => ( member @ A @ X @ B7 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_111_insert__ident,axiom,
% 0.32/1.54 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ~ ( member @ A @ X @ A4 )
% 0.32/1.54 => ( ~ ( member @ A @ X @ B5 )
% 0.32/1.54 => ( ( ( insert @ A @ X @ A4 )
% 0.32/1.54 = ( insert @ A @ X @ B5 ) )
% 0.32/1.54 = ( A4 = B5 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_112_insert__absorb,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,A4: set @ A] :
% 0.32/1.54 ( ( member @ A @ A3 @ A4 )
% 0.32/1.54 => ( ( insert @ A @ A3 @ A4 )
% 0.32/1.54 = A4 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_113_insert__eq__iff,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,A4: set @ A,B4: A,B5: set @ A] :
% 0.32/1.54 ( ~ ( member @ A @ A3 @ A4 )
% 0.32/1.54 => ( ~ ( member @ A @ B4 @ B5 )
% 0.32/1.54 => ( ( ( insert @ A @ A3 @ A4 )
% 0.32/1.54 = ( insert @ A @ B4 @ B5 ) )
% 0.32/1.54 = ( ( ( A3 = B4 )
% 0.32/1.54 => ( A4 = B5 ) )
% 0.32/1.54 & ( ( A3 != B4 )
% 0.32/1.54 => ? [C5: set @ A] :
% 0.32/1.54 ( ( A4
% 0.32/1.54 = ( insert @ A @ B4 @ C5 ) )
% 0.32/1.54 & ~ ( member @ A @ B4 @ C5 )
% 0.32/1.54 & ( B5
% 0.32/1.54 = ( insert @ A @ A3 @ C5 ) )
% 0.32/1.54 & ~ ( member @ A @ A3 @ C5 ) ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_114_insert__commute,axiom,
% 0.32/1.54 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 0.32/1.54 ( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
% 0.32/1.54 = ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_115_mk__disjoint__insert,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,A4: set @ A] :
% 0.32/1.54 ( ( member @ A @ A3 @ A4 )
% 0.32/1.54 => ? [B7: set @ A] :
% 0.32/1.54 ( ( A4
% 0.32/1.54 = ( insert @ A @ A3 @ B7 ) )
% 0.32/1.54 & ~ ( member @ A @ A3 @ B7 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_116_UnE,axiom,
% 0.32/1.54 ! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/1.54 => ( ~ ( member @ A @ C4 @ A4 )
% 0.32/1.54 => ( member @ A @ C4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_117_UnI1,axiom,
% 0.32/1.54 ! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( member @ A @ C4 @ A4 )
% 0.32/1.54 => ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_118_UnI2,axiom,
% 0.32/1.54 ! [A: $tType,C4: A,B5: set @ A,A4: set @ A] :
% 0.32/1.54 ( ( member @ A @ C4 @ B5 )
% 0.32/1.54 => ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_119_bex__Un,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
% 0.32/1.54 ( ( ? [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/1.54 & ( P @ X4 ) ) )
% 0.32/1.54 = ( ? [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ A4 )
% 0.32/1.54 & ( P @ X4 ) )
% 0.32/1.54 | ? [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ B5 )
% 0.32/1.54 & ( P @ X4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_120_ball__Un,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
% 0.32/1.54 ( ( ! [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/1.54 => ( P @ X4 ) ) )
% 0.32/1.54 = ( ! [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ A4 )
% 0.32/1.54 => ( P @ X4 ) )
% 0.32/1.54 & ! [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ B5 )
% 0.32/1.54 => ( P @ X4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_121_Un__assoc,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
% 0.32/1.54 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_122_Un__absorb,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 0.32/1.54 = A4 ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_123_Un__commute,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_124_Un__left__absorb,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/1.54 = ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_125_Un__left__commute,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
% 0.32/1.54 = ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_126_empty__def,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bot_bot @ ( set @ A ) )
% 0.32/1.54 = ( collect @ A
% 0.32/1.54 @ ^ [X4: A] : $false ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_127_Collect__subset,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,P: A > $o] :
% 0.32/1.54 ( ord_less_eq @ ( set @ A )
% 0.32/1.54 @ ( collect @ A
% 0.32/1.54 @ ^ [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ A4 )
% 0.32/1.54 & ( P @ X4 ) ) )
% 0.32/1.54 @ A4 ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_128_insert__compr,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( insert @ A )
% 0.32/1.54 = ( ^ [A2: A,B6: set @ A] :
% 0.32/1.54 ( collect @ A
% 0.32/1.54 @ ^ [X4: A] :
% 0.32/1.54 ( ( X4 = A2 )
% 0.32/1.54 | ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_129_insert__Collect,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,P: A > $o] :
% 0.32/1.54 ( ( insert @ A @ A3 @ ( collect @ A @ P ) )
% 0.32/1.54 = ( collect @ A
% 0.32/1.54 @ ^ [U: A] :
% 0.32/1.54 ( ( U != A3 )
% 0.32/1.54 => ( P @ U ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_130_Un__def,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( collect @ A
% 0.32/1.54 @ ^ [X4: A] :
% 0.32/1.54 ( ( member @ A @ X4 @ A5 )
% 0.32/1.54 | ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_131_Collect__disj__eq,axiom,
% 0.32/1.54 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.54 ( ( collect @ A
% 0.32/1.54 @ ^ [X4: A] :
% 0.32/1.54 ( ( P @ X4 )
% 0.32/1.54 | ( Q @ X4 ) ) )
% 0.32/1.54 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_132_singletonD,axiom,
% 0.32/1.54 ! [A: $tType,B4: A,A3: A] :
% 0.32/1.54 ( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 => ( B4 = A3 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_133_singleton__iff,axiom,
% 0.32/1.54 ! [A: $tType,B4: A,A3: A] :
% 0.32/1.54 ( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 = ( B4 = A3 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_134_doubleton__eq__iff,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,B4: A,C4: A,D3: A] :
% 0.32/1.54 ( ( ( insert @ A @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 = ( insert @ A @ C4 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/1.54 = ( ( ( A3 = C4 )
% 0.32/1.54 & ( B4 = D3 ) )
% 0.32/1.54 | ( ( A3 = D3 )
% 0.32/1.54 & ( B4 = C4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_135_insert__not__empty,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,A4: set @ A] :
% 0.32/1.54 ( ( insert @ A @ A3 @ A4 )
% 0.32/1.54 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_136_singleton__inject,axiom,
% 0.32/1.54 ! [A: $tType,A3: A,B4: A] :
% 0.32/1.54 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.32/1.54 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 => ( A3 = B4 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_137_insert__mono,axiom,
% 0.32/1.54 ! [A: $tType,C3: set @ A,D4: set @ A,A3: A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ C3 @ D4 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C3 ) @ ( insert @ A @ A3 @ D4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_138_subset__insert,axiom,
% 0.32/1.54 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ~ ( member @ A @ X @ A4 )
% 0.32/1.54 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
% 0.32/1.54 = ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_139_subset__insertI,axiom,
% 0.32/1.54 ! [A: $tType,B5: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A3 @ B5 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_140_subset__insertI2,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,B4: A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_141_Un__empty__left,axiom,
% 0.32/1.54 ! [A: $tType,B5: set @ A] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
% 0.32/1.54 = B5 ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_142_Un__empty__right,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.32/1.54 = A4 ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_143_Un__mono,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,C3: set @ A,B5: set @ A,D4: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.32/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_144_Un__least,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,C3: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.32/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
