%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP030^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n018.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 08:54:59 AM UTC 2026
% Result : ContradictoryAxioms 2.92s 1.64s
% Output : Refutation 2.92s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP030^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n018.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 08:23:46 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.27/0.47 ---- Original DTF file ---
% 0.27/0.47 thf(spec,logic,$$dhol).
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % File : ITP030^2 : TPTP v9.2.1. Released v7.5.0.
% 0.27/0.47 % Domain : Interactive Theorem Proving
% 0.27/0.47 % Problem : Sledgehammer BinaryTree problem prob_220__3252356_1
% 0.27/0.47 % Version : Especial.
% 0.27/0.47 % English :
% 0.27/0.47
% 0.27/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.27/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.27/0.47 % Source : [Des21]
% 0.27/0.47 % Names : BinaryTree/prob_220__3252356_1 [Des21]
% 0.27/0.47
% 0.27/0.47 % Status : ContradictoryAxioms
% 0.27/0.47 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.27/0.47 % Syntax : Number of formulae : 362 ( 161 unt; 60 typ; 0 def)
% 0.27/0.47 % Number of atoms : 735 ( 355 equ; 0 cnn)
% 0.27/0.47 % Maximal formula atoms : 7 ( 2 avg)
% 0.27/0.47 % Number of connectives : 3204 ( 106 ~; 20 |; 71 &;2715 @)
% 0.27/0.47 % ( 0 <=>; 292 =>; 0 <=; 0 <~>)
% 0.27/0.47 % Maximal formula depth : 18 ( 6 avg)
% 0.27/0.47 % Number of types : 3 ( 2 usr)
% 0.27/0.47 % Number of type conns : 185 ( 185 >; 0 *; 0 +; 0 <<)
% 0.27/0.47 % Number of symbols : 61 ( 58 usr; 4 con; 0-5 aty)
% 0.27/0.47 % Number of variables : 1066 ( 103 ^; 880 !; 28 ?;1066 :)
% 0.27/0.47 % ( 55 !>; 0 ?*; 0 @-; 0 @+)
% 0.27/0.47 % SPC : TH1_THM_EQU_NAR
% 0.27/0.47
% 0.27/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:15:42.144
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % Could-be-implicit typings (5)
% 0.27/0.47 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
% 0.27/0.47 binary1291135688e_Tree: $tType > $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_t_Option_Ooption,type,
% 0.27/0.47 option: $tType > $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_t_Set_Oset,type,
% 0.27/0.47 set: $tType > $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_t_Int_Oint,type,
% 0.27/0.47 int: $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_tf_a,type,
% 0.27/0.47 a: $tType ).
% 0.27/0.47
% 0.27/0.47 % Explicit typings (55)
% 0.27/0.47 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.27/0.47 bounded_lattice:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_HOL_Otype,type,
% 0.27/0.47 type:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Obot,type,
% 0.27/0.47 bot:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.27/0.47 ord:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Otop,type,
% 0.27/0.47 top:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.27/0.47 order:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Ono__bot,type,
% 0.27/0.47 no_bot:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Ono__top,type,
% 0.27/0.47 no_top:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.27/0.47 linorder:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.27/0.47 preorder:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.27/0.47 order_bot:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.27/0.47 wellorder:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Complete__Lattices_OInf,type,
% 0.27/0.47 complete_Inf:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Odense__order,type,
% 0.27/0.47 dense_order:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.27/0.47 linordered_idom:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Fields_Olinordered__field,type,
% 0.27/0.47 linordered_field:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.27/0.47 semilattice_sup:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.27/0.47 comple187826305attice:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 0.27/0.47 comple1035589618norder:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.27/0.47 bounde1808546759up_bot:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.27/0.47 condit1656338222tinuum:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
% 0.27/0.47 binary210054475elle_T:
% 0.27/0.47 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
% 0.27/0.47 binary1746293266le_Tip:
% 0.27/0.47 !>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Oset__Tree,type,
% 0.27/0.47 binary2130109271t_Tree:
% 0.27/0.47 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oeqs,type,
% 0.27/0.47 binary64540844le_eqs:
% 0.27/0.47 !>[A: $tType] : ( ( A > int ) > A > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
% 0.27/0.47 binary827270440e_memb:
% 0.27/0.47 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
% 0.27/0.47 binary1653327646_setOf:
% 0.27/0.47 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
% 0.27/0.47 binary1610619414edTree:
% 0.27/0.47 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
% 0.27/0.47 binary231205461t_pred:
% 0.27/0.47 !>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Otlookup,type,
% 0.27/0.47 binary1265063667lookup:
% 0.27/0.47 !>[A: $tType] : ( ( A > int ) > int > ( binary1291135688e_Tree @ A ) > ( option @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
% 0.27/0.47 complete_Inf_Inf:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HOL_OThe,type,
% 0.27/0.47 the:
% 0.27/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HOL_Oundefined,type,
% 0.27/0.47 undefined:
% 0.27/0.47 !>[A: $tType] : A ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_If,type,
% 0.27/0.47 if:
% 0.27/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.27/0.47 sup_sup:
% 0.27/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Option_Ooption_ONone,type,
% 0.27/0.47 none:
% 0.27/0.47 !>[A: $tType] : ( option @ A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Option_Ooption_OSome,type,
% 0.27/0.47 some:
% 0.27/0.47 !>[A: $tType] : ( A > ( option @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Option_Ooption_Ocase__option,type,
% 0.27/0.47 case_option:
% 0.27/0.47 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Option_Ooption_Oset__option,type,
% 0.27/0.47 set_option:
% 0.27/0.47 !>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Option_Ooption_Othe,type,
% 0.27/0.47 the2:
% 0.27/0.47 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Option_Othese,type,
% 0.27/0.47 these:
% 0.27/0.47 !>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.27/0.47 bot_bot:
% 0.27/0.47 !>[A: $tType] : A ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.27/0.47 ord_less:
% 0.27/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.27/0.47 top_top:
% 0.27/0.47 !>[A: $tType] : A ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_OCollect,type,
% 0.27/0.47 collect:
% 0.27/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_Obind,type,
% 0.27/0.47 bind:
% 0.27/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_Oimage,type,
% 0.27/0.47 image:
% 0.27/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_Oinsert,type,
% 0.27/0.47 insert:
% 0.27/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_Ois__empty,type,
% 0.27/0.47 is_empty:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_Ois__singleton,type,
% 0.27/0.47 is_singleton:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_Othe__elem,type,
% 0.27/0.47 the_elem:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_member,type,
% 0.27/0.47 member:
% 0.27/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_h,type,
% 0.27/0.47 h: a > int ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_t,type,
% 0.27/0.47 t: binary1291135688e_Tree @ a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_x,type,
% 0.27/0.47 x: a ).
% 0.27/0.47
% 0.27/0.47 % Relevant facts (255)
% 0.27/0.47 thf(fact_0_h,axiom,
% 0.27/0.47 member @ a @ x @ ( binary1653327646_setOf @ a @ t ) ).
% 0.27/0.47
% 0.27/0.47 % h
% 0.27/0.47 thf(fact_1_sorted__distinct,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int,A2: A,B2: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A2 @ B2 @ T ) ).
% 0.27/0.47
% 0.27/0.47 % sorted_distinct
% 0.27/0.47 thf(fact_2_s,axiom,
% 0.27/0.47 binary1610619414edTree @ a @ h @ t ).
% 0.27/0.47
% 0.27/0.47 % s
% 0.27/0.47 thf(fact_3_res,axiom,
% 0.27/0.47 ~ ( binary827270440e_memb @ a @ h @ x @ t ) ).
% 0.27/0.47
% 0.27/0.47 % res
% 0.27/0.47 thf(fact_4__092_060open_062h_Ax_A_092_060noteq_062_Ah_Ax_092_060close_062,axiom,
% 0.27/0.47 ( ( h @ x )
% 0.27/0.47 != ( h @ x ) ) ).
% 0.27/0.47
% 0.27/0.47 % \<open>h x \<noteq> h x\<close>
% 0.27/0.47 thf(fact_5_o1,axiom,
% 0.27/0.47 ! [X: a] :
% 0.27/0.47 ( ( member @ a @ X @ ( binary1653327646_setOf @ a @ t ) )
% 0.27/0.47 => ( ( h @ X )
% 0.27/0.47 != ( h @ x ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % o1
% 0.27/0.47 thf(fact_6_sorted__distinct__pred__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( binary231205461t_pred @ A )
% 0.27/0.47 = ( ^ [H2: A > int,A3: A,B3: A,T2: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( ( binary1610619414edTree @ A @ H2 @ T2 )
% 0.27/0.47 & ( member @ A @ A3 @ ( binary1653327646_setOf @ A @ T2 ) )
% 0.27/0.47 & ( member @ A @ B3 @ ( binary1653327646_setOf @ A @ T2 ) )
% 0.27/0.47 & ( ( H2 @ A3 )
% 0.27/0.47 = ( H2 @ B3 ) ) )
% 0.27/0.47 => ( A3 = B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sorted_distinct_pred_def
% 0.27/0.47 thf(fact_7_tNone,axiom,
% 0.27/0.47 ( ( binary1265063667lookup @ a @ h @ ( h @ x ) @ t )
% 0.27/0.47 = ( none @ a ) ) ).
% 0.27/0.47
% 0.27/0.47 % tNone
% 0.27/0.47 thf(fact_8_sortedTree_Osimps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % sortedTree.simps(1)
% 0.27/0.47 thf(fact_9_sortLemmaL,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 0.27/0.47 => ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% 0.27/0.47
% 0.27/0.47 % sortLemmaL
% 0.27/0.47 thf(fact_10_sortLemmaR,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 0.27/0.47 => ( binary1610619414edTree @ A @ H @ T22 ) ) ).
% 0.27/0.47
% 0.27/0.47 % sortLemmaR
% 0.27/0.47 thf(fact_11_tlookup__none,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,K: int] :
% 0.27/0.47 ( ( ( binary1610619414edTree @ A @ H @ T )
% 0.27/0.47 & ( ( binary1265063667lookup @ A @ H @ K @ T )
% 0.27/0.47 = ( none @ A ) ) )
% 0.27/0.47 => ! [X: A] :
% 0.27/0.47 ( ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) )
% 0.27/0.47 => ( ( H @ X )
% 0.27/0.47 != K ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % tlookup_none
% 0.27/0.47 thf(fact_12_tlookup__some,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,K: int,X2: A] :
% 0.27/0.47 ( ( ( binary1610619414edTree @ A @ H @ T )
% 0.27/0.47 & ( ( binary1265063667lookup @ A @ H @ K @ T )
% 0.27/0.47 = ( some @ A @ X2 ) ) )
% 0.27/0.47 => ( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T ) )
% 0.27/0.47 & ( ( H @ X2 )
% 0.27/0.47 = K ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % tlookup_some
% 0.27/0.47 thf(fact_13_tlookup__finds,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,Y: A] :
% 0.27/0.47 ( ( ( binary1610619414edTree @ A @ H @ T )
% 0.27/0.47 & ( member @ A @ Y @ ( binary1653327646_setOf @ A @ T ) ) )
% 0.27/0.47 => ( ( binary1265063667lookup @ A @ H @ ( H @ Y ) @ T )
% 0.27/0.47 = ( some @ A @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % tlookup_finds
% 0.27/0.47 thf(fact_14_sortedTree_Osimps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 0.27/0.47 = ( ( binary1610619414edTree @ A @ H @ T1 )
% 0.27/0.47 & ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T1 ) )
% 0.27/0.47 => ( ord_less @ int @ ( H @ X3 ) @ ( H @ X2 ) ) )
% 0.27/0.47 & ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T22 ) )
% 0.27/0.47 => ( ord_less @ int @ ( H @ X2 ) @ ( H @ X3 ) ) )
% 0.27/0.47 & ( binary1610619414edTree @ A @ H @ T22 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sortedTree.simps(2)
% 0.27/0.47 thf(fact_15_setOf_Osimps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % setOf.simps(1)
% 0.27/0.47 thf(fact_16_memb__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( binary827270440e_memb @ A )
% 0.27/0.47 = ( ^ [H2: A > int,X3: A,T2: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( case_option @ $o @ A @ $false
% 0.27/0.47 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.27/0.47 @ X3 )
% 0.27/0.47 @ ( binary1265063667lookup @ A @ H2 @ ( H2 @ X3 ) @ T2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % memb_def
% 0.27/0.47 thf(fact_17_Tree_Oinject,axiom,
% 0.27/0.47 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
% 0.27/0.47 = ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
% 0.27/0.47 = ( ( X21 = Y21 )
% 0.27/0.47 & ( X22 = Y22 )
% 0.27/0.47 & ( X23 = Y23 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Tree.inject
% 0.27/0.47 thf(fact_18_Tree_Oexhaust,axiom,
% 0.27/0.47 ! [A: $tType,Y: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( Y
% 0.27/0.47 != ( binary1746293266le_Tip @ A ) )
% 0.27/0.47 => ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Tree.exhaust
% 0.27/0.47 thf(fact_19_Tree_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( P @ ( binary1746293266le_Tip @ A ) )
% 0.27/0.47 => ( ! [X1: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( P @ X1 )
% 0.27/0.47 => ( ( P @ X32 )
% 0.27/0.47 => ( P @ ( binary210054475elle_T @ A @ X1 @ X24 @ X32 ) ) ) )
% 0.27/0.47 => ( P @ Tree ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Tree.induct
% 0.27/0.47 thf(fact_20_tlookup_Osimps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,H: A > int,K: int] :
% 0.27/0.47 ( ( binary1265063667lookup @ A @ H @ K @ ( binary1746293266le_Tip @ A ) )
% 0.27/0.47 = ( none @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % tlookup.simps(1)
% 0.27/0.47 thf(fact_21_tlookup_Osimps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,K: int,H: A > int,X2: A,T1: binary1291135688e_Tree @ A,T22: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( ( ord_less @ int @ K @ ( H @ X2 ) )
% 0.27/0.47 => ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 0.27/0.47 = ( binary1265063667lookup @ A @ H @ K @ T1 ) ) )
% 0.27/0.47 & ( ~ ( ord_less @ int @ K @ ( H @ X2 ) )
% 0.27/0.47 => ( ( ( ord_less @ int @ ( H @ X2 ) @ K )
% 0.27/0.47 => ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 0.27/0.47 = ( binary1265063667lookup @ A @ H @ K @ T22 ) ) )
% 0.27/0.47 & ( ~ ( ord_less @ int @ ( H @ X2 ) @ K )
% 0.27/0.47 => ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 0.27/0.47 = ( some @ A @ X2 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % tlookup.simps(2)
% 0.27/0.47 thf(fact_22_Tree_Odistinct_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( binary1746293266le_Tip @ A )
% 0.27/0.47 != ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Tree.distinct(1)
% 0.27/0.47 thf(fact_23_not__Some__eq,axiom,
% 0.27/0.47 ! [A: $tType,X2: option @ A] :
% 0.27/0.47 ( ( ! [Y3: A] :
% 0.27/0.47 ( X2
% 0.27/0.47 != ( some @ A @ Y3 ) ) )
% 0.27/0.47 = ( X2
% 0.27/0.47 = ( none @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % not_Some_eq
% 0.27/0.47 thf(fact_24_not__None__eq,axiom,
% 0.27/0.47 ! [A: $tType,X2: option @ A] :
% 0.27/0.47 ( ( X2
% 0.27/0.47 != ( none @ A ) )
% 0.27/0.47 = ( ? [Y3: A] :
% 0.27/0.47 ( X2
% 0.27/0.47 = ( some @ A @ Y3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % not_None_eq
% 0.27/0.47 thf(fact_25_case__optionE,axiom,
% 0.27/0.47 ! [A: $tType,P: $o,Q: A > $o,X2: option @ A] :
% 0.27/0.47 ( ( case_option @ $o @ A @ P @ Q @ X2 )
% 0.27/0.47 => ( ( ( X2
% 0.27/0.47 = ( none @ A ) )
% 0.27/0.47 => ~ P )
% 0.27/0.47 => ~ ! [Y4: A] :
% 0.27/0.47 ( ( X2
% 0.27/0.47 = ( some @ A @ Y4 ) )
% 0.27/0.47 => ~ ( Q @ Y4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % case_optionE
% 0.27/0.47 thf(fact_26_disjE__realizer2,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: $o,Q: A > $o,X2: option @ A,R: B > $o,F: B,G: A > B] :
% 0.27/0.47 ( ( case_option @ $o @ A @ P @ Q @ X2 )
% 0.27/0.47 => ( ( P
% 0.27/0.47 => ( R @ F ) )
% 0.27/0.47 => ( ! [Q2: A] :
% 0.27/0.47 ( ( Q @ Q2 )
% 0.27/0.47 => ( R @ ( G @ Q2 ) ) )
% 0.27/0.47 => ( R @ ( case_option @ B @ A @ F @ G @ X2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % disjE_realizer2
% 0.27/0.47 thf(fact_27_option_Odisc__eq__case_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,Option: option @ A] :
% 0.27/0.47 ( ( Option
% 0.27/0.47 = ( none @ A ) )
% 0.27/0.47 = ( case_option @ $o @ A @ $true
% 0.27/0.47 @ ^ [Uu: A] : $false
% 0.27/0.47 @ Option ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.disc_eq_case(1)
% 0.27/0.47 thf(fact_28_option_Odisc__eq__case_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,Option: option @ A] :
% 0.27/0.47 ( ( Option
% 0.27/0.47 != ( none @ A ) )
% 0.27/0.47 = ( case_option @ $o @ A @ $false
% 0.27/0.47 @ ^ [Uu: A] : $true
% 0.27/0.47 @ Option ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.disc_eq_case(2)
% 0.27/0.47 thf(fact_29_option_Oinject,axiom,
% 0.27/0.47 ! [A: $tType,X25: A,Y24: A] :
% 0.27/0.47 ( ( ( some @ A @ X25 )
% 0.27/0.47 = ( some @ A @ Y24 ) )
% 0.27/0.47 = ( X25 = Y24 ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.inject
% 0.27/0.47 thf(fact_30_empty__Collect__eq,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o] :
% 0.27/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.27/0.47 = ( collect @ A @ P ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ~ ( P @ X3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % empty_Collect_eq
% 0.27/0.47 thf(fact_31_Collect__empty__eq,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o] :
% 0.27/0.47 ( ( ( collect @ A @ P )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ~ ( P @ X3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_empty_eq
% 0.27/0.47 thf(fact_32_all__not__in__conv,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A] :
% 0.27/0.47 ( ( ! [X3: A] :
% 0.27/0.47 ~ ( member @ A @ X3 @ A4 ) )
% 0.27/0.47 = ( A4
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % all_not_in_conv
% 0.27/0.47 thf(fact_33_empty__iff,axiom,
% 0.27/0.47 ! [A: $tType,C: A] :
% 0.27/0.47 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % empty_iff
% 0.27/0.47 thf(fact_34_emptyE,axiom,
% 0.27/0.47 ! [A: $tType,A2: A] :
% 0.27/0.47 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % emptyE
% 0.27/0.47 thf(fact_35_equals0D,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,A2: A] :
% 0.27/0.47 ( ( A4
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % equals0D
% 0.27/0.47 thf(fact_36_equals0I,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A] :
% 0.27/0.47 ( ! [Y4: A] :
% 0.27/0.47 ~ ( member @ A @ Y4 @ A4 )
% 0.27/0.47 => ( A4
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % equals0I
% 0.27/0.47 thf(fact_37_ex__in__conv,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A] :
% 0.27/0.47 ( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.27/0.47 = ( A4
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ex_in_conv
% 0.27/0.47 thf(fact_38_empty__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.27/0.47 = ( collect @ A
% 0.27/0.47 @ ^ [X3: A] : $false ) ) ).
% 0.27/0.47
% 0.27/0.47 % empty_def
% 0.27/0.47 thf(fact_39_option_Ocase__distrib,axiom,
% 0.27/0.47 ! [C2: $tType,B: $tType,A: $tType,H: B > C2,F1: B,F2: A > B,Option: option @ A] :
% 0.27/0.47 ( ( H @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
% 0.27/0.47 = ( case_option @ C2 @ A @ ( H @ F1 )
% 0.27/0.47 @ ^ [X3: A] : ( H @ ( F2 @ X3 ) )
% 0.27/0.47 @ Option ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.case_distrib
% 0.27/0.47 thf(fact_40_option_Odistinct_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,X25: A] :
% 0.27/0.47 ( ( none @ A )
% 0.27/0.47 != ( some @ A @ X25 ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.distinct(1)
% 0.27/0.47 thf(fact_41_option_OdiscI,axiom,
% 0.27/0.47 ! [A: $tType,Option: option @ A,X25: A] :
% 0.27/0.47 ( ( Option
% 0.27/0.47 = ( some @ A @ X25 ) )
% 0.27/0.47 => ( Option
% 0.27/0.47 != ( none @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.discI
% 0.27/0.47 thf(fact_42_option_Oexhaust,axiom,
% 0.27/0.47 ! [A: $tType,Y: option @ A] :
% 0.27/0.47 ( ( Y
% 0.27/0.47 != ( none @ A ) )
% 0.27/0.47 => ~ ! [X24: A] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( some @ A @ X24 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.exhaust
% 0.27/0.47 thf(fact_43_option_Oinducts,axiom,
% 0.27/0.47 ! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
% 0.27/0.47 ( ( P @ ( none @ A ) )
% 0.27/0.47 => ( ! [X4: A] : ( P @ ( some @ A @ X4 ) )
% 0.27/0.47 => ( P @ Option ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.inducts
% 0.27/0.47 thf(fact_44_mem__Collect__eq,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.27/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.27/0.47 = ( P @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % mem_Collect_eq
% 0.27/0.47 thf(fact_45_Collect__mem__eq,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A] :
% 0.27/0.47 ( ( collect @ A
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.27/0.47 = A4 ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_mem_eq
% 0.27/0.47 thf(fact_46_Collect__cong,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ( P @ X4 )
% 0.27/0.47 = ( Q @ X4 ) )
% 0.27/0.47 => ( ( collect @ A @ P )
% 0.27/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_cong
% 0.27/0.47 thf(fact_47_ext,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ( F @ X4 )
% 0.27/0.47 = ( G @ X4 ) )
% 0.27/0.47 => ( F = G ) ) ).
% 0.27/0.47
% 0.27/0.47 % ext
% 0.27/0.47 thf(fact_48_split__option__ex,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ^ [P2: ( option @ A ) > $o] :
% 0.27/0.47 ? [X5: option @ A] : ( P2 @ X5 ) )
% 0.27/0.47 = ( ^ [P3: ( option @ A ) > $o] :
% 0.27/0.47 ( ( P3 @ ( none @ A ) )
% 0.27/0.47 | ? [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % split_option_ex
% 0.27/0.47 thf(fact_49_split__option__all,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ^ [P2: ( option @ A ) > $o] :
% 0.27/0.47 ! [X5: option @ A] : ( P2 @ X5 ) )
% 0.27/0.47 = ( ^ [P3: ( option @ A ) > $o] :
% 0.27/0.47 ( ( P3 @ ( none @ A ) )
% 0.27/0.47 & ! [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % split_option_all
% 0.27/0.47 thf(fact_50_combine__options__cases,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,X2: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
% 0.27/0.47 ( ( ( X2
% 0.27/0.47 = ( none @ A ) )
% 0.27/0.47 => ( P @ X2 @ Y ) )
% 0.27/0.47 => ( ( ( Y
% 0.27/0.47 = ( none @ B ) )
% 0.27/0.47 => ( P @ X2 @ Y ) )
% 0.27/0.47 => ( ! [A5: A,B4: B] :
% 0.27/0.47 ( ( X2
% 0.27/0.47 = ( some @ A @ A5 ) )
% 0.27/0.47 => ( ( Y
% 0.27/0.47 = ( some @ B @ B4 ) )
% 0.27/0.47 => ( P @ X2 @ Y ) ) )
% 0.27/0.47 => ( P @ X2 @ Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % combine_options_cases
% 0.27/0.47 thf(fact_51_option_Osimps_I4_J,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,F1: B,F2: A > B] :
% 0.27/0.47 ( ( case_option @ B @ A @ F1 @ F2 @ ( none @ A ) )
% 0.27/0.47 = F1 ) ).