% 0.32/1.54 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_145_Un__upper1,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_146_Un__upper2,axiom,
% 0.32/1.54 ! [A: $tType,B5: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_147_Un__absorb1,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 = B5 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_148_Un__absorb2,axiom,
% 0.32/1.54 ! [A: $tType,B5: set @ A,A4: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
% 0.32/1.54 => ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 = A4 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_149_subset__UnE,axiom,
% 0.32/1.54 ! [A: $tType,C3: set @ A,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/1.54 => ~ ! [A6: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A6 @ A4 )
% 0.32/1.54 => ! [B8: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ B8 @ B5 )
% 0.32/1.54 => ( C3
% 0.32/1.54 != ( sup_sup @ ( set @ A ) @ A6 @ B8 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_150_subset__Un__eq,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
% 0.32/1.54 = B6 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_151_Collect__conv__if,axiom,
% 0.32/1.54 ! [A: $tType,P: A > $o,A3: A] :
% 0.32/1.54 ( ( ( P @ A3 )
% 0.32/1.54 => ( ( collect @ A
% 0.32/1.54 @ ^ [X4: A] :
% 0.32/1.54 ( ( X4 = A3 )
% 0.32/1.54 & ( P @ X4 ) ) )
% 0.32/1.54 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/1.54 & ( ~ ( P @ A3 )
% 0.32/1.54 => ( ( collect @ A
% 0.32/1.54 @ ^ [X4: A] :
% 0.32/1.54 ( ( X4 = A3 )
% 0.32/1.54 & ( P @ X4 ) ) )
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_152_Collect__conv__if2,axiom,
% 0.32/1.54 ! [A: $tType,P: A > $o,A3: A] :
% 0.32/1.54 ( ( ( P @ A3 )
% 0.32/1.54 => ( ( collect @ A
% 0.32/1.54 @ ^ [X4: A] :
% 0.32/1.54 ( ( A3 = X4 )
% 0.32/1.54 & ( P @ X4 ) ) )
% 0.32/1.54 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/1.54 & ( ~ ( P @ A3 )
% 0.32/1.54 => ( ( collect @ A
% 0.32/1.54 @ ^ [X4: A] :
% 0.32/1.54 ( ( A3 = X4 )
% 0.32/1.54 & ( P @ X4 ) ) )
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_153_insert__def,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( insert @ A )
% 0.32/1.54 = ( ^ [A2: A] :
% 0.32/1.54 ( sup_sup @ ( set @ A )
% 0.32/1.54 @ ( collect @ A
% 0.32/1.54 @ ^ [X4: A] : ( X4 = A2 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_154_subset__singletonD,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,X: A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 => ( ( A4
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) )
% 0.32/1.54 | ( A4
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_155_subset__singleton__iff,axiom,
% 0.32/1.54 ! [A: $tType,X5: set @ A,A3: A] :
% 0.32/1.54 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 = ( ( X5
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) )
% 0.32/1.54 | ( X5
% 0.32/1.54 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_156_insert__is__Un,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( insert @ A )
% 0.32/1.54 = ( ^ [A2: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_157_Un__singleton__iff,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
% 0.32/1.54 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 = ( ( ( A4
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) )
% 0.32/1.54 & ( B5
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/1.54 | ( ( A4
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 & ( B5
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 | ( ( A4
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 & ( B5
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_158_singleton__Un__iff,axiom,
% 0.32/1.54 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.32/1.54 ( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
% 0.32/1.54 = ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.32/1.54 = ( ( ( A4
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) )
% 0.32/1.54 & ( B5
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.32/1.54 | ( ( A4
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 & ( B5
% 0.32/1.54 = ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 | ( ( A4
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 & ( B5
% 0.32/1.54 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_159_sup__bot__left,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bounde1808546759up_bot @ A )
% 0.32/1.54 => ! [X: A] :
% 0.32/1.54 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_160_sup__bot__right,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bounde1808546759up_bot @ A )
% 0.32/1.54 => ! [X: A] :
% 0.32/1.54 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_161_bot__eq__sup__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bounde1808546759up_bot @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ( ( bot_bot @ A )
% 0.32/1.54 = ( sup_sup @ A @ X @ Y ) )
% 0.32/1.54 = ( ( X
% 0.32/1.54 = ( bot_bot @ A ) )
% 0.32/1.54 & ( Y
% 0.32/1.54 = ( bot_bot @ A ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_162_sup__eq__bot__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bounde1808546759up_bot @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ( ( sup_sup @ A @ X @ Y )
% 0.32/1.54 = ( bot_bot @ A ) )
% 0.32/1.54 = ( ( X
% 0.32/1.54 = ( bot_bot @ A ) )
% 0.32/1.54 & ( Y
% 0.32/1.54 = ( bot_bot @ A ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_163_sup__bot_Oeq__neutr__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bounde1808546759up_bot @ A )
% 0.32/1.54 => ! [A3: A,B4: A] :
% 0.32/1.54 ( ( ( sup_sup @ A @ A3 @ B4 )
% 0.32/1.54 = ( bot_bot @ A ) )
% 0.32/1.54 = ( ( A3
% 0.32/1.54 = ( bot_bot @ A ) )
% 0.32/1.54 & ( B4
% 0.32/1.54 = ( bot_bot @ A ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_164_sup__bot_Oleft__neutral,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bounde1808546759up_bot @ A )
% 0.32/1.54 => ! [A3: A] :
% 0.32/1.54 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
% 0.32/1.54 = A3 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_165_sup__bot_Oneutr__eq__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bounde1808546759up_bot @ A )
% 0.32/1.54 => ! [A3: A,B4: A] :
% 0.32/1.54 ( ( ( bot_bot @ A )
% 0.32/1.54 = ( sup_sup @ A @ A3 @ B4 ) )
% 0.32/1.54 = ( ( A3
% 0.32/1.54 = ( bot_bot @ A ) )
% 0.32/1.54 & ( B4
% 0.32/1.54 = ( bot_bot @ A ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_166_sup_Oright__idem,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,B4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B4 ) @ B4 )
% 0.32/1.54 = ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_167_sup__left__idem,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.32/1.54 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_168_sup_Oleft__idem,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,B4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B4 ) )
% 0.32/1.54 = ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_169_sup__idem,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A] :
% 0.32/1.54 ( ( sup_sup @ A @ X @ X )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_170_sup_Oidem,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A] :
% 0.32/1.54 ( ( sup_sup @ A @ A3 @ A3 )
% 0.32/1.54 = A3 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_171_sup__apply,axiom,
% 0.32/1.54 ! [B2: $tType,A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ B2 )
% 0.32/1.54 => ( ( sup_sup @ ( A > B2 ) )
% 0.32/1.54 = ( ^ [F3: A > B2,G2: A > B2,X4: A] : ( sup_sup @ B2 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_172_sup_Obounded__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B4: A,C4: A,A3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
% 0.32/1.54 = ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/1.54 & ( ord_less_eq @ A @ C4 @ A3 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_173_le__sup__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,Y: A,Z4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
% 0.32/1.54 = ( ( ord_less_eq @ A @ X @ Z4 )
% 0.32/1.54 & ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_174_sup__bot_Oright__neutral,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bounde1808546759up_bot @ A )
% 0.32/1.54 => ! [A3: A] :
% 0.32/1.54 ( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
% 0.32/1.54 = A3 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_175_less__set__def,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( ord_less @ ( A > $o )
% 0.32/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ A5 )
% 0.32/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_176_psubset__trans,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
% 0.32/1.54 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_177_psubsetD,axiom,
% 0.32/1.54 ! [A: $tType,A4: set @ A,B5: set @ A,C4: A] :
% 0.32/1.54 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.32/1.54 => ( ( member @ A @ C4 @ A4 )
% 0.32/1.54 => ( member @ A @ C4 @ B5 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_178_bot__set__def,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( bot_bot @ ( set @ A ) )
% 0.32/1.54 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_179_sup__set__def,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( sup_sup @ ( set @ A ) )
% 0.32/1.54 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.32/1.54 ( collect @ A
% 0.32/1.54 @ ( sup_sup @ ( A > $o )
% 0.32/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ A5 )
% 0.32/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_180_sup__left__commute,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,Y: A,Z4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) )
% 0.32/1.54 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_181_sup_Oleft__commute,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B4: A,A3: A,C4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ B4 @ ( sup_sup @ A @ A3 @ C4 ) )
% 0.32/1.54 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B4 @ C4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_182_sup__commute,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ( ( sup_sup @ A )
% 0.32/1.54 = ( ^ [X4: A,Y4: A] : ( sup_sup @ A @ Y4 @ X4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_183_sup_Ocommute,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ( ( sup_sup @ A )
% 0.32/1.54 = ( ^ [A2: A,B: A] : ( sup_sup @ A @ B @ A2 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_184_sup__assoc,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,Y: A,Z4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
% 0.32/1.54 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_185_sup_Oassoc,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,B4: A,C4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B4 ) @ C4 )
% 0.32/1.54 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B4 @ C4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_186_boolean__algebra__cancel_Osup2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B5: A,K: A,B4: A,A3: A] :
% 0.32/1.54 ( ( B5
% 0.32/1.54 = ( sup_sup @ A @ K @ B4 ) )
% 0.32/1.54 => ( ( sup_sup @ A @ A3 @ B5 )
% 0.32/1.54 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_187_boolean__algebra__cancel_Osup1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A4: A,K: A,A3: A,B4: A] :
% 0.32/1.54 ( ( A4
% 0.32/1.54 = ( sup_sup @ A @ K @ A3 ) )
% 0.32/1.54 => ( ( sup_sup @ A @ A4 @ B4 )
% 0.32/1.54 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_188_sup__fun__def,axiom,
% 0.32/1.54 ! [B2: $tType,A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ B2 )
% 0.32/1.54 => ( ( sup_sup @ ( A > B2 ) )
% 0.32/1.54 = ( ^ [F3: A > B2,G2: A > B2,X4: A] : ( sup_sup @ B2 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_189_inf__sup__aci_I5_J,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( lattice @ A )
% 0.32/1.54 => ( ( sup_sup @ A )
% 0.32/1.54 = ( ^ [X4: A,Y4: A] : ( sup_sup @ A @ Y4 @ X4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_190_inf__sup__aci_I6_J,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( lattice @ A )
% 0.32/1.54 => ! [X: A,Y: A,Z4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
% 0.32/1.54 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_191_inf__sup__aci_I7_J,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( lattice @ A )
% 0.32/1.54 => ! [X: A,Y: A,Z4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) )
% 0.32/1.54 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_192_inf__sup__aci_I8_J,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( lattice @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.32/1.54 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_193_sup_OcoboundedI2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [C4: A,B4: A,A3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ C4 @ B4 )
% 0.32/1.54 => ( ord_less_eq @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_194_sup_OcoboundedI1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [C4: A,A3: A,B4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ C4 @ A3 )
% 0.32/1.54 => ( ord_less_eq @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_195_sup_Oabsorb__iff2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ( ( ord_less_eq @ A )
% 0.32/1.54 = ( ^ [A2: A,B: A] :
% 0.32/1.54 ( ( sup_sup @ A @ A2 @ B )
% 0.32/1.54 = B ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_196_sup_Oabsorb__iff1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ( ( ord_less_eq @ A )
% 0.32/1.54 = ( ^ [B: A,A2: A] :
% 0.32/1.54 ( ( sup_sup @ A @ A2 @ B )
% 0.32/1.54 = A2 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_197_sup_Ocobounded2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B4: A,A3: A] : ( ord_less_eq @ A @ B4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_198_sup_Ocobounded1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,B4: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_199_sup_Oorder__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ( ( ord_less_eq @ A )
% 0.32/1.54 = ( ^ [B: A,A2: A] :
% 0.32/1.54 ( A2
% 0.32/1.54 = ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_200_sup_OboundedI,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B4: A,A3: A,C4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/1.54 => ( ( ord_less_eq @ A @ C4 @ A3 )
% 0.32/1.54 => ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_201_sup_OboundedE,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B4: A,C4: A,A3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
% 0.32/1.54 => ~ ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/1.54 => ~ ( ord_less_eq @ A @ C4 @ A3 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_202_sup__absorb2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.54 => ( ( sup_sup @ A @ X @ Y )
% 0.32/1.54 = Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_203_sup__absorb1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [Y: A,X: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/1.54 => ( ( sup_sup @ A @ X @ Y )
% 0.32/1.54 = X ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_204_sup_Oabsorb2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,B4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/1.54 => ( ( sup_sup @ A @ A3 @ B4 )
% 0.32/1.54 = B4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_205_sup_Oabsorb1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B4: A,A3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/1.54 => ( ( sup_sup @ A @ A3 @ B4 )
% 0.32/1.54 = A3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_206_sup__unique,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [F2: A > A > A,X: A,Y: A] :
% 0.32/1.54 ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y3 ) )
% 0.32/1.54 => ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F2 @ X3 @ Y3 ) )
% 0.32/1.54 => ( ! [X3: A,Y3: A,Z: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.32/1.54 => ( ( ord_less_eq @ A @ Z @ X3 )
% 0.32/1.54 => ( ord_less_eq @ A @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
% 0.32/1.54 => ( ( sup_sup @ A @ X @ Y )
% 0.32/1.54 = ( F2 @ X @ Y ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_207_sup_OorderI,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,B4: A] :