% 0.27/0.47
% 0.27/0.47 % option.simps(4)
% 0.27/0.47 thf(fact_52_option_Osimps_I5_J,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,F1: B,F2: A > B,X25: A] :
% 0.27/0.47 ( ( case_option @ B @ A @ F1 @ F2 @ ( some @ A @ X25 ) )
% 0.27/0.47 = ( F2 @ X25 ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.simps(5)
% 0.27/0.47 thf(fact_53_bot__apply,axiom,
% 0.27/0.47 ! [C2: $tType,D: $tType] :
% 0.27/0.47 ( ( bot @ C2 )
% 0.27/0.47 => ( ( bot_bot @ ( D > C2 ) )
% 0.27/0.47 = ( ^ [X3: D] : ( bot_bot @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_apply
% 0.27/0.47 thf(fact_54_bot_Oextremum__strict,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order_bot @ A )
% 0.27/0.47 => ! [A2: A] :
% 0.27/0.47 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot.extremum_strict
% 0.27/0.47 thf(fact_55_bot_Onot__eq__extremum,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order_bot @ A )
% 0.27/0.47 => ! [A2: A] :
% 0.27/0.47 ( ( A2
% 0.27/0.47 != ( bot_bot @ A ) )
% 0.27/0.47 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot.not_eq_extremum
% 0.27/0.47 thf(fact_56_Set_Ois__empty__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( is_empty @ A )
% 0.27/0.47 = ( ^ [A6: set @ A] :
% 0.27/0.47 ( A6
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Set.is_empty_def
% 0.27/0.47 thf(fact_57_eqs__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( binary64540844le_eqs @ A )
% 0.27/0.47 = ( ^ [H2: A > int,X3: A] :
% 0.27/0.47 ( collect @ A
% 0.27/0.47 @ ^ [Y3: A] :
% 0.27/0.47 ( ( H2 @ Y3 )
% 0.27/0.47 = ( H2 @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eqs_def
% 0.27/0.47 thf(fact_58_option_Osplit__sel,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > B,Option: option @ A] :
% 0.27/0.47 ( ( P @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
% 0.27/0.47 = ( ( ( Option
% 0.27/0.47 = ( none @ A ) )
% 0.27/0.47 => ( P @ F1 ) )
% 0.27/0.47 & ( ( Option
% 0.27/0.47 = ( some @ A @ ( the2 @ A @ Option ) ) )
% 0.27/0.47 => ( P @ ( F2 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.split_sel
% 0.27/0.47 thf(fact_59_option_Osplit__sel__asm,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > B,Option: option @ A] :
% 0.27/0.47 ( ( P @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
% 0.27/0.47 = ( ~ ( ( ( Option
% 0.27/0.47 = ( none @ A ) )
% 0.27/0.47 & ~ ( P @ F1 ) )
% 0.27/0.47 | ( ( Option
% 0.27/0.47 = ( some @ A @ ( the2 @ A @ Option ) ) )
% 0.27/0.47 & ~ ( P @ ( F2 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.split_sel_asm
% 0.27/0.47 thf(fact_60_set__empty__eq,axiom,
% 0.27/0.47 ! [A: $tType,Xo: option @ A] :
% 0.27/0.47 ( ( ( set_option @ A @ Xo )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( Xo
% 0.27/0.47 = ( none @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % set_empty_eq
% 0.27/0.47 thf(fact_61_bot__fun__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( bot @ B )
% 0.27/0.47 => ( ( bot_bot @ ( A > B ) )
% 0.27/0.47 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_fun_def
% 0.27/0.47 thf(fact_62_elem__set,axiom,
% 0.27/0.47 ! [A: $tType,X2: A,Xo: option @ A] :
% 0.27/0.47 ( ( member @ A @ X2 @ ( set_option @ A @ Xo ) )
% 0.27/0.47 = ( Xo
% 0.27/0.47 = ( some @ A @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % elem_set
% 0.27/0.47 thf(fact_63_option_Ocollapse,axiom,
% 0.27/0.47 ! [A: $tType,Option: option @ A] :
% 0.27/0.47 ( ( Option
% 0.27/0.47 != ( none @ A ) )
% 0.27/0.47 => ( ( some @ A @ ( the2 @ A @ Option ) )
% 0.27/0.47 = Option ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.collapse
% 0.27/0.47 thf(fact_64_option_Oset__sel,axiom,
% 0.27/0.47 ! [A: $tType,A2: option @ A] :
% 0.27/0.47 ( ( A2
% 0.27/0.47 != ( none @ A ) )
% 0.27/0.47 => ( member @ A @ ( the2 @ A @ A2 ) @ ( set_option @ A @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.set_sel
% 0.27/0.47 thf(fact_65_bot__set__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.27/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_set_def
% 0.27/0.47 thf(fact_66_not__psubset__empty,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A] :
% 0.27/0.47 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % not_psubset_empty
% 0.27/0.47 thf(fact_67_option_Oexpand,axiom,
% 0.27/0.47 ! [A: $tType,Option: option @ A,Option2: option @ A] :
% 0.27/0.47 ( ( ( Option
% 0.27/0.47 = ( none @ A ) )
% 0.27/0.47 = ( Option2
% 0.27/0.47 = ( none @ A ) ) )
% 0.27/0.47 => ( ( ( Option
% 0.27/0.47 != ( none @ A ) )
% 0.27/0.47 => ( ( Option2
% 0.27/0.47 != ( none @ A ) )
% 0.27/0.47 => ( ( the2 @ A @ Option )
% 0.27/0.47 = ( the2 @ A @ Option2 ) ) ) )
% 0.27/0.47 => ( Option = Option2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.expand
% 0.27/0.47 thf(fact_68_option_Osel,axiom,
% 0.27/0.47 ! [A: $tType,X25: A] :
% 0.27/0.47 ( ( the2 @ A @ ( some @ A @ X25 ) )
% 0.27/0.47 = X25 ) ).
% 0.27/0.47
% 0.27/0.47 % option.sel
% 0.27/0.47 thf(fact_69_option_Oset__intros,axiom,
% 0.27/0.47 ! [A: $tType,X25: A] : ( member @ A @ X25 @ ( set_option @ A @ ( some @ A @ X25 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.set_intros
% 0.27/0.47 thf(fact_70_option_Oset__cases,axiom,
% 0.27/0.47 ! [A: $tType,E: A,A2: option @ A] :
% 0.27/0.47 ( ( member @ A @ E @ ( set_option @ A @ A2 ) )
% 0.27/0.47 => ( A2
% 0.27/0.47 = ( some @ A @ E ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.set_cases
% 0.27/0.47 thf(fact_71_ospec,axiom,
% 0.27/0.47 ! [A: $tType,A4: option @ A,P: A > $o,X2: A] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ ( set_option @ A @ A4 ) )
% 0.27/0.47 => ( P @ X4 ) )
% 0.27/0.47 => ( ( A4
% 0.27/0.47 = ( some @ A @ X2 ) )
% 0.27/0.47 => ( P @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ospec
% 0.27/0.47 thf(fact_72_option_Oexhaust__sel,axiom,
% 0.27/0.47 ! [A: $tType,Option: option @ A] :
% 0.27/0.47 ( ( Option
% 0.27/0.47 != ( none @ A ) )
% 0.27/0.47 => ( Option
% 0.27/0.47 = ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.exhaust_sel
% 0.27/0.47 thf(fact_73_option_Osimps_I14_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( set_option @ A @ ( none @ A ) )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.simps(14)
% 0.27/0.47 thf(fact_74_option_Ocase__eq__if,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( case_option @ B @ A )
% 0.27/0.47 = ( ^ [F12: B,F22: A > B,Option3: option @ A] :
% 0.27/0.47 ( if @ B
% 0.27/0.47 @ ( Option3
% 0.27/0.47 = ( none @ A ) )
% 0.27/0.47 @ F12
% 0.27/0.47 @ ( F22 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.case_eq_if
% 0.27/0.47 thf(fact_75_dual__order_Ostrict__implies__not__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [B2: A,A2: A] :
% 0.27/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.27/0.47 => ( A2 != B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.strict_implies_not_eq
% 0.27/0.47 thf(fact_76_order_Ostrict__implies__not__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A2: A,B2: A] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.27/0.47 => ( A2 != B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order.strict_implies_not_eq
% 0.27/0.47 thf(fact_77_not__less__iff__gr__or__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.27/0.47 = ( ( ord_less @ A @ Y @ X2 )
% 0.27/0.47 | ( X2 = Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % not_less_iff_gr_or_eq
% 0.27/0.47 thf(fact_78_dual__order_Ostrict__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [B2: A,A2: A,C: A] :
% 0.27/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.27/0.47 => ( ( ord_less @ A @ C @ B2 )
% 0.27/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.strict_trans
% 0.27/0.47 thf(fact_79_linorder__less__wlog,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.27/0.47 ( ! [A5: A,B4: A] :
% 0.27/0.47 ( ( ord_less @ A @ A5 @ B4 )
% 0.27/0.47 => ( P @ A5 @ B4 ) )
% 0.27/0.47 => ( ! [A5: A] : ( P @ A5 @ A5 )
% 0.27/0.47 => ( ! [A5: A,B4: A] :
% 0.27/0.47 ( ( P @ B4 @ A5 )
% 0.27/0.47 => ( P @ A5 @ B4 ) )
% 0.27/0.47 => ( P @ A2 @ B2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % linorder_less_wlog
% 0.27/0.47 thf(fact_80_exists__least__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( wellorder @ A )
% 0.27/0.47 => ( ( ^ [P2: A > $o] :
% 0.27/0.47 ? [X5: A] : ( P2 @ X5 ) )
% 0.27/0.47 = ( ^ [P3: A > $o] :
% 0.27/0.47 ? [N: A] :
% 0.27/0.47 ( ( P3 @ N )
% 0.27/0.47 & ! [M: A] :
% 0.27/0.47 ( ( ord_less @ A @ M @ N )
% 0.27/0.47 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % exists_least_iff
% 0.27/0.47 thf(fact_81_less__imp__not__less,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_imp_not_less
% 0.27/0.47 thf(fact_82_order_Ostrict__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A2: A,B2: A,C: A] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.27/0.47 => ( ( ord_less @ A @ B2 @ C )
% 0.27/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order.strict_trans
% 0.27/0.47 thf(fact_83_dual__order_Oirrefl,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A2: A] :
% 0.27/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.irrefl
% 0.27/0.47 thf(fact_84_linorder__cases,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ~ ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ( ( X2 != Y )
% 0.27/0.47 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % linorder_cases
% 0.27/0.47 thf(fact_85_less__imp__triv,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A,P: $o] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ( ( ord_less @ A @ Y @ X2 )
% 0.27/0.47 => P ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_imp_triv
% 0.27/0.47 thf(fact_86_less__imp__not__eq2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ( Y != X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_imp_not_eq2
% 0.27/0.47 thf(fact_87_antisym__conv3,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [Y: A,X2: A] :
% 0.27/0.47 ( ~ ( ord_less @ A @ Y @ X2 )
% 0.27/0.47 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.27/0.47 = ( X2 = Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym_conv3
% 0.27/0.47 thf(fact_88_less__induct,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( wellorder @ A )
% 0.27/0.47 => ! [P: A > $o,A2: A] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ! [Y5: A] :
% 0.27/0.47 ( ( ord_less @ A @ Y5 @ X4 )
% 0.27/0.47 => ( P @ Y5 ) )
% 0.27/0.47 => ( P @ X4 ) )
% 0.27/0.47 => ( P @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_induct
% 0.27/0.47 thf(fact_89_less__not__sym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_not_sym
% 0.27/0.47 thf(fact_90_less__imp__not__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ( X2 != Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_imp_not_eq
% 0.27/0.47 thf(fact_91_dual__order_Oasym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [B2: A,A2: A] :
% 0.27/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.27/0.47 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.asym
% 0.27/0.47 thf(fact_92_ord__less__eq__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord @ A )
% 0.27/0.47 => ! [A2: A,B2: A,C: A] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.27/0.47 => ( ( B2 = C )
% 0.27/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_less_eq_trans
% 0.27/0.47 thf(fact_93_ord__eq__less__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord @ A )
% 0.27/0.47 => ! [A2: A,B2: A,C: A] :
% 0.27/0.47 ( ( A2 = B2 )
% 0.27/0.47 => ( ( ord_less @ A @ B2 @ C )
% 0.27/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_less_trans
% 0.27/0.47 thf(fact_94_less__irrefl,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A] :
% 0.27/0.47 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_irrefl
% 0.27/0.47 thf(fact_95_less__linear,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 | ( X2 = Y )
% 0.27/0.47 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_linear
% 0.27/0.47 thf(fact_96_less__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A,Z2: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ( ( ord_less @ A @ Y @ Z2 )
% 0.27/0.47 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_trans
% 0.27/0.47 thf(fact_97_less__asym_H,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [A2: A,B2: A] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.27/0.47 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_asym'
% 0.27/0.47 thf(fact_98_less__asym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_asym
% 0.27/0.47 thf(fact_99_less__imp__neq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ( X2 != Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_imp_neq
% 0.27/0.47 thf(fact_100_dense,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( dense_order @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ? [Z3: A] :
% 0.27/0.47 ( ( ord_less @ A @ X2 @ Z3 )
% 0.27/0.47 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dense
% 0.27/0.47 thf(fact_101_order_Oasym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A2: A,B2: A] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.27/0.47 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order.asym
% 0.27/0.47 thf(fact_102_neq__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( X2 != Y )
% 0.27/0.47 = ( ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 | ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % neq_iff
% 0.27/0.47 thf(fact_103_neqE,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( X2 != Y )
% 0.27/0.47 => ( ~ ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % neqE
% 0.27/0.47 thf(fact_104_gt__ex,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( no_top @ A )
% 0.27/0.47 => ! [X2: A] :
% 0.27/0.47 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 0.27/0.47
% 0.27/0.47 % gt_ex
% 0.27/0.47 thf(fact_105_lt__ex,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( no_bot @ A )
% 0.27/0.47 => ! [X2: A] :
% 0.27/0.47 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % lt_ex
% 0.27/0.47 thf(fact_106_order__less__subst2,axiom,
% 0.27/0.47 ! [A: $tType,C2: $tType] :
% 0.27/0.47 ( ( ( order @ C2 )
% 0.27/0.47 & ( order @ A ) )
% 0.27/0.47 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.27/0.47 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.27/0.47 => ( ! [X4: A,Y4: A] :
% 0.27/0.47 ( ( ord_less @ A @ X4 @ Y4 )
% 0.27/0.47 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.27/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_less_subst2
% 0.27/0.47 thf(fact_107_order__less__subst1,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( order @ B )
% 0.27/0.47 & ( order @ A ) )
% 0.27/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.27/0.47 => ( ( ord_less @ B @ B2 @ C )
% 0.27/0.47 => ( ! [X4: B,Y4: B] :
% 0.27/0.47 ( ( ord_less @ B @ X4 @ Y4 )
% 0.27/0.47 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.27/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_less_subst1
% 0.27/0.47 thf(fact_108_ord__less__eq__subst,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( ord @ B )
% 0.27/0.47 & ( ord @ A ) )
% 0.27/0.47 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.27/0.47 => ( ( ( F @ B2 )
% 0.27/0.47 = C )
% 0.27/0.47 => ( ! [X4: A,Y4: A] :
% 0.27/0.47 ( ( ord_less @ A @ X4 @ Y4 )
% 0.27/0.47 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.27/0.47 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_less_eq_subst
% 0.27/0.47 thf(fact_109_ord__eq__less__subst,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( ord @ B )
% 0.27/0.47 & ( ord @ A ) )
% 0.27/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.27/0.47 ( ( A2
% 0.27/0.47 = ( F @ B2 ) )
% 0.27/0.47 => ( ( ord_less @ B @ B2 @ C )
% 0.27/0.47 => ( ! [X4: B,Y4: B] :
% 0.27/0.47 ( ( ord_less @ B @ X4 @ Y4 )
% 0.27/0.47 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.27/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_less_subst
% 0.27/0.47 thf(fact_110_option_Osimps_I15_J,axiom,
% 0.27/0.47 ! [A: $tType,X25: A] :
% 0.27/0.47 ( ( set_option @ A @ ( some @ A @ X25 ) )
% 0.27/0.47 = ( insert @ A @ X25 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.simps(15)
% 0.27/0.47 thf(fact_111_option_Othe__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( the2 @ A )
% 0.27/0.47 = ( case_option @ A @ A @ ( undefined @ A )
% 0.27/0.47 @ ^ [X26: A] : X26 ) ) ).