% 0.32/1.54 ( ( A3
% 0.32/1.54 = ( sup_sup @ A @ A3 @ B4 ) )
% 0.32/1.54 => ( ord_less_eq @ A @ B4 @ A3 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_208_sup_OorderE,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B4: A,A3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/1.54 => ( A3
% 0.32/1.54 = ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_209_le__iff__sup,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ( ( ord_less_eq @ A )
% 0.32/1.54 = ( ^ [X4: A,Y4: A] :
% 0.32/1.54 ( ( sup_sup @ A @ X4 @ Y4 )
% 0.32/1.54 = Y4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_210_sup__least,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [Y: A,X: A,Z4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/1.54 => ( ( ord_less_eq @ A @ Z4 @ X )
% 0.32/1.54 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z4 ) @ X ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_211_sup__mono,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,C4: A,B4: A,D3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ C4 )
% 0.32/1.54 => ( ( ord_less_eq @ A @ B4 @ D3 )
% 0.32/1.54 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ ( sup_sup @ A @ C4 @ D3 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_212_sup_Omono,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [C4: A,A3: A,D3: A,B4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ C4 @ A3 )
% 0.32/1.54 => ( ( ord_less_eq @ A @ D3 @ B4 )
% 0.32/1.54 => ( ord_less_eq @ A @ ( sup_sup @ A @ C4 @ D3 ) @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_213_le__supI2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,B4: A,A3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X @ B4 )
% 0.32/1.54 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_214_le__supI1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,A3: A,B4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X @ A3 )
% 0.32/1.54 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_215_sup__ge2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_216_sup__ge1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_217_le__supI,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,X: A,B4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ X )
% 0.32/1.54 => ( ( ord_less_eq @ A @ B4 @ X )
% 0.32/1.54 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ X ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_218_le__supE,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [A3: A,B4: A,X: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ X )
% 0.32/1.54 => ~ ( ( ord_less_eq @ A @ A3 @ X )
% 0.32/1.54 => ~ ( ord_less_eq @ A @ B4 @ X ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_219_inf__sup__ord_I3_J,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( lattice @ A )
% 0.32/1.54 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_220_inf__sup__ord_I4_J,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( lattice @ A )
% 0.32/1.54 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_221_less__supI1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,A3: A,B4: A] :
% 0.32/1.54 ( ( ord_less @ A @ X @ A3 )
% 0.32/1.54 => ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_222_less__supI2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [X: A,B4: A,A3: A] :
% 0.32/1.54 ( ( ord_less @ A @ X @ B4 )
% 0.32/1.54 => ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_223_sup_Ostrict__boundedE,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [B4: A,C4: A,A3: A] :
% 0.32/1.54 ( ( ord_less @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
% 0.32/1.54 => ~ ( ( ord_less @ A @ B4 @ A3 )
% 0.32/1.54 => ~ ( ord_less @ A @ C4 @ A3 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_224_sup_Ostrict__order__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ( ( ord_less @ A )
% 0.32/1.54 = ( ^ [B: A,A2: A] :
% 0.32/1.54 ( ( A2
% 0.32/1.54 = ( sup_sup @ A @ A2 @ B ) )
% 0.32/1.54 & ( A2 != B ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_225_sup_Ostrict__coboundedI1,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [C4: A,A3: A,B4: A] :
% 0.32/1.54 ( ( ord_less @ A @ C4 @ A3 )
% 0.32/1.54 => ( ord_less @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_226_sup_Ostrict__coboundedI2,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A )
% 0.32/1.54 => ! [C4: A,B4: A,A3: A] :
% 0.32/1.54 ( ( ord_less @ A @ C4 @ B4 )
% 0.32/1.54 => ( ord_less @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_227_predicate1I,axiom,
% 0.32/1.54 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.54 ( ! [X3: A] :
% 0.32/1.54 ( ( P @ X3 )
% 0.32/1.54 => ( Q @ X3 ) )
% 0.32/1.54 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_228_pred__subset__eq,axiom,
% 0.32/1.54 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.32/1.54 ( ( ord_less_eq @ ( A > $o )
% 0.32/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ R )
% 0.32/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ S ) )
% 0.32/1.54 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_229_the__elem__eq,axiom,
% 0.32/1.54 ! [A: $tType,X: A] :
% 0.32/1.54 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.32/1.54 = X ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_230_order__refl,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( preorder @ A )
% 0.32/1.54 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_231_sup1CI,axiom,
% 0.32/1.54 ! [A: $tType,B5: A > $o,X: A,A4: A > $o] :
% 0.32/1.54 ( ( ~ ( B5 @ X )
% 0.32/1.54 => ( A4 @ X ) )
% 0.32/1.54 => ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_232_bot__apply,axiom,
% 0.32/1.54 ! [C: $tType,D: $tType] :
% 0.32/1.54 ( ( bot @ C )
% 0.32/1.54 => ( ( bot_bot @ ( D > C ) )
% 0.32/1.54 = ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_233_sup1I2,axiom,
% 0.32/1.54 ! [A: $tType,B5: A > $o,X: A,A4: A > $o] :
% 0.32/1.54 ( ( B5 @ X )
% 0.32/1.54 => ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_234_sup1I1,axiom,
% 0.32/1.54 ! [A: $tType,A4: A > $o,X: A,B5: A > $o] :
% 0.32/1.54 ( ( A4 @ X )
% 0.32/1.54 => ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_235_sup1E,axiom,
% 0.32/1.54 ! [A: $tType,A4: A > $o,B5: A > $o,X: A] :
% 0.32/1.54 ( ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X )
% 0.32/1.54 => ( ~ ( A4 @ X )
% 0.32/1.54 => ( B5 @ X ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_236_order__subst1,axiom,
% 0.32/1.54 ! [A: $tType,B2: $tType] :
% 0.32/1.54 ( ( ( order @ B2 )
% 0.32/1.54 & ( order @ A ) )
% 0.32/1.54 => ! [A3: A,F2: B2 > A,B4: B2,C4: B2] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ ( F2 @ B4 ) )
% 0.32/1.54 => ( ( ord_less_eq @ B2 @ B4 @ C4 )
% 0.32/1.54 => ( ! [X3: B2,Y3: B2] :
% 0.32/1.54 ( ( ord_less_eq @ B2 @ X3 @ Y3 )
% 0.32/1.54 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.32/1.54 => ( ord_less_eq @ A @ A3 @ ( F2 @ C4 ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_237_order__subst2,axiom,
% 0.32/1.54 ! [A: $tType,C: $tType] :
% 0.32/1.54 ( ( ( order @ C )
% 0.32/1.54 & ( order @ A ) )
% 0.32/1.54 => ! [A3: A,B4: A,F2: A > C,C4: C] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/1.54 => ( ( ord_less_eq @ C @ ( F2 @ B4 ) @ C4 )
% 0.32/1.54 => ( ! [X3: A,Y3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.32/1.54 => ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.32/1.54 => ( ord_less_eq @ C @ ( F2 @ A3 ) @ C4 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_238_ord__eq__le__subst,axiom,
% 0.32/1.54 ! [A: $tType,B2: $tType] :
% 0.32/1.54 ( ( ( ord @ B2 )
% 0.32/1.54 & ( ord @ A ) )
% 0.32/1.54 => ! [A3: A,F2: B2 > A,B4: B2,C4: B2] :
% 0.32/1.54 ( ( A3
% 0.32/1.54 = ( F2 @ B4 ) )
% 0.32/1.54 => ( ( ord_less_eq @ B2 @ B4 @ C4 )
% 0.32/1.54 => ( ! [X3: B2,Y3: B2] :
% 0.32/1.54 ( ( ord_less_eq @ B2 @ X3 @ Y3 )
% 0.32/1.54 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.32/1.54 => ( ord_less_eq @ A @ A3 @ ( F2 @ C4 ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_239_ord__le__eq__subst,axiom,
% 0.32/1.54 ! [A: $tType,B2: $tType] :
% 0.32/1.54 ( ( ( ord @ B2 )
% 0.32/1.54 & ( ord @ A ) )
% 0.32/1.54 => ! [A3: A,B4: A,F2: A > B2,C4: B2] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/1.54 => ( ( ( F2 @ B4 )
% 0.32/1.54 = C4 )
% 0.32/1.54 => ( ! [X3: A,Y3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.32/1.54 => ( ord_less_eq @ B2 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.32/1.54 => ( ord_less_eq @ B2 @ ( F2 @ A3 ) @ C4 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_240_eq__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( order @ A )
% 0.32/1.54 => ( ( ^ [Y2: A,Z3: A] : ( Y2 = Z3 ) )
% 0.32/1.54 = ( ^ [X4: A,Y4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.32/1.54 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_241_antisym,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( order @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.54 => ( ( ord_less_eq @ A @ Y @ X )
% 0.32/1.54 => ( X = Y ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_242_linear,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( linorder @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.54 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_243_eq__refl,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( preorder @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ( X = Y )
% 0.32/1.54 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_244_le__cases,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( linorder @ A )
% 0.32/1.54 => ! [X: A,Y: A] :
% 0.32/1.54 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.32/1.54 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_245_order_Otrans,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( order @ A )
% 0.32/1.54 => ! [A3: A,B4: A,C4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/1.54 => ( ( ord_less_eq @ A @ B4 @ C4 )
% 0.32/1.54 => ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_246_le__cases3,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( linorder @ A )
% 0.32/1.54 => ! [X: A,Y: A,Z4: A] :
% 0.32/1.54 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.54 => ~ ( ord_less_eq @ A @ Y @ Z4 ) )
% 0.32/1.54 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.32/1.54 => ~ ( ord_less_eq @ A @ X @ Z4 ) )
% 0.32/1.54 => ( ( ( ord_less_eq @ A @ X @ Z4 )
% 0.32/1.54 => ~ ( ord_less_eq @ A @ Z4 @ Y ) )
% 0.32/1.54 => ( ( ( ord_less_eq @ A @ Z4 @ Y )
% 0.32/1.54 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.32/1.54 => ( ( ( ord_less_eq @ A @ Y @ Z4 )
% 0.32/1.54 => ~ ( ord_less_eq @ A @ Z4 @ X ) )
% 0.32/1.54 => ~ ( ( ord_less_eq @ A @ Z4 @ X )
% 0.32/1.54 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_247_antisym__conv,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( order @ A )
% 0.32/1.54 => ! [Y: A,X: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/1.54 => ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.54 = ( X = Y ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( order @ A )
% 0.32/1.54 => ( ( ^ [Y2: A,Z3: A] : ( Y2 = Z3 ) )
% 0.32/1.54 = ( ^ [A2: A,B: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/1.54 & ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_249_ord__eq__le__trans,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord @ A )
% 0.32/1.54 => ! [A3: A,B4: A,C4: A] :
% 0.32/1.54 ( ( A3 = B4 )
% 0.32/1.54 => ( ( ord_less_eq @ A @ B4 @ C4 )
% 0.32/1.54 => ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_250_ord__le__eq__trans,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( ord @ A )
% 0.32/1.54 => ! [A3: A,B4: A,C4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/1.54 => ( ( B4 = C4 )
% 0.32/1.54 => ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_251_order__class_Oorder_Oantisym,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( order @ A )
% 0.32/1.54 => ! [A3: A,B4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.32/1.54 => ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.32/1.54 => ( A3 = B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_252_order__trans,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( preorder @ A )
% 0.32/1.54 => ! [X: A,Y: A,Z4: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.54 => ( ( ord_less_eq @ A @ Y @ Z4 )
% 0.32/1.54 => ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_253_dual__order_Orefl,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( order @ A )
% 0.32/1.54 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_254_linorder__wlog,axiom,
% 0.32/1.54 ! [A: $tType] :
% 0.32/1.54 ( ( linorder @ A )
% 0.32/1.54 => ! [P: A > A > $o,A3: A,B4: A] :
% 0.32/1.54 ( ! [A7: A,B3: A] :
% 0.32/1.54 ( ( ord_less_eq @ A @ A7 @ B3 )
% 0.32/1.54 => ( P @ A7 @ B3 ) )
% 0.32/1.54 => ( ! [A7: A,B3: A] :
% 0.32/1.54 ( ( P @ B3 @ A7 )
% 0.32/1.54 => ( P @ A7 @ B3 ) )
% 0.32/1.54 => ( P @ A3 @ B4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.32/1.54 bounded_lattice @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.32/1.54 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.32/1.54 ! [A8: $tType,A9: $tType] :
% 0.32/1.54 ( ( bounded_lattice @ A9 )
% 0.32/1.54 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.32/1.54 ! [A8: $tType,A9: $tType] :
% 0.32/1.54 ( ( bounded_lattice @ A9 )
% 0.32/1.54 => ( bounde1808546759up_bot @ ( A8 > A9 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.32/1.54 ! [A8: $tType,A9: $tType] :
% 0.32/1.54 ( ( semilattice_sup @ A9 )
% 0.32/1.54 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/1.54 ! [A8: $tType,A9: $tType] :
% 0.32/1.54 ( ( preorder @ A9 )
% 0.32/1.54 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.32/1.54 ! [A8: $tType,A9: $tType] :
% 0.32/1.54 ( ( lattice @ A9 )
% 0.32/1.54 => ( lattice @ ( A8 > A9 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/1.54 ! [A8: $tType,A9: $tType] :
% 0.32/1.54 ( ( order @ A9 )
% 0.32/1.54 => ( order @ ( A8 > A9 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/1.54 ! [A8: $tType,A9: $tType] :
% 0.32/1.54 ( ( ord @ A9 )
% 0.32/1.54 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_fun___Orderings_Obot,axiom,
% 0.32/1.54 ! [A8: $tType,A9: $tType] :
% 0.32/1.54 ( ( bot @ A9 )
% 0.32/1.54 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.32/1.54 condit1037483654norder @ int ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
% 0.32/1.54 semilattice_sup @ int ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.32/1.54 linordered_idom @ int ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Int_Oint___Orderings_Opreorder_4,axiom,
% 0.32/1.54 preorder @ int ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.32/1.54 linorder @ int ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Int_Oint___Lattices_Olattice_5,axiom,
% 0.32/1.54 lattice @ int ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Int_Oint___Orderings_Oorder_6,axiom,
% 0.32/1.54 order @ int ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Int_Oint___Orderings_Oord_7,axiom,
% 0.32/1.54 ord @ int ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_8,axiom,
% 0.32/1.54 ! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_9,axiom,
% 0.32/1.54 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_10,axiom,
% 0.32/1.54 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Lattices_Olattice_11,axiom,
% 0.32/1.54 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Orderings_Oorder_12,axiom,
% 0.32/1.54 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Orderings_Oord_13,axiom,
% 0.32/1.54 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Orderings_Obot_14,axiom,
% 0.32/1.54 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_15,axiom,
% 0.32/1.54 bounde1808546759up_bot @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_16,axiom,
% 0.32/1.54 semilattice_sup @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_17,axiom,
% 0.32/1.54 preorder @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder_18,axiom,
% 0.32/1.54 linorder @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Lattices_Olattice_19,axiom,
% 0.32/1.54 lattice @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_20,axiom,
% 0.32/1.54 order @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Oord_21,axiom,
% 0.32/1.54 ord @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Obot_22,axiom,
% 0.32/1.54 bot @ $o ).