% 0.27/0.47
% 0.27/0.47 % option.the_def
% 0.27/0.47 thf(fact_112_bot__empty__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.27/0.47 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_empty_eq
% 0.27/0.47 thf(fact_113_Collect__empty__eq__bot,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o] :
% 0.27/0.47 ( ( ( collect @ A @ P )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( P
% 0.27/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_empty_eq_bot
% 0.27/0.47 thf(fact_114_ex__gt__or__lt,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( condit1656338222tinuum @ A )
% 0.27/0.47 => ! [A2: A] :
% 0.27/0.47 ? [B4: A] :
% 0.27/0.47 ( ( ord_less @ A @ A2 @ B4 )
% 0.27/0.47 | ( ord_less @ A @ B4 @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ex_gt_or_lt
% 0.27/0.47 thf(fact_115_linorder__neqE__linordered__idom,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linordered_idom @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( X2 != Y )
% 0.27/0.47 => ( ~ ( ord_less @ A @ X2 @ Y )
% 0.27/0.47 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % linorder_neqE_linordered_idom
% 0.27/0.47 thf(fact_116_linordered__field__no__ub,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linordered_field @ A )
% 0.27/0.47 => ! [X: A] :
% 0.27/0.47 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.27/0.47
% 0.27/0.47 % linordered_field_no_ub
% 0.27/0.47 thf(fact_117_insert__absorb2,axiom,
% 0.27/0.47 ! [A: $tType,X2: A,A4: set @ A] :
% 0.27/0.47 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A4 ) )
% 0.27/0.47 = ( insert @ A @ X2 @ A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_absorb2
% 0.27/0.47 thf(fact_118_insert__iff,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.27/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 0.27/0.47 = ( ( A2 = B2 )
% 0.27/0.47 | ( member @ A @ A2 @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_iff
% 0.27/0.47 thf(fact_119_insertCI,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 0.27/0.47 ( ( ~ ( member @ A @ A2 @ B5 )
% 0.27/0.47 => ( A2 = B2 ) )
% 0.27/0.47 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insertCI
% 0.27/0.47 thf(fact_120_singletonI,axiom,
% 0.27/0.47 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % singletonI
% 0.27/0.47 thf(fact_121_singleton__conv,axiom,
% 0.27/0.47 ! [A: $tType,A2: A] :
% 0.27/0.47 ( ( collect @ A
% 0.27/0.47 @ ^ [X3: A] : ( X3 = A2 ) )
% 0.27/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % singleton_conv
% 0.27/0.47 thf(fact_122_singleton__conv2,axiom,
% 0.27/0.47 ! [A: $tType,A2: A] :
% 0.27/0.47 ( ( collect @ A
% 0.27/0.47 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.27/0.47 @ A2 ) )
% 0.27/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % singleton_conv2
% 0.27/0.47 thf(fact_123_psubset__trans,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.27/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.27/0.47 => ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
% 0.27/0.47 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % psubset_trans
% 0.27/0.47 thf(fact_124_less__set__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord_less @ ( set @ A ) )
% 0.27/0.47 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.27/0.47 ( ord_less @ ( A > $o )
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A6 )
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_set_def
% 0.27/0.47 thf(fact_125_psubsetD,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C: A] :
% 0.27/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.27/0.47 => ( ( member @ A @ C @ A4 )
% 0.27/0.47 => ( member @ A @ C @ B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % psubsetD
% 0.27/0.47 thf(fact_126_mk__disjoint__insert,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,A4: set @ A] :
% 0.27/0.47 ( ( member @ A @ A2 @ A4 )
% 0.27/0.47 => ? [B7: set @ A] :
% 0.27/0.47 ( ( A4
% 0.27/0.47 = ( insert @ A @ A2 @ B7 ) )
% 0.27/0.47 & ~ ( member @ A @ A2 @ B7 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % mk_disjoint_insert
% 0.27/0.47 thf(fact_127_insert__commute,axiom,
% 0.27/0.47 ! [A: $tType,X2: A,Y: A,A4: set @ A] :
% 0.27/0.47 ( ( insert @ A @ X2 @ ( insert @ A @ Y @ A4 ) )
% 0.27/0.47 = ( insert @ A @ Y @ ( insert @ A @ X2 @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_commute
% 0.27/0.47 thf(fact_128_insert__eq__iff,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,A4: set @ A,B2: A,B5: set @ A] :
% 0.27/0.47 ( ~ ( member @ A @ A2 @ A4 )
% 0.27/0.47 => ( ~ ( member @ A @ B2 @ B5 )
% 0.27/0.47 => ( ( ( insert @ A @ A2 @ A4 )
% 0.27/0.47 = ( insert @ A @ B2 @ B5 ) )
% 0.27/0.47 = ( ( ( A2 = B2 )
% 0.27/0.47 => ( A4 = B5 ) )
% 0.27/0.47 & ( ( A2 != B2 )
% 0.27/0.47 => ? [C4: set @ A] :
% 0.27/0.47 ( ( A4
% 0.27/0.47 = ( insert @ A @ B2 @ C4 ) )
% 0.27/0.47 & ~ ( member @ A @ B2 @ C4 )
% 0.27/0.47 & ( B5
% 0.27/0.47 = ( insert @ A @ A2 @ C4 ) )
% 0.27/0.47 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_eq_iff
% 0.27/0.47 thf(fact_129_insert__absorb,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,A4: set @ A] :
% 0.27/0.47 ( ( member @ A @ A2 @ A4 )
% 0.27/0.47 => ( ( insert @ A @ A2 @ A4 )
% 0.27/0.47 = A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_absorb
% 0.27/0.47 thf(fact_130_insert__ident,axiom,
% 0.27/0.47 ! [A: $tType,X2: A,A4: set @ A,B5: set @ A] :
% 0.27/0.47 ( ~ ( member @ A @ X2 @ A4 )
% 0.27/0.47 => ( ~ ( member @ A @ X2 @ B5 )
% 0.27/0.47 => ( ( ( insert @ A @ X2 @ A4 )
% 0.27/0.47 = ( insert @ A @ X2 @ B5 ) )
% 0.27/0.47 = ( A4 = B5 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_ident
% 0.27/0.47 thf(fact_131_Set_Oset__insert,axiom,
% 0.27/0.47 ! [A: $tType,X2: A,A4: set @ A] :
% 0.27/0.47 ( ( member @ A @ X2 @ A4 )
% 0.27/0.47 => ~ ! [B7: set @ A] :
% 0.27/0.47 ( ( A4
% 0.27/0.47 = ( insert @ A @ X2 @ B7 ) )
% 0.27/0.47 => ( member @ A @ X2 @ B7 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Set.set_insert
% 0.27/0.47 thf(fact_132_insertI2,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 0.27/0.47 ( ( member @ A @ A2 @ B5 )
% 0.27/0.47 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insertI2
% 0.27/0.47 thf(fact_133_insertI1,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% 0.27/0.47
% 0.27/0.47 % insertI1
% 0.27/0.47 thf(fact_134_insertE,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.27/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 0.27/0.47 => ( ( A2 != B2 )
% 0.27/0.47 => ( member @ A @ A2 @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insertE
% 0.27/0.47 thf(fact_135_insert__compr,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( insert @ A )
% 0.27/0.47 = ( ^ [A3: A,B6: set @ A] :
% 0.27/0.47 ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( X3 = A3 )
% 0.27/0.47 | ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_compr
% 0.27/0.47 thf(fact_136_insert__Collect,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.27/0.47 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 0.27/0.47 = ( collect @ A
% 0.27/0.47 @ ^ [U: A] :
% 0.27/0.47 ( ( U != A2 )
% 0.27/0.47 => ( P @ U ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_Collect
% 0.27/0.47 thf(fact_137_singletonD,axiom,
% 0.27/0.47 ! [A: $tType,B2: A,A2: A] :
% 0.27/0.47 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.47 => ( B2 = A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % singletonD
% 0.27/0.47 thf(fact_138_singleton__iff,axiom,
% 0.27/0.47 ! [A: $tType,B2: A,A2: A] :
% 0.27/0.47 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.47 = ( B2 = A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % singleton_iff
% 0.27/0.47 thf(fact_139_doubleton__eq__iff,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,B2: A,C: A,D2: A] :
% 0.27/0.47 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.47 = ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.27/0.47 = ( ( ( A2 = C )
% 0.27/0.47 & ( B2 = D2 ) )
% 0.27/0.47 | ( ( A2 = D2 )
% 0.27/0.47 & ( B2 = C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % doubleton_eq_iff
% 0.27/0.47 thf(fact_140_insert__not__empty,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,A4: set @ A] :
% 0.27/0.47 ( ( insert @ A @ A2 @ A4 )
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_not_empty
% 0.27/0.47 thf(fact_141_singleton__inject,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,B2: A] :
% 0.27/0.47 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.47 => ( A2 = B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % singleton_inject
% 0.27/0.47 thf(fact_142_Collect__conv__if,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,A2: A] :
% 0.27/0.47 ( ( ( P @ A2 )
% 0.27/0.47 => ( ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( X3 = A2 )
% 0.27/0.47 & ( P @ X3 ) ) )
% 0.27/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.27/0.47 & ( ~ ( P @ A2 )
% 0.27/0.47 => ( ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( X3 = A2 )
% 0.27/0.47 & ( P @ X3 ) ) )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_conv_if
% 0.27/0.47 thf(fact_143_Collect__conv__if2,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,A2: A] :
% 0.27/0.47 ( ( ( P @ A2 )
% 0.27/0.47 => ( ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( A2 = X3 )
% 0.27/0.47 & ( P @ X3 ) ) )
% 0.27/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.27/0.47 & ( ~ ( P @ A2 )
% 0.27/0.47 => ( ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( A2 = X3 )
% 0.27/0.47 & ( P @ X3 ) ) )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_conv_if2
% 0.27/0.47 thf(fact_144_linordered__field__no__lb,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linordered_field @ A )
% 0.27/0.47 => ! [X: A] :
% 0.27/0.47 ? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % linordered_field_no_lb
% 0.27/0.47 thf(fact_145_the__elem__eq,axiom,
% 0.27/0.47 ! [A: $tType,X2: A] :
% 0.27/0.47 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.47 = X2 ) ).