% 0.32/1.54
% 0.32/1.54 thf(conj_0,conjecture,
% 0.32/1.54 ord_less @ int @ ( h @ l ) @ ( h @ ( binary576689334lle_rm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ) ) ).
% 0.32/1.54 ------------------------
% 0.32/1.56 % (18475)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_18142 for (2999ds/27Mi)
% 0.32/1.56 % (18474)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_18142 for (2999ds/4Mi)
% 0.32/1.56 % (18476)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18142 for (2999ds/2Mi)
% 0.32/1.56 % (18476)Instruction limit reached!
% 0.32/1.56 % (18476)------------------------------
% 0.32/1.56 % (18476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.56 % (18476)Termination reason: Unknown
% 0.32/1.56 % (18476)Termination phase: shuffling
% 0.32/1.56
% 0.32/1.56 % (18476)Memory used [KB]: 1279
% 0.32/1.56 % (18476)Time elapsed: 0.002 s
% 0.32/1.56 % (18476)Instructions burned: 3 (million)
% 0.32/1.56 % (18476)------------------------------
% 0.32/1.56 % (18476)------------------------------
% 0.32/1.56 % (18474)Instruction limit reached!
% 0.32/1.56 % (18474)------------------------------
% 0.32/1.56 % (18474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.56 % (18474)Termination reason: Unknown
% 0.32/1.56 % (18474)Termination phase: shuffling
% 0.32/1.56
% 0.32/1.56 % (18474)Memory used [KB]: 1407
% 0.32/1.56 % (18474)Time elapsed: 0.003 s
% 0.32/1.56 % (18474)Instructions burned: 6 (million)
% 0.32/1.56 % (18474)------------------------------
% 0.32/1.56 % (18474)------------------------------
% 0.32/1.56 % (18473)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18142 for (2999ds/183Mi)
% 0.32/1.56 % (18477)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_18142 for (2999ds/2Mi)
% 0.32/1.56 % (18478)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_18142 for (2999ds/275Mi)
% 0.32/1.56 % (18479)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18142 for (2999ds/18Mi)
% 0.32/1.57 % (18475)Instruction limit reached!
% 0.32/1.57 % (18475)------------------------------
% 0.32/1.57 % (18475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (18475)Termination reason: Unknown
% 0.32/1.57 % (18475)Termination phase: Property scanning
% 0.32/1.57
% 0.32/1.57 % (18475)Memory used [KB]: 1791
% 0.32/1.57 % (18475)Time elapsed: 0.010 s
% 0.32/1.57 % (18475)Instructions burned: 28 (million)
% 0.32/1.57 % (18475)------------------------------
% 0.32/1.57 % (18475)------------------------------
% 0.32/1.57 % (18477)Instruction limit reached!
% 0.32/1.57 % (18477)------------------------------
% 0.32/1.57 % (18477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (18477)Termination reason: Unknown
% 0.32/1.57 % (18477)Termination phase: shuffling
% 0.32/1.57
% 0.32/1.57 % (18477)Memory used [KB]: 1279
% 0.32/1.57 % (18477)Time elapsed: 0.004 s
% 0.32/1.57 % (18477)Instructions burned: 3 (million)
% 0.32/1.57 % (18477)------------------------------
% 0.32/1.57 % (18477)------------------------------
% 0.32/1.57 % (18480)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18142 for (2999ds/3Mi)
% 0.32/1.57 % (18481)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_18142 for (2999ds/37Mi)
% 0.32/1.57 % (18480)Instruction limit reached!
% 0.32/1.57 % (18480)------------------------------
% 0.32/1.57 % (18480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (18480)Termination reason: Unknown
% 0.32/1.57 % (18480)Termination phase: shuffling
% 0.32/1.57
% 0.32/1.57 % (18480)Memory used [KB]: 1279
% 0.32/1.57 % (18480)Time elapsed: 0.003 s
% 0.32/1.57 % (18480)Instructions burned: 4 (million)
% 0.32/1.57 % (18480)------------------------------
% 0.32/1.57 % (18480)------------------------------
% 0.32/1.57 % (18482)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_18142 for (2999ds/15Mi)
% 0.32/1.58 % (18479)Instruction limit reached!
% 0.32/1.58 % (18479)------------------------------
% 0.32/1.58 % (18479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (18479)Termination reason: Unknown
% 0.32/1.58 % (18479)Termination phase: shuffling
% 0.32/1.58
% 0.32/1.58 % (18479)Memory used [KB]: 1663
% 0.32/1.58 % (18479)Time elapsed: 0.014 s
% 0.32/1.58 % (18479)Instructions burned: 19 (million)
% 0.32/1.58 % (18479)------------------------------
% 0.32/1.58 % (18479)------------------------------
% 0.32/1.58 % (18482)Instruction limit reached!
% 0.32/1.58 % (18482)------------------------------
% 0.32/1.58 % (18482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (18482)Termination reason: Unknown
% 0.32/1.58 % (18482)Termination phase: shuffling
% 0.32/1.58
% 0.32/1.58 % (18482)Memory used [KB]: 1663
% 0.32/1.58 % (18482)Time elapsed: 0.006 s
% 0.32/1.58 % (18482)Instructions burned: 15 (million)
% 0.32/1.58 % (18482)------------------------------
% 0.32/1.58 % (18482)------------------------------
% 0.32/1.58 % (18484)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_18142 for (2999ds/1041Mi)
% 0.32/1.58 % (18481)Instruction limit reached!
% 0.32/1.58 % (18481)------------------------------
% 0.32/1.58 % (18481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (18481)Termination reason: Unknown
% 0.32/1.58 % (18481)Termination phase: Property scanning
% 0.32/1.58
% 0.32/1.58 % (18481)Memory used [KB]: 2046
% 0.32/1.58 % (18481)Time elapsed: 0.013 s
% 0.32/1.58 % (18481)Instructions burned: 38 (million)
% 0.32/1.58 % (18481)------------------------------
% 0.32/1.58 % (18481)------------------------------
% 0.32/1.58 % (18483)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_18142 for (2999ds/3Mi)
% 0.32/1.59 % (18483)Instruction limit reached!