% 0.27/0.47
% 0.27/0.47 % the_elem_eq
% 0.27/0.47 thf(fact_146_is__singletonI,axiom,
% 0.27/0.47 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_singletonI
% 0.27/0.47 thf(fact_147_these__insert__Some,axiom,
% 0.27/0.47 ! [A: $tType,X2: A,A4: set @ ( option @ A )] :
% 0.27/0.47 ( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X2 ) @ A4 ) )
% 0.27/0.47 = ( insert @ A @ X2 @ ( these @ A @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % these_insert_Some
% 0.27/0.47 thf(fact_148_is__singletonE,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A] :
% 0.27/0.47 ( ( is_singleton @ A @ A4 )
% 0.27/0.47 => ~ ! [X4: A] :
% 0.27/0.47 ( A4
% 0.27/0.47 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_singletonE
% 0.27/0.47 thf(fact_149_these__empty,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % these_empty
% 0.27/0.47 thf(fact_150_these__insert__None,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ ( option @ A )] :
% 0.27/0.47 ( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A4 ) )
% 0.27/0.47 = ( these @ A @ A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % these_insert_None
% 0.27/0.47 thf(fact_151_in__these__eq,axiom,
% 0.27/0.47 ! [A: $tType,X2: A,A4: set @ ( option @ A )] :
% 0.27/0.47 ( ( member @ A @ X2 @ ( these @ A @ A4 ) )
% 0.27/0.47 = ( member @ ( option @ A ) @ ( some @ A @ X2 ) @ A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % in_these_eq
% 0.27/0.47 thf(fact_152_is__singleton__the__elem,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( is_singleton @ A )
% 0.27/0.47 = ( ^ [A6: set @ A] :
% 0.27/0.47 ( A6
% 0.27/0.47 = ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_singleton_the_elem
% 0.27/0.47 thf(fact_153_is__singletonI_H,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A] :
% 0.27/0.47 ( ( A4
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( ! [X4: A,Y4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A4 )
% 0.27/0.47 => ( ( member @ A @ Y4 @ A4 )
% 0.27/0.47 => ( X4 = Y4 ) ) )
% 0.27/0.47 => ( is_singleton @ A @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_singletonI'
% 0.27/0.47 thf(fact_154_these__not__empty__eq,axiom,
% 0.27/0.47 ! [A: $tType,B5: set @ ( option @ A )] :
% 0.27/0.47 ( ( ( these @ A @ B5 )
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( ( B5
% 0.27/0.47 != ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 0.27/0.47 & ( B5
% 0.27/0.47 != ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % these_not_empty_eq
% 0.27/0.47 thf(fact_155_these__empty__eq,axiom,
% 0.27/0.47 ! [A: $tType,B5: set @ ( option @ A )] :
% 0.27/0.47 ( ( ( these @ A @ B5 )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( ( B5
% 0.27/0.47 = ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 0.27/0.47 | ( B5
% 0.27/0.47 = ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % these_empty_eq
% 0.27/0.47 thf(fact_156_is__singleton__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( is_singleton @ A )
% 0.27/0.47 = ( ^ [A6: set @ A] :
% 0.27/0.47 ? [X3: A] :
% 0.27/0.47 ( A6
% 0.27/0.47 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_singleton_def
% 0.27/0.47 thf(fact_157_the__elem__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( the_elem @ A )
% 0.27/0.47 = ( ^ [X6: set @ A] :
% 0.27/0.47 ( the @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( X6
% 0.27/0.47 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % the_elem_def
% 0.27/0.47 thf(fact_158_setOf_Osimps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
% 0.27/0.47 ( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 0.27/0.47 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T22 ) ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % setOf.simps(2)
% 0.27/0.47 thf(fact_159_Option_Othese__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( these @ A )
% 0.27/0.47 = ( ^ [A6: set @ ( option @ A )] :
% 0.27/0.47 ( image @ ( option @ A ) @ A @ ( the2 @ A )
% 0.27/0.47 @ ( collect @ ( option @ A )
% 0.27/0.47 @ ^ [X3: option @ A] :
% 0.27/0.47 ( ( member @ ( option @ A ) @ X3 @ A6 )
% 0.27/0.47 & ( X3
% 0.27/0.47 != ( none @ A ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Option.these_def
% 0.27/0.47 thf(fact_160_image__eqI,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A4: set @ B] :
% 0.27/0.47 ( ( B2
% 0.27/0.47 = ( F @ X2 ) )
% 0.27/0.47 => ( ( member @ B @ X2 @ A4 )
% 0.27/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % image_eqI
% 0.27/0.47 thf(fact_161_Un__iff,axiom,
% 0.27/0.47 ! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
% 0.27/0.47 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.27/0.47 = ( ( member @ A @ C @ A4 )
% 0.27/0.47 | ( member @ A @ C @ B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_iff
% 0.27/0.47 thf(fact_162_UnCI,axiom,
% 0.27/0.47 ! [A: $tType,C: A,B5: set @ A,A4: set @ A] :
% 0.27/0.47 ( ( ~ ( member @ A @ C @ B5 )
% 0.27/0.47 => ( member @ A @ C @ A4 ) )
% 0.27/0.47 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % UnCI
% 0.27/0.47 thf(fact_163_image__ident,axiom,
% 0.27/0.47 ! [A: $tType,Y6: set @ A] :
% 0.27/0.47 ( ( image @ A @ A
% 0.27/0.47 @ ^ [X3: A] : X3
% 0.27/0.47 @ Y6 )
% 0.27/0.47 = Y6 ) ).
% 0.27/0.47
% 0.27/0.47 % image_ident
% 0.27/0.47 thf(fact_164_image__is__empty,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
% 0.27/0.47 ( ( ( image @ B @ A @ F @ A4 )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( A4
% 0.27/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % image_is_empty
% 0.27/0.47 thf(fact_165_empty__is__image,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
% 0.27/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.27/0.47 = ( image @ B @ A @ F @ A4 ) )
% 0.27/0.47 = ( A4
% 0.27/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % empty_is_image
% 0.27/0.47 thf(fact_166_image__empty,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,F: B > A] :
% 0.27/0.47 ( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % image_empty
% 0.27/0.47 thf(fact_167_image__insert,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,F: B > A,A2: B,B5: set @ B] :
% 0.27/0.47 ( ( image @ B @ A @ F @ ( insert @ B @ A2 @ B5 ) )
% 0.27/0.47 = ( insert @ A @ ( F @ A2 ) @ ( image @ B @ A @ F @ B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % image_insert
% 0.27/0.47 thf(fact_168_insert__image,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
% 0.27/0.47 ( ( member @ A @ X2 @ A4 )
% 0.27/0.47 => ( ( insert @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) )
% 0.27/0.47 = ( image @ A @ B @ F @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_image
% 0.27/0.47 thf(fact_169_Un__empty,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.27/0.47 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( ( A4
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 & ( B5
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_empty
% 0.27/0.47 thf(fact_170_Un__insert__left,axiom,
% 0.27/0.47 ! [A: $tType,A2: A,B5: set @ A,C3: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B5 ) @ C3 )
% 0.27/0.47 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_insert_left
% 0.27/0.47 thf(fact_171_Un__insert__right,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
% 0.27/0.47 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_insert_right
% 0.27/0.47 thf(fact_172_insert__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( insert @ A )
% 0.27/0.47 = ( ^ [A3: A] :
% 0.27/0.47 ( sup_sup @ ( set @ A )
% 0.27/0.47 @ ( collect @ A
% 0.27/0.47 @ ^ [X3: A] : ( X3 = A3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_def
% 0.27/0.47 thf(fact_173_Un__left__commute,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
% 0.27/0.47 = ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_left_commute
% 0.27/0.47 thf(fact_174_Un__left__absorb,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.27/0.47 = ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_left_absorb
% 0.27/0.47 thf(fact_175_rev__image__eqI,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,X2: A,A4: set @ A,B2: B,F: A > B] :
% 0.27/0.47 ( ( member @ A @ X2 @ A4 )
% 0.27/0.47 => ( ( B2
% 0.27/0.47 = ( F @ X2 ) )
% 0.27/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rev_image_eqI
% 0.27/0.47 thf(fact_176_ball__imageD,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A4 ) )
% 0.27/0.47 => ( P @ X4 ) )
% 0.27/0.47 => ! [X: B] :
% 0.27/0.47 ( ( member @ B @ X @ A4 )
% 0.27/0.47 => ( P @ ( F @ X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ball_imageD
% 0.27/0.47 thf(fact_177_image__cong,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,M2: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 0.27/0.47 ( ( M2 = N2 )
% 0.27/0.47 => ( ! [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ N2 )
% 0.27/0.47 => ( ( F @ X4 )
% 0.27/0.47 = ( G @ X4 ) ) )
% 0.27/0.47 => ( ( image @ A @ B @ F @ M2 )
% 0.27/0.47 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % image_cong
% 0.27/0.47 thf(fact_178_bex__imageD,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 0.27/0.47 ( ? [X: A] :
% 0.27/0.47 ( ( member @ A @ X @ ( image @ B @ A @ F @ A4 ) )
% 0.27/0.47 & ( P @ X ) )
% 0.27/0.47 => ? [X4: B] :
% 0.27/0.47 ( ( member @ B @ X4 @ A4 )
% 0.27/0.47 & ( P @ ( F @ X4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bex_imageD
% 0.27/0.47 thf(fact_179_Un__commute,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) )
% 0.27/0.47 = ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_commute
% 0.27/0.47 thf(fact_180_image__iff,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,Z2: A,F: B > A,A4: set @ B] :
% 0.27/0.47 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A4 ) )
% 0.27/0.47 = ( ? [X3: B] :
% 0.27/0.47 ( ( member @ B @ X3 @ A4 )
% 0.27/0.47 & ( Z2
% 0.27/0.47 = ( F @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % image_iff
% 0.27/0.47 thf(fact_181_Un__absorb,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 0.27/0.47 = A4 ) ).
% 0.27/0.47
% 0.27/0.47 % Un_absorb
% 0.27/0.47 thf(fact_182_image__Un,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,B5: set @ B] :
% 0.27/0.47 ( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A4 @ B5 ) )
% 0.27/0.47 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A4 ) @ ( image @ B @ A @ F @ B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % image_Un
% 0.27/0.47 thf(fact_183_Un__assoc,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
% 0.27/0.47 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_assoc
% 0.27/0.47 thf(fact_184_ball__Un,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
% 0.27/0.47 ( ( ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.27/0.47 => ( P @ X3 ) ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A4 )
% 0.27/0.47 => ( P @ X3 ) )
% 0.27/0.47 & ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ B5 )
% 0.27/0.47 => ( P @ X3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ball_Un
% 0.27/0.47 thf(fact_185_imageI,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
% 0.27/0.47 ( ( member @ A @ X2 @ A4 )
% 0.27/0.47 => ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % imageI
% 0.27/0.47 thf(fact_186_bex__Un,axiom,
% 0.27/0.47 ! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
% 0.27/0.48 ( ( ? [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.27/0.48 & ( P @ X3 ) ) )
% 0.27/0.48 = ( ? [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A4 )
% 0.27/0.48 & ( P @ X3 ) )
% 0.27/0.48 | ? [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ B5 )
% 0.27/0.48 & ( P @ X3 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % bex_Un
% 0.27/0.48 thf(fact_187_UnI2,axiom,
% 0.27/0.48 ! [A: $tType,C: A,B5: set @ A,A4: set @ A] :
% 0.27/0.48 ( ( member @ A @ C @ B5 )
% 0.27/0.48 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % UnI2
% 0.27/0.48 thf(fact_188_UnI1,axiom,
% 0.27/0.48 ! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
% 0.27/0.48 ( ( member @ A @ C @ A4 )
% 0.27/0.48 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % UnI1
% 0.27/0.48 thf(fact_189_UnE,axiom,
% 0.27/0.48 ! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
% 0.27/0.48 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.27/0.48 => ( ~ ( member @ A @ C @ A4 )
% 0.27/0.48 => ( member @ A @ C @ B5 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % UnE
% 0.27/0.48 thf(fact_190_Un__def,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( sup_sup @ ( set @ A ) )
% 0.27/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.27/0.48 ( collect @ A
% 0.27/0.48 @ ^ [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A6 )
% 0.27/0.48 | ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Un_def
% 0.27/0.48 thf(fact_191_imageE,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType,B2: A,F: B > A,A4: set @ B] :
% 0.27/0.48 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) )
% 0.27/0.48 => ~ ! [X4: B] :
% 0.27/0.48 ( ( B2
% 0.27/0.48 = ( F @ X4 ) )
% 0.27/0.48 => ~ ( member @ B @ X4 @ A4 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % imageE
% 0.27/0.48 thf(fact_192_image__image,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType,C2: $tType,F: B > A,G: C2 > B,A4: set @ C2] :
% 0.27/0.48 ( ( image @ B @ A @ F @ ( image @ C2 @ B @ G @ A4 ) )
% 0.27/0.48 = ( image @ C2 @ A
% 0.27/0.48 @ ^ [X3: C2] : ( F @ ( G @ X3 ) )
% 0.27/0.48 @ A4 ) ) ).
% 0.27/0.48
% 0.27/0.48 % image_image
% 0.27/0.48 thf(fact_193_Compr__image__eq,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 0.27/0.48 ( ( collect @ A
% 0.27/0.48 @ ^ [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A4 ) )
% 0.27/0.48 & ( P @ X3 ) ) )
% 0.27/0.48 = ( image @ B @ A @ F
% 0.27/0.48 @ ( collect @ B
% 0.27/0.48 @ ^ [X3: B] :
% 0.27/0.48 ( ( member @ B @ X3 @ A4 )
% 0.27/0.48 & ( P @ ( F @ X3 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Compr_image_eq
% 0.27/0.48 thf(fact_194_Collect__disj__eq,axiom,
% 0.27/0.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.48 ( ( collect @ A
% 0.27/0.48 @ ^ [X3: A] :
% 0.27/0.48 ( ( P @ X3 )
% 0.27/0.48 | ( Q @ X3 ) ) )
% 0.27/0.48 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Collect_disj_eq
% 0.27/0.48 thf(fact_195_Un__empty__right,axiom,
% 0.27/0.48 ! [A: $tType,A4: set @ A] :
% 0.27/0.48 ( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 = A4 ) ).
% 0.27/0.48
% 0.27/0.48 % Un_empty_right
% 0.27/0.48 thf(fact_196_Un__empty__left,axiom,
% 0.27/0.48 ! [A: $tType,B5: set @ A] :
% 0.27/0.48 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
% 0.27/0.48 = B5 ) ).
% 0.27/0.48
% 0.27/0.48 % Un_empty_left
% 0.27/0.48 thf(fact_197_singleton__Un__iff,axiom,
% 0.27/0.48 ! [A: $tType,X2: A,A4: set @ A,B5: set @ A] :
% 0.27/0.48 ( ( ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 = ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 0.27/0.48 = ( ( ( A4
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 & ( B5
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.27/0.48 | ( ( A4
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 & ( B5
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 | ( ( A4
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 & ( B5
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % singleton_Un_iff
% 0.27/0.48 thf(fact_198_Un__singleton__iff,axiom,
% 0.27/0.48 ! [A: $tType,A4: set @ A,B5: set @ A,X2: A] :
% 0.27/0.48 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 = ( ( ( A4
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 & ( B5
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.27/0.48 | ( ( A4
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 & ( B5
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 | ( ( A4
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 & ( B5
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Un_singleton_iff
% 0.27/0.48 thf(fact_199_insert__is__Un,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( insert @ A )
% 0.27/0.48 = ( ^ [A3: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % insert_is_Un
% 0.27/0.48 thf(fact_200_image__constant,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType,X2: A,A4: set @ A,C: B] :
% 0.27/0.48 ( ( member @ A @ X2 @ A4 )
% 0.27/0.48 => ( ( image @ A @ B
% 0.27/0.48 @ ^ [X3: A] : C
% 0.27/0.48 @ A4 )
% 0.27/0.48 = ( insert @ B @ C @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % image_constant
% 0.27/0.48 thf(fact_201_image__constant__conv,axiom,
% 0.27/0.48 ! [B: $tType,A: $tType,A4: set @ B,C: A] :
% 0.27/0.48 ( ( ( A4
% 0.27/0.48 = ( bot_bot @ ( set @ B ) ) )
% 0.27/0.48 => ( ( image @ B @ A
% 0.27/0.48 @ ^ [X3: B] : C
% 0.27/0.48 @ A4 )
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 & ( ( A4
% 0.27/0.48 != ( bot_bot @ ( set @ B ) ) )
% 0.27/0.48 => ( ( image @ B @ A
% 0.27/0.48 @ ^ [X3: B] : C
% 0.27/0.48 @ A4 )
% 0.27/0.48 = ( insert @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % image_constant_conv
% 0.27/0.48 thf(fact_202_the__elem__image__unique,axiom,
% 0.27/0.48 ! [B: $tType,A: $tType,A4: set @ A,F: A > B,X2: A] :
% 0.27/0.48 ( ( A4
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ! [Y4: A] :
% 0.27/0.48 ( ( member @ A @ Y4 @ A4 )
% 0.27/0.48 => ( ( F @ Y4 )
% 0.27/0.48 = ( F @ X2 ) ) )
% 0.27/0.48 => ( ( the_elem @ B @ ( image @ A @ B @ F @ A4 ) )
% 0.27/0.48 = ( F @ X2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % the_elem_image_unique
% 0.27/0.48 thf(fact_203_sup__bot__left,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( bounde1808546759up_bot @ A )
% 0.27/0.48 => ! [X2: A] :
% 0.27/0.48 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X2 )
% 0.27/0.48 = X2 ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_bot_left
% 0.27/0.48 thf(fact_204_sup__bot__right,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( bounde1808546759up_bot @ A )
% 0.27/0.48 => ! [X2: A] :
% 0.27/0.48 ( ( sup_sup @ A @ X2 @ ( bot_bot @ A ) )
% 0.27/0.48 = X2 ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_bot_right
% 0.27/0.48 thf(fact_205_bot__eq__sup__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( bounde1808546759up_bot @ A )
% 0.27/0.48 => ! [X2: A,Y: A] :
% 0.27/0.48 ( ( ( bot_bot @ A )
% 0.27/0.48 = ( sup_sup @ A @ X2 @ Y ) )
% 0.27/0.48 = ( ( X2
% 0.27/0.48 = ( bot_bot @ A ) )
% 0.27/0.48 & ( Y
% 0.27/0.48 = ( bot_bot @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % bot_eq_sup_iff
% 0.27/0.48 thf(fact_206_sup__bot_Oright__neutral,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( bounde1808546759up_bot @ A )
% 0.27/0.48 => ! [A2: A] :
% 0.27/0.48 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 0.27/0.48 = A2 ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_bot.right_neutral
% 0.27/0.48 thf(fact_207_sup__bot_Oneutr__eq__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( bounde1808546759up_bot @ A )
% 0.27/0.48 => ! [A2: A,B2: A] :
% 0.27/0.48 ( ( ( bot_bot @ A )
% 0.27/0.48 = ( sup_sup @ A @ A2 @ B2 ) )
% 0.27/0.48 = ( ( A2
% 0.27/0.48 = ( bot_bot @ A ) )
% 0.27/0.48 & ( B2
% 0.27/0.48 = ( bot_bot @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_bot.neutr_eq_iff
% 0.27/0.48 thf(fact_208_sup__bot_Oleft__neutral,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( bounde1808546759up_bot @ A )
% 0.27/0.48 => ! [A2: A] :
% 0.27/0.48 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 0.27/0.48 = A2 ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_bot.left_neutral
% 0.27/0.48 thf(fact_209_sup__bot_Oeq__neutr__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( bounde1808546759up_bot @ A )
% 0.27/0.48 => ! [A2: A,B2: A] :
% 0.27/0.48 ( ( ( sup_sup @ A @ A2 @ B2 )
% 0.27/0.48 = ( bot_bot @ A ) )
% 0.27/0.48 = ( ( A2
% 0.27/0.48 = ( bot_bot @ A ) )
% 0.27/0.48 & ( B2
% 0.27/0.48 = ( bot_bot @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_bot.eq_neutr_iff
% 0.27/0.48 thf(fact_210_sup__eq__bot__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( bounde1808546759up_bot @ A )
% 0.27/0.48 => ! [X2: A,Y: A] :
% 0.27/0.48 ( ( ( sup_sup @ A @ X2 @ Y )
% 0.27/0.48 = ( bot_bot @ A ) )
% 0.27/0.48 = ( ( X2
% 0.27/0.48 = ( bot_bot @ A ) )
% 0.27/0.48 & ( Y
% 0.27/0.48 = ( bot_bot @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_eq_bot_iff
% 0.27/0.48 thf(fact_211_these__image__Some__eq,axiom,
% 0.27/0.48 ! [A: $tType,A4: set @ A] :
% 0.27/0.48 ( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) )
% 0.27/0.48 = A4 ) ).
% 0.27/0.48
% 0.27/0.48 % these_image_Some_eq
% 0.27/0.48 thf(fact_212_sup__set__def,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( sup_sup @ ( set @ A ) )
% 0.27/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.27/0.48 ( collect @ A
% 0.27/0.48 @ ( sup_sup @ ( A > $o )
% 0.27/0.48 @ ^ [X3: A] : ( member @ A @ X3 @ A6 )
% 0.27/0.48 @ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_set_def
% 0.27/0.48 thf(fact_213_sup__Un__eq,axiom,
% 0.27/0.48 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.27/0.48 ( ( sup_sup @ ( A > $o )
% 0.27/0.48 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 0.27/0.48 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 0.27/0.48 = ( ^ [X3: A] : ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_Un_eq
% 0.27/0.48 thf(fact_214_None__notin__image__Some,axiom,
% 0.27/0.48 ! [A: $tType,A4: set @ A] :
% 0.27/0.48 ~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ).
% 0.27/0.48
% 0.27/0.48 % None_notin_image_Some
% 0.27/0.48 thf(fact_215_Some__image__these__eq,axiom,
% 0.27/0.48 ! [A: $tType,A4: set @ ( option @ A )] :
% 0.27/0.48 ( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
% 0.27/0.48 = ( collect @ ( option @ A )
% 0.27/0.48 @ ^ [X3: option @ A] :
% 0.27/0.48 ( ( member @ ( option @ A ) @ X3 @ A4 )
% 0.27/0.48 & ( X3
% 0.27/0.48 != ( none @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Some_image_these_eq
% 0.27/0.48 thf(fact_216_sup_Ostrict__coboundedI2,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [C: A,B2: A,A2: A] :
% 0.27/0.48 ( ( ord_less @ A @ C @ B2 )
% 0.27/0.48 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup.strict_coboundedI2
% 0.27/0.48 thf(fact_217_sup_Ostrict__coboundedI1,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [C: A,A2: A,B2: A] :
% 0.27/0.48 ( ( ord_less @ A @ C @ A2 )
% 0.27/0.48 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup.strict_coboundedI1
% 0.27/0.48 thf(fact_218_sup_Ostrict__order__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ( ( ord_less @ A )
% 0.27/0.48 = ( ^ [B3: A,A3: A] :
% 0.27/0.48 ( ( A3
% 0.27/0.48 = ( sup_sup @ A @ A3 @ B3 ) )
% 0.27/0.48 & ( A3 != B3 ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup.strict_order_iff
% 0.27/0.48 thf(fact_219_sup_Ostrict__boundedE,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [B2: A,C: A,A2: A] :
% 0.27/0.48 ( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
% 0.27/0.48 => ~ ( ( ord_less @ A @ B2 @ A2 )
% 0.27/0.48 => ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup.strict_boundedE
% 0.27/0.48 thf(fact_220_less__supI2,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [X2: A,B2: A,A2: A] :
% 0.27/0.48 ( ( ord_less @ A @ X2 @ B2 )
% 0.27/0.48 => ( ord_less @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % less_supI2
% 0.27/0.48 thf(fact_221_less__supI1,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [X2: A,A2: A,B2: A] :
% 0.27/0.48 ( ( ord_less @ A @ X2 @ A2 )
% 0.27/0.48 => ( ord_less @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % less_supI1
% 0.27/0.48 thf(fact_222_the__equality,axiom,
% 0.27/0.48 ! [A: $tType,P: A > $o,A2: A] :
% 0.27/0.48 ( ( P @ A2 )
% 0.27/0.48 => ( ! [X4: A] :
% 0.27/0.48 ( ( P @ X4 )
% 0.27/0.48 => ( X4 = A2 ) )
% 0.27/0.48 => ( ( the @ A @ P )
% 0.27/0.48 = A2 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % the_equality
% 0.27/0.48 thf(fact_223_the__eq__trivial,axiom,
% 0.27/0.48 ! [A: $tType,A2: A] :
% 0.27/0.48 ( ( the @ A
% 0.27/0.48 @ ^ [X3: A] : ( X3 = A2 ) )
% 0.27/0.48 = A2 ) ).
% 0.27/0.48
% 0.27/0.48 % the_eq_trivial
% 0.27/0.48 thf(fact_224_the__sym__eq__trivial,axiom,
% 0.27/0.48 ! [A: $tType,X2: A] :
% 0.27/0.48 ( ( the @ A
% 0.27/0.48 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.27/0.48 @ X2 ) )
% 0.27/0.48 = X2 ) ).
% 0.27/0.48
% 0.27/0.48 % the_sym_eq_trivial
% 0.27/0.48 thf(fact_225_the1__equality,axiom,
% 0.27/0.48 ! [A: $tType,P: A > $o,A2: A] :
% 0.27/0.48 ( ? [X: A] :
% 0.27/0.48 ( ( P @ X )
% 0.27/0.48 & ! [Y4: A] :
% 0.27/0.48 ( ( P @ Y4 )
% 0.27/0.48 => ( Y4 = X ) ) )
% 0.27/0.48 => ( ( P @ A2 )
% 0.27/0.48 => ( ( the @ A @ P )
% 0.27/0.48 = A2 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % the1_equality
% 0.27/0.48 thf(fact_226_the1I2,axiom,
% 0.27/0.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.48 ( ? [X: A] :
% 0.27/0.48 ( ( P @ X )
% 0.27/0.48 & ! [Y4: A] :
% 0.27/0.48 ( ( P @ Y4 )
% 0.27/0.48 => ( Y4 = X ) ) )
% 0.27/0.48 => ( ! [X4: A] :
% 0.27/0.48 ( ( P @ X4 )
% 0.27/0.48 => ( Q @ X4 ) )
% 0.27/0.48 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % the1I2
% 0.27/0.48 thf(fact_227_If__def,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( if @ A )
% 0.27/0.48 = ( ^ [P3: $o,X3: A,Y3: A] :
% 0.27/0.48 ( the @ A
% 0.27/0.48 @ ^ [Z4: A] :
% 0.27/0.48 ( ( P3
% 0.27/0.48 => ( Z4 = X3 ) )
% 0.27/0.48 & ( ~ P3
% 0.27/0.48 => ( Z4 = Y3 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % If_def
% 0.27/0.48 thf(fact_228_theI2,axiom,
% 0.27/0.48 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 0.27/0.48 ( ( P @ A2 )
% 0.27/0.48 => ( ! [X4: A] :
% 0.27/0.48 ( ( P @ X4 )
% 0.27/0.48 => ( X4 = A2 ) )
% 0.27/0.48 => ( ! [X4: A] :
% 0.27/0.48 ( ( P @ X4 )
% 0.27/0.48 => ( Q @ X4 ) )
% 0.27/0.48 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % theI2
% 0.27/0.48 thf(fact_229_theI_H,axiom,
% 0.27/0.48 ! [A: $tType,P: A > $o] :
% 0.27/0.48 ( ? [X: A] :
% 0.27/0.48 ( ( P @ X )
% 0.27/0.48 & ! [Y4: A] :
% 0.27/0.48 ( ( P @ Y4 )
% 0.27/0.48 => ( Y4 = X ) ) )
% 0.27/0.48 => ( P @ ( the @ A @ P ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % theI'
% 0.27/0.48 thf(fact_230_theI,axiom,
% 0.27/0.48 ! [A: $tType,P: A > $o,A2: A] :
% 0.27/0.48 ( ( P @ A2 )
% 0.27/0.48 => ( ! [X4: A] :
% 0.27/0.48 ( ( P @ X4 )
% 0.27/0.48 => ( X4 = A2 ) )
% 0.27/0.48 => ( P @ ( the @ A @ P ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % theI
% 0.27/0.48 thf(fact_231_Tree_Osimps_I15_J,axiom,
% 0.27/0.48 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.27/0.48 ( ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
% 0.27/0.48 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary2130109271t_Tree @ A @ X21 ) @ ( insert @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( binary2130109271t_Tree @ A @ X23 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Tree.simps(15)
% 0.27/0.48 thf(fact_232_bind__singleton__conv__image,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType,A4: set @ B,F: B > A] :
% 0.27/0.48 ( ( bind @ B @ A @ A4
% 0.27/0.48 @ ^ [X3: B] : ( insert @ A @ ( F @ X3 ) @ ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 = ( image @ B @ A @ F @ A4 ) ) ).