% 0.32/1.59 % (18483)------------------------------
% 0.32/1.59 % (18483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.59 % (18483)Termination reason: Unknown
% 0.32/1.59 % (18483)Termination phase: shuffling
% 0.32/1.59
% 0.32/1.59 % (18483)Memory used [KB]: 1407
% 0.32/1.59 % (18483)Time elapsed: 0.004 s
% 0.32/1.59 % (18483)Instructions burned: 3 (million)
% 0.32/1.59 % (18483)------------------------------
% 0.32/1.59 % (18483)------------------------------
% 0.32/1.59 % (18487)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_18142 for (2999ds/3Mi)
% 0.32/1.59 % (18486)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18142 for (2999ds/16Mi)
% 0.32/1.59 % (18487)Instruction limit reached!
% 0.32/1.59 % (18487)------------------------------
% 0.32/1.59 % (18487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.59 % (18487)Termination reason: Unknown
% 0.32/1.59 % (18487)Termination phase: shuffling
% 0.32/1.59
% 0.32/1.59 % (18487)Memory used [KB]: 1279
% 0.32/1.59 % (18487)Time elapsed: 0.003 s
% 0.32/1.59 % (18487)Instructions burned: 4 (million)
% 0.32/1.59 % (18487)------------------------------
% 0.32/1.59 % (18487)------------------------------
% 0.32/1.59 % (18485)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18142 for (2999ds/7Mi)
% 2.73/1.60 % (18486)Instruction limit reached!
% 2.73/1.60 % (18486)------------------------------
% 2.73/1.60 % (18486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (18486)Termination reason: Unknown
% 2.73/1.60 % (18486)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (18486)Memory used [KB]: 1663
% 2.73/1.60 % (18486)Time elapsed: 0.009 s
% 2.73/1.60 % (18486)Instructions burned: 17 (million)
% 2.73/1.60 % (18486)------------------------------
% 2.73/1.60 % (18486)------------------------------
% 2.73/1.60 % (18485)Instruction limit reached!
% 2.73/1.60 % (18485)------------------------------
% 2.73/1.60 % (18485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (18485)Termination reason: Unknown
% 2.73/1.60 % (18485)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (18485)Memory used [KB]: 1407
% 2.73/1.60 % (18485)Time elapsed: 0.006 s
% 2.73/1.60 % (18485)Instructions burned: 7 (million)
% 2.73/1.60 % (18485)------------------------------
% 2.73/1.60 % (18485)------------------------------
% 2.73/1.60 % (18489)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18142 for (2999ds/7Mi)
% 2.73/1.60 % (18488)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_18142 for (2999ds/3Mi)
% 2.73/1.61 % (18489)Instruction limit reached!
% 2.73/1.61 % (18489)------------------------------
% 2.73/1.61 % (18489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (18489)Termination reason: Unknown
% 2.73/1.61 % (18489)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (18489)Memory used [KB]: 1535
% 2.73/1.61 % (18489)Time elapsed: 0.005 s
% 2.73/1.61 % (18489)Instructions burned: 9 (million)
% 2.73/1.61 % (18489)------------------------------
% 2.73/1.61 % (18489)------------------------------
% 2.73/1.61 % (18488)Instruction limit reached!
% 2.73/1.61 % (18488)------------------------------
% 2.73/1.61 % (18488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (18488)Termination reason: Unknown
% 2.73/1.61 % (18488)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (18488)Memory used [KB]: 1279
% 2.73/1.61 % (18488)Time elapsed: 0.004 s
% 2.73/1.61 % (18488)Instructions burned: 4 (million)
% 2.73/1.61 % (18488)------------------------------
% 2.73/1.61 % (18488)------------------------------
% 2.73/1.61 % (18490)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18142 for (2999ds/3Mi)
% 2.73/1.61 % (18490)Instruction limit reached!
% 2.73/1.61 % (18490)------------------------------
% 2.73/1.61 % (18490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (18490)Termination reason: Unknown
% 2.73/1.61 % (18490)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (18490)Memory used [KB]: 1407
% 2.73/1.61 % (18490)Time elapsed: 0.003 s
% 2.73/1.61 % (18490)Instructions burned: 4 (million)
% 2.73/1.61 % (18490)------------------------------
% 2.73/1.61 % (18490)------------------------------
% 2.73/1.61 % (18478)First to succeed.
% 2.73/1.61 % (18478)Refutation found. Thanks to Tanya!
% 2.73/1.61 % SZS status Theorem for DTF2THF_18142
% 2.73/1.61 % SZS output start Proof for DTF2THF_18142
% 2.73/1.61 thf(type_def_5, type, binary1291135688e_Tree: ($tType) > $tType).
% 2.73/1.61 thf(type_def_7, type, int: $tType).
% 2.73/1.61 thf(type_def_8, type, set: ($tType) > $tType).
% 2.73/1.61 thf(type_def_9, type, a: $tType).
% 2.73/1.61 thf(func_def_0, type, binary1291135688e_Tree: ($tType) > $tType).
% 2.73/1.61 thf(func_def_1, type, set: ($tType) > $tType).
% 2.73/1.61 thf(func_def_2, type, int: $tType).
% 2.73/1.61 thf(func_def_3, type, a: $tType).
% 2.73/1.61 thf(func_def_4, type, bounded_lattice: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_6, type, bot: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_7, type, ord: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_8, type, order: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_9, type, lattice: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_10, type, linorder: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_11, type, preorder: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_12, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_13, type, semilattice_sup: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_14, type, bounde1808546759up_bot: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_15, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_16, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.73/1.61 thf(func_def_17, type, binary210054475elle_T: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_18, type, binary1746293266le_Tip: !>[X0: $tType]:(binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_19, type, binary1830089824insert: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_20, type, binary827270440e_memb: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > $o)).
% 2.73/1.61 thf(func_def_21, type, binary997842527remove: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_22, type, binary576689334lle_rm: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > X0)).
% 2.73/1.61 thf(func_def_23, type, binary1653327646_setOf: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > set @ X0)).
% 2.73/1.61 thf(func_def_24, type, binary1610619414edTree: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > $o)).
% 2.73/1.61 thf(func_def_25, type, binary231205461t_pred: !>[X0: $tType]:((X0 > int) > X0 > X0 > binary1291135688e_Tree @ X0 > $o)).
% 2.73/1.61 thf(func_def_26, type, binary213313527le_wrm: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_27, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.73/1.61 thf(func_def_28, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.73/1.61 thf(func_def_29, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.73/1.61 thf(func_def_30, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.73/1.61 thf(func_def_31, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.73/1.61 thf(func_def_32, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.73/1.61 thf(func_def_33, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.73/1.61 thf(func_def_34, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.73/1.61 thf(func_def_35, type, h: a > int).
% 2.73/1.61 thf(func_def_36, type, l: a).
% 2.73/1.61 thf(func_def_37, type, t1: binary1291135688e_Tree @ a).