% 0.27/0.48
% 0.27/0.48 % bind_singleton_conv_image
% 0.27/0.48 thf(fact_233_empty__bind,axiom,
% 0.27/0.48 ! [B: $tType,A: $tType,F: B > ( set @ A )] :
% 0.27/0.48 ( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % empty_bind
% 0.27/0.48 thf(fact_234_Set_Obind__bind,axiom,
% 0.27/0.48 ! [C2: $tType,B: $tType,A: $tType,A4: set @ A,B5: A > ( set @ C2 ),C3: C2 > ( set @ B )] :
% 0.27/0.48 ( ( bind @ C2 @ B @ ( bind @ A @ C2 @ A4 @ B5 ) @ C3 )
% 0.27/0.48 = ( bind @ A @ B @ A4
% 0.27/0.48 @ ^ [X3: A] : ( bind @ C2 @ B @ ( B5 @ X3 ) @ C3 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Set.bind_bind
% 0.27/0.48 thf(fact_235_Tree_Oset__intros_I3_J,axiom,
% 0.27/0.48 ! [A: $tType,Ya: A,X23: binary1291135688e_Tree @ A,X21: binary1291135688e_Tree @ A,X22: A] :
% 0.27/0.48 ( ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ X23 ) )
% 0.27/0.48 => ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Tree.set_intros(3)
% 0.27/0.48 thf(fact_236_Tree_Oset__intros_I2_J,axiom,
% 0.27/0.48 ! [A: $tType,X22: A,X21: binary1291135688e_Tree @ A,X23: binary1291135688e_Tree @ A] : ( member @ A @ X22 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Tree.set_intros(2)
% 0.27/0.48 thf(fact_237_Tree_Oset__intros_I1_J,axiom,
% 0.27/0.48 ! [A: $tType,Y: A,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.27/0.48 ( ( member @ A @ Y @ ( binary2130109271t_Tree @ A @ X21 ) )
% 0.27/0.48 => ( member @ A @ Y @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Tree.set_intros(1)
% 0.27/0.48 thf(fact_238_Tree_Oset__cases,axiom,
% 0.27/0.48 ! [A: $tType,E: A,A2: binary1291135688e_Tree @ A] :
% 0.27/0.48 ( ( member @ A @ E @ ( binary2130109271t_Tree @ A @ A2 ) )
% 0.27/0.48 => ( ! [Z1: binary1291135688e_Tree @ A] :
% 0.27/0.48 ( ? [Z22: A,Z32: binary1291135688e_Tree @ A] :
% 0.27/0.48 ( A2
% 0.27/0.48 = ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
% 0.27/0.48 => ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z1 ) ) )
% 0.27/0.48 => ( ! [Z1: binary1291135688e_Tree @ A,Z32: binary1291135688e_Tree @ A] :
% 0.27/0.48 ( A2
% 0.27/0.48 != ( binary210054475elle_T @ A @ Z1 @ E @ Z32 ) )
% 0.27/0.48 => ~ ! [Z1: binary1291135688e_Tree @ A,Z22: A,Z32: binary1291135688e_Tree @ A] :
% 0.27/0.48 ( ( A2
% 0.27/0.48 = ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
% 0.27/0.48 => ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z32 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Tree.set_cases
% 0.27/0.48 thf(fact_239_bind__const,axiom,
% 0.27/0.48 ! [B: $tType,A: $tType,A4: set @ B,B5: set @ A] :
% 0.27/0.48 ( ( ( A4
% 0.27/0.48 = ( bot_bot @ ( set @ B ) ) )
% 0.27/0.48 => ( ( bind @ B @ A @ A4
% 0.27/0.48 @ ^ [Uu: B] : B5 )
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) ) )
% 0.27/0.48 & ( ( A4
% 0.27/0.48 != ( bot_bot @ ( set @ B ) ) )
% 0.27/0.48 => ( ( bind @ B @ A @ A4
% 0.27/0.48 @ ^ [Uu: B] : B5 )
% 0.27/0.48 = B5 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % bind_const
% 0.27/0.48 thf(fact_240_nonempty__bind__const,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType,A4: set @ A,B5: set @ B] :
% 0.27/0.48 ( ( A4
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( bind @ A @ B @ A4
% 0.27/0.48 @ ^ [Uu: A] : B5 )
% 0.27/0.48 = B5 ) ) ).
% 0.27/0.48
% 0.27/0.48 % nonempty_bind_const
% 0.27/0.48 thf(fact_241_Tree_Osimps_I14_J,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( binary2130109271t_Tree @ A @ ( binary1746293266le_Tip @ A ) )
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Tree.simps(14)
% 0.27/0.48 thf(fact_242_Inf_OINF__identity__eq,axiom,
% 0.27/0.48 ! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
% 0.27/0.48 ( ( Inf
% 0.27/0.48 @ ( image @ A @ A
% 0.27/0.48 @ ^ [X3: A] : X3
% 0.27/0.48 @ A4 ) )
% 0.27/0.48 = ( Inf @ A4 ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf.INF_identity_eq
% 0.27/0.48 thf(fact_243_Sup_OSUP__identity__eq,axiom,
% 0.27/0.48 ! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
% 0.27/0.48 ( ( Sup
% 0.27/0.48 @ ( image @ A @ A
% 0.27/0.48 @ ^ [X3: A] : X3
% 0.27/0.48 @ A4 ) )
% 0.27/0.48 = ( Sup @ A4 ) ) ).
% 0.27/0.48
% 0.27/0.48 % Sup.SUP_identity_eq
% 0.27/0.48 thf(fact_244_range__constant,axiom,
% 0.27/0.48 ! [B: $tType,A: $tType,X2: A] :
% 0.27/0.48 ( ( image @ B @ A
% 0.27/0.48 @ ^ [Uu: B] : X2
% 0.27/0.48 @ ( top_top @ ( set @ B ) ) )
% 0.27/0.48 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % range_constant
% 0.27/0.48 thf(fact_245_INF__eq__bot__iff,axiom,
% 0.27/0.48 ! [B: $tType,A: $tType] :
% 0.27/0.48 ( ( comple1035589618norder @ A )
% 0.27/0.48 => ! [F: B > A,A4: set @ B] :
% 0.27/0.48 ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A4 ) )
% 0.27/0.48 = ( bot_bot @ A ) )
% 0.27/0.48 = ( ! [X3: A] :
% 0.27/0.48 ( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
% 0.27/0.48 => ? [Y3: B] :
% 0.27/0.48 ( ( member @ B @ Y3 @ A4 )
% 0.27/0.48 & ( ord_less @ A @ ( F @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % INF_eq_bot_iff
% 0.27/0.48 thf(fact_246_top__apply,axiom,
% 0.27/0.48 ! [C2: $tType,D: $tType] :
% 0.27/0.48 ( ( top @ C2 )
% 0.27/0.48 => ( ( top_top @ ( D > C2 ) )
% 0.27/0.48 = ( ^ [X3: D] : ( top_top @ C2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % top_apply
% 0.27/0.48 thf(fact_247_UNIV__I,axiom,
% 0.27/0.48 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % UNIV_I
% 0.27/0.48 thf(fact_248_Inf__apply,axiom,
% 0.27/0.48 ! [B: $tType,A: $tType] :
% 0.27/0.48 ( ( complete_Inf @ B )
% 0.27/0.48 => ( ( complete_Inf_Inf @ ( A > B ) )
% 0.27/0.48 = ( ^ [A6: set @ ( A > B ),X3: A] :
% 0.27/0.48 ( complete_Inf_Inf @ B
% 0.27/0.48 @ ( image @ ( A > B ) @ B
% 0.27/0.48 @ ^ [F3: A > B] : ( F3 @ X3 )
% 0.27/0.48 @ A6 ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_apply
% 0.27/0.48 thf(fact_249_INT__constant,axiom,
% 0.27/0.48 ! [B: $tType,A: $tType,A4: set @ B,C: set @ A] :
% 0.27/0.48 ( ( ( A4
% 0.27/0.48 = ( bot_bot @ ( set @ B ) ) )
% 0.27/0.48 => ( ( complete_Inf_Inf @ ( set @ A )
% 0.27/0.48 @ ( image @ B @ ( set @ A )
% 0.27/0.48 @ ^ [Y3: B] : C
% 0.27/0.48 @ A4 ) )
% 0.27/0.48 = ( top_top @ ( set @ A ) ) ) )
% 0.27/0.48 & ( ( A4
% 0.27/0.48 != ( bot_bot @ ( set @ B ) ) )
% 0.27/0.48 => ( ( complete_Inf_Inf @ ( set @ A )
% 0.27/0.48 @ ( image @ B @ ( set @ A )
% 0.27/0.48 @ ^ [Y3: B] : C
% 0.27/0.48 @ A4 ) )
% 0.27/0.48 = C ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % INT_constant
% 0.27/0.48 thf(fact_250_Collect__const,axiom,
% 0.27/0.48 ! [A: $tType,P: $o] :
% 0.27/0.48 ( ( P
% 0.27/0.48 => ( ( collect @ A
% 0.27/0.48 @ ^ [S2: A] : P )
% 0.27/0.48 = ( top_top @ ( set @ A ) ) ) )
% 0.27/0.48 & ( ~ P
% 0.27/0.48 => ( ( collect @ A
% 0.27/0.48 @ ^ [S2: A] : P )
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Collect_const
% 0.27/0.48 thf(fact_251_INF__apply,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType,C2: $tType] :
% 0.27/0.48 ( ( complete_Inf @ A )
% 0.27/0.48 => ! [F: C2 > B > A,A4: set @ C2,X2: B] :
% 0.27/0.48 ( ( complete_Inf_Inf @ ( B > A ) @ ( image @ C2 @ ( B > A ) @ F @ A4 ) @ X2 )
% 0.27/0.48 = ( complete_Inf_Inf @ A
% 0.27/0.48 @ ( image @ C2 @ A
% 0.27/0.48 @ ^ [Y3: C2] : ( F @ Y3 @ X2 )
% 0.27/0.48 @ A4 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % INF_apply
% 0.27/0.48 thf(fact_252_INF__identity__eq,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( complete_Inf @ A )
% 0.27/0.48 => ! [A4: set @ A] :
% 0.27/0.48 ( ( complete_Inf_Inf @ A
% 0.27/0.48 @ ( image @ A @ A
% 0.27/0.48 @ ^ [X3: A] : X3
% 0.27/0.48 @ A4 ) )
% 0.27/0.48 = ( complete_Inf_Inf @ A @ A4 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % INF_identity_eq
% 0.27/0.48 thf(fact_253_Inf__eq__bot__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( comple1035589618norder @ A )
% 0.27/0.48 => ! [A4: set @ A] :
% 0.27/0.48 ( ( ( complete_Inf_Inf @ A @ A4 )
% 0.27/0.48 = ( bot_bot @ A ) )
% 0.27/0.48 = ( ! [X3: A] :
% 0.27/0.48 ( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
% 0.27/0.48 => ? [Y3: A] :
% 0.27/0.48 ( ( member @ A @ Y3 @ A4 )
% 0.27/0.48 & ( ord_less @ A @ Y3 @ X3 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_eq_bot_iff
% 0.27/0.48 thf(fact_254_Inf__empty,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( comple187826305attice @ A )
% 0.27/0.48 => ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 = ( top_top @ A ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_empty
% 0.27/0.48
% 0.27/0.48 % Type constructors (43)
% 0.27/0.48 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.27/0.48 bounded_lattice @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.27/0.48 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( bounded_lattice @ A8 )
% 0.27/0.48 => ( bounded_lattice @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( bounded_lattice @ A8 )
% 0.27/0.48 => ( bounde1808546759up_bot @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( comple187826305attice @ A8 )
% 0.27/0.48 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A8 )
% 0.27/0.48 => ( semilattice_sup @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Complete__Lattices_OInf,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( complete_Inf @ A8 )
% 0.27/0.48 => ( complete_Inf @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( order_bot @ A8 )
% 0.27/0.48 => ( order_bot @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( preorder @ A8 )
% 0.27/0.48 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( order @ A8 )
% 0.27/0.48 => ( order @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Otop,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( top @ A8 )
% 0.27/0.48 => ( top @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( ord @ A8 )
% 0.27/0.48 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Obot,axiom,
% 0.27/0.48 ! [A7: $tType,A8: $tType] :
% 0.27/0.48 ( ( bot @ A8 )
% 0.27/0.48 => ( bot @ ( A7 > A8 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
% 0.27/0.48 semilattice_sup @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.27/0.48 linordered_idom @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Complete__Lattices_OInf_4,axiom,
% 0.27/0.48 complete_Inf @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Orderings_Opreorder_5,axiom,
% 0.27/0.48 preorder @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.27/0.48 linorder @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.27/0.48 no_top @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.27/0.48 no_bot @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Orderings_Oorder_6,axiom,
% 0.27/0.48 order @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Int_Oint___Orderings_Oord_7,axiom,
% 0.27/0.48 ord @ int ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_8,axiom,
% 0.27/0.48 ! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_9,axiom,
% 0.27/0.48 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10,axiom,
% 0.27/0.48 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Complete__Lattices_OInf_11,axiom,
% 0.27/0.48 ! [A7: $tType] : ( complete_Inf @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Oorder__bot_12,axiom,
% 0.27/0.48 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_13,axiom,
% 0.27/0.48 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
% 0.27/0.48 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Otop_15,axiom,
% 0.27/0.48 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
% 0.27/0.48 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Obot_17,axiom,
% 0.27/0.48 ! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_18,axiom,
% 0.27/0.48 bounde1808546759up_bot @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_19,axiom,
% 0.27/0.48 comple187826305attice @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_20,axiom,
% 0.27/0.48 semilattice_sup @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Complete__Lattices_OInf_21,axiom,
% 0.27/0.48 complete_Inf @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Oorder__bot_22,axiom,
% 0.27/0.48 order_bot @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
% 0.27/0.48 preorder @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder_24,axiom,
% 0.27/0.48 linorder @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_25,axiom,
% 0.27/0.48 order @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Otop_26,axiom,
% 0.27/0.48 top @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
% 0.27/0.48 ord @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Obot_28,axiom,
% 0.27/0.48 bot @ $o ).
% 0.27/0.48
% 0.27/0.48 % Helper facts (3)
% 0.27/0.48 thf(help_If_3_1_T,axiom,
% 0.27/0.48 ! [P: $o] :
% 0.27/0.48 ( ( P = $true )
% 0.27/0.48 | ( P = $false ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(help_If_2_1_T,axiom,
% 0.27/0.48 ! [A: $tType,X2: A,Y: A] :
% 0.27/0.48 ( ( if @ A @ $false @ X2 @ Y )
% 0.27/0.48 = Y ) ).
% 0.27/0.48
% 0.27/0.48 thf(help_If_1_1_T,axiom,
% 0.27/0.48 ! [A: $tType,X2: A,Y: A] :
% 0.27/0.48 ( ( if @ A @ $true @ X2 @ Y )
% 0.27/0.48 = X2 ) ).
% 0.27/0.48
% 0.27/0.48 % Conjectures (1)
% 0.27/0.48 thf(conj_0,conjecture,
% 0.27/0.48 $false ).
% 0.27/0.48
% 0.27/0.48 %------------------------------------------------------------------------------
% 0.27/0.48 ------------------------
% 2.83/1.52 ---- Embedded in THF ---
% 2.83/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.83/1.52 %%% Generated on Tue Feb 24 08:23:47 EST 2026
% 2.83/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.83/1.52 %%% Logic specification used:
% 2.83/1.52 %%% thf(spec, logic, $$dhol).
% 2.83/1.52
% 2.83/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.UyQQfMpErQ/DTF2DTF25161.p
% 2.83/1.52 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree, type, binary1291135688e_Tree: ($tType > $tType)).
% 2.83/1.52 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 2.83/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.83/1.52 thf(ty_t_Int_Oint, type, int: $tType).
% 2.83/1.52 thf(ty_tf_a, type, a: $tType).
% 2.83/1.52 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Complete__Lattices_OInf, type, complete_Inf: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Complete__Lattices_Ocomplete__linorder, type, comple1035589618norder: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde1808546759up_bot: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT, type, binary210054475elle_T: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip, type, binary1746293266le_Tip: (!> [A:$tType]: ((binary1291135688e_Tree @ A)))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Oset__Tree, type, binary2130109271t_Tree: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (set @ A))))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oeqs, type, binary64540844le_eqs: (!> [A:$tType]: (((A > int) > (A > (set @ A)))))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb, type, binary827270440e_memb: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > $o)))))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf, type, binary1653327646_setOf: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (set @ A))))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree, type, binary1610619414edTree: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > $o))))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred, type, binary231205461t_pred: (!> [A:$tType]: (((A > int) > (A > (A > ((binary1291135688e_Tree @ A) > $o))))))).
% 2.83/1.52 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Otlookup, type, binary1265063667lookup: (!> [A:$tType]: (((A > int) > (int > ((binary1291135688e_Tree @ A) > (option @ A))))))).
% 2.83/1.52 thf(sy_c_Complete__Lattices_OInf__class_OInf, type, complete_Inf_Inf: (!> [A:$tType]: (((set @ A) > A)))).
% 2.83/1.52 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 2.83/1.52 thf(sy_c_HOL_Oundefined, type, undefined: (!> [A:$tType]: (A))).
% 2.83/1.52 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.83/1.52 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.83/1.52 thf(sy_c_Option_Ooption_ONone, type, none: (!> [A:$tType]: ((option @ A)))).
% 2.83/1.52 thf(sy_c_Option_Ooption_OSome, type, some: (!> [A:$tType]: ((A > (option @ A))))).
% 2.83/1.52 thf(sy_c_Option_Ooption_Ocase__option, type, case_option: (!> [B:$tType,A:$tType]: ((B > ((A > B) > ((option @ A) > B)))))).
% 2.83/1.52 thf(sy_c_Option_Ooption_Oset__option, type, set_option: (!> [A:$tType]: (((option @ A) > (set @ A))))).
% 2.83/1.52 thf(sy_c_Option_Ooption_Othe, type, the2: (!> [A:$tType]: (((option @ A) > A)))).
% 2.83/1.52 thf(sy_c_Option_Othese, type, these: (!> [A:$tType]: (((set @ (option @ A)) > (set @ A))))).
% 2.83/1.52 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.83/1.52 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.83/1.52 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.83/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.83/1.52 thf(sy_c_Set_Obind, type, bind: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ B)))))).
% 2.83/1.52 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.83/1.52 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.83/1.52 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.83/1.52 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.83/1.52 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.83/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.83/1.52 thf(sy_v_h, type, h: (a > int)).
% 2.83/1.52 thf(sy_v_t, type, t: (binary1291135688e_Tree @ a)).
% 2.83/1.52 thf(sy_v_x, type, x: a).
% 2.83/1.52 thf(fact_0_h, axiom, (((member @ a) @ x) @ ((binary1653327646_setOf @ a) @ t))).
% 2.83/1.52 thf(fact_1_sorted__distinct, axiom, (! [A:$tType,H:(A > int),A2:A,B2:A,T:(binary1291135688e_Tree @ A)]: ((((((binary231205461t_pred @ A) @ H) @ A2) @ B2) @ T)))).
% 2.83/1.52 thf(fact_2_s, axiom, (((binary1610619414edTree @ a) @ h) @ t)).
% 2.83/1.52 thf(fact_3_res, axiom, (~ (((((binary827270440e_memb @ a) @ h) @ x) @ t)))).
% 2.83/1.52 thf(fact_4__092_060open_062h_Ax_A_092_060noteq_062_Ah_Ax_092_060close_062, axiom, ((h @ x) != (h @ x))).
% 2.83/1.52 thf(fact_5_o1, axiom, (! [X:a]: (((((member @ a) @ X) @ ((binary1653327646_setOf @ a) @ t)) => ((h @ X) != (h @ x)))))).
% 2.83/1.52 thf(fact_6_sorted__distinct__pred__def, axiom, (! [A:$tType]: (((binary231205461t_pred @ A) = (^ [H2:(A > int),A3:A,B3:A,T2:(binary1291135688e_Tree @ A)]: ((((((binary1610619414edTree @ A) @ H2) @ T2) & ((((member @ A) @ A3) @ ((binary1653327646_setOf @ A) @ T2)) & ((((member @ A) @ B3) @ ((binary1653327646_setOf @ A) @ T2)) & ((H2 @ A3) = (H2 @ B3))))) => (A3 = B3)))))))).
% 2.83/1.52 thf(fact_7_tNone, axiom, (((((binary1265063667lookup @ a) @ h) @ (h @ x)) @ t) = (none @ a))).
% 2.83/1.52 thf(fact_8_sortedTree_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int)]: ((((binary1610619414edTree @ A) @ H) @ (binary1746293266le_Tip @ A))))).
% 2.83/1.52 thf(fact_9_sortLemmaL, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X2:A,T22:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X2) @ T22)) => (((binary1610619414edTree @ A) @ H) @ T1))))).
% 2.83/1.52 thf(fact_10_sortLemmaR, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X2:A,T22:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X2) @ T22)) => (((binary1610619414edTree @ A) @ H) @ T22))))).
% 2.83/1.52 thf(fact_11_tlookup__none, axiom, (! [A:$tType,H:(A > int),T:(binary1291135688e_Tree @ A),K:int]: ((((((binary1610619414edTree @ A) @ H) @ T) & (((((binary1265063667lookup @ A) @ H) @ K) @ T) = (none @ A))) => (! [X:A]: (((((member @ A) @ X) @ ((binary1653327646_setOf @ A) @ T)) => ((H @ X) != K)))))))).
% 2.83/1.52 thf(fact_12_tlookup__some, axiom, (! [A:$tType,H:(A > int),T:(binary1291135688e_Tree @ A),K:int,X2:A]: ((((((binary1610619414edTree @ A) @ H) @ T) & (((((binary1265063667lookup @ A) @ H) @ K) @ T) = ((some @ A) @ X2))) => ((((member @ A) @ X2) @ ((binary1653327646_setOf @ A) @ T)) & ((H @ X2) = K)))))).
% 2.83/1.52 thf(fact_13_tlookup__finds, axiom, (! [A:$tType,H:(A > int),T:(binary1291135688e_Tree @ A),Y:A]: ((((((binary1610619414edTree @ A) @ H) @ T) & (((member @ A) @ Y) @ ((binary1653327646_setOf @ A) @ T))) => (((((binary1265063667lookup @ A) @ H) @ (H @ Y)) @ T) = ((some @ A) @ Y)))))).
% 2.83/1.52 thf(fact_14_sortedTree_Osimps_I2_J, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X2:A,T22:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X2) @ T22)) = ((((binary1610619414edTree @ A) @ H) @ T1) & ((! [X3:A]: (((((member @ A) @ X3) @ ((binary1653327646_setOf @ A) @ T1)) => (((ord_less @ int) @ (H @ X3)) @ (H @ X2))))) & ((! [X3:A]: (((((member @ A) @ X3) @ ((binary1653327646_setOf @ A) @ T22)) => (((ord_less @ int) @ (H @ X2)) @ (H @ X3))))) & (((binary1610619414edTree @ A) @ H) @ T22)))))))).
% 2.83/1.52 thf(fact_15_setOf_Osimps_I1_J, axiom, (! [A:$tType]: ((((binary1653327646_setOf @ A) @ (binary1746293266le_Tip @ A)) = (bot_bot @ (set @ A)))))).
% 2.83/1.52 thf(fact_16_memb__def, axiom, (! [A:$tType]: (((binary827270440e_memb @ A) = (^ [H2:(A > int),X3:A,T2:(binary1291135688e_Tree @ A)]: ((((((case_option @ $o) @ A) @ $false) @ ((^ [Y2:A,Z:A]: ((Y2 = Z))) @ X3)) @ ((((binary1265063667lookup @ A) @ H2) @ (H2 @ X3)) @ T2)))))))).
% 2.83/1.52 thf(fact_17_Tree_Oinject, axiom, (! [A:$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))))))).
% 2.83/1.52 thf(fact_18_Tree_Oexhaust, axiom, (! [A:$tType,Y:(binary1291135688e_Tree @ A)]: (((Y != (binary1746293266le_Tip @ A)) => (~ ((! [X212:(binary1291135688e_Tree @ A),X222:A,X232:(binary1291135688e_Tree @ A)]: ((Y != ((((binary210054475elle_T @ A) @ X212) @ X222) @ X232)))))))))).
% 2.83/1.52 thf(fact_19_Tree_Oinduct, axiom, (! [A:$tType,P:((binary1291135688e_Tree