% 2.73/1.61 thf(func_def_38, type, t2: binary1291135688e_Tree @ a).
% 2.73/1.61 thf(func_def_39, type, x: a).
% 2.73/1.61 thf(func_def_54, type, sP0: !>[X0: $tType]:(set @ X0 > X0 > set @ X0 > $o)).
% 2.73/1.61 thf(func_def_55, type, sP1: !>[X0: $tType]:(set @ X0 > X0 > set @ X0 > $o)).
% 2.73/1.61 thf(func_def_56, type, sP2: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > $o)).
% 2.73/1.61 thf(func_def_57, type, sP3: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > $o)).
% 2.73/1.61 thf(func_def_58, type, sK4: !>[X0: $tType]:(set @ X0 > X0)).
% 2.73/1.61 thf(func_def_59, type, sK5: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.73/1.61 thf(func_def_60, type, sK6: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_61, type, sK7: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0)).
% 2.73/1.61 thf(func_def_62, type, sK8: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_63, type, sK9: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > (X0 > int) > X0)).
% 2.73/1.61 thf(func_def_64, type, sK10: !>[X0: $tType]:(X0 > binary1291135688e_Tree @ X0 > (X0 > int) > X0)).
% 2.73/1.61 thf(func_def_65, type, sK11: !>[X0: $tType]:(set @ X0 > X0)).
% 2.73/1.61 thf(func_def_66, type, sK12: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.73/1.61 thf(func_def_67, type, sK13: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.73/1.61 thf(func_def_68, type, sK14: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.73/1.61 thf(func_def_69, type, sK15: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.73/1.61 thf(func_def_70, type, sK16: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.73/1.61 thf(func_def_71, type, sK17: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.73/1.61 thf(func_def_72, type, sK18: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.73/1.61 thf(func_def_73, type, sK19: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.73/1.61 thf(func_def_74, type, sK20: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.73/1.61 thf(func_def_75, type, sK21: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.73/1.61 thf(func_def_76, type, sK22: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.73/1.61 thf(func_def_77, type, sK23: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.73/1.61 thf(func_def_78, type, sK24: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.73/1.61 thf(func_def_79, type, sK25: !>[X0: $tType]:(set @ X0 > X0)).
% 2.73/1.61 thf(func_def_80, type, sK26: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_81, type, sK27: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > X0)).
% 2.73/1.61 thf(func_def_82, type, sK28: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > binary1291135688e_Tree @ X0)).
% 2.73/1.61 thf(func_def_83, type, sK29: !>[X0: $tType]:(set @ X0 > set @ X0 > set @ X0 > set @ X0)).
% 2.73/1.61 thf(func_def_84, type, sK30: !>[X0: $tType]:(set @ X0 > set @ X0 > set @ X0 > set @ X0)).
% 2.73/1.61 thf(func_def_85, type, sK31: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > set @ X0)).
% 2.73/1.61 thf(func_def_86, type, sK32: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.73/1.61 thf(func_def_87, type, sK33: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.73/1.61 thf(func_def_88, type, sK34: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.73/1.61 thf(func_def_89, type, sK35: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.73/1.61 thf(func_def_91, type, ph37: !>[X0: $tType]:(X0)).
% 2.73/1.61 thf(f1403,plain,(
% 2.73/1.61 $false),
% 2.73/1.61 inference(subsumption_resolution,[status(thm)],[f1402,f1244])).
% 2.73/1.61 thf(f1244,plain,(
% 2.73/1.61 ((ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ t2))) = $true)),
% 2.73/1.61 inference(cnf_transformation,[status(thm)],[f533])).
% 2.73/1.61 thf(f533,plain,(
% 2.73/1.61 ((ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ t2))) = $true)),
% 2.73/1.61 inference(fool_elimination,[status(thm)],[f532])).
% 2.73/1.61 thf(f532,plain,(
% 2.73/1.61 (ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ t2)))),
% 2.73/1.61 inference(rectify,[status(thm)],[f1])).
% 2.73/1.61 thf(f1,axiom,(
% 2.73/1.61 (ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ t2)))),
% 2.73/1.61 file('/export/starexec/sandbox/tmp/tmp.7sjdMpQ3z4/DTF2THF_18142.p',fact_0_hLess)).
% 2.73/1.61 thf(f1402,plain,(
% 2.73/1.61 ((ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ t2))) != $true)),
% 2.73/1.61 inference(forward_demodulation,[status(thm)],[f1104,f1269])).
% 2.73/1.61 thf(f1269,plain,(
% 2.73/1.61 ((binary576689334lle_rm @ a @ h @ t2) = (binary576689334lle_rm @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2)))),
% 2.73/1.61 inference(cnf_transformation,[status(thm)],[f2])).
% 2.73/1.61 thf(f2,axiom,(
% 2.73/1.61 ((binary576689334lle_rm @ a @ h @ t2) = (binary576689334lle_rm @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2)))),
% 2.73/1.61 file('/export/starexec/sandbox/tmp/tmp.7sjdMpQ3z4/DTF2THF_18142.p',fact_1_rm__res)).
% 2.73/1.61 thf(f1104,plain,(
% 2.73/1.61 ((ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2)))) != $true)),
% 2.73/1.61 inference(cnf_transformation,[status(thm)],[f813])).
% 2.73/1.61 thf(f813,plain,(
% 2.73/1.61 ((ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2)))) != $true)),
% 2.73/1.61 inference(flattening,[status(thm)],[f601])).
% 2.73/1.61 thf(f601,plain,(
% 2.73/1.61 ~((ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2)))) = $true)),
% 2.73/1.61 inference(fool_elimination,[status(thm)],[f600])).
% 2.73/1.61 thf(f600,plain,(
% 2.73/1.61 ~(ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2))))),
% 2.73/1.61 inference(rectify,[status(thm)],[f290])).
% 2.73/1.61 thf(f290,negated_conjecture,(
% 2.73/1.61 ~(ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2))))),
% 2.73/1.61 inference(negated_conjecture,[status(cth)],[f289])).
% 2.73/1.61 thf(f289,conjecture,(
% 2.73/1.61 (ord_less @ int @ (h @ l) @ (h @ (binary576689334lle_rm @ a @ h @ (binary210054475elle_T @ a @ t1 @ x @ t2))))),
% 2.73/1.61 file('/export/starexec/sandbox/tmp/tmp.7sjdMpQ3z4/DTF2THF_18142.p',conj_0)).
% 2.73/1.61 % SZS output end Proof for DTF2THF_18142
% 2.73/1.61 % (18478)------------------------------
% 2.73/1.61 % (18478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (18478)Termination reason: Refutation
% 2.73/1.61
% 2.73/1.61 % (18478)Memory used [KB]: 6780
% 2.73/1.61 % (18478)Time elapsed: 0.049 s
% 2.73/1.61 % (18478)Instructions burned: 90 (million)
% 2.73/1.61 % (18478)------------------------------
% 2.73/1.61 % (18478)------------------------------
% 2.73/1.61 % (18472)Success in time 0.068 s
%------------------------------------------------------------------------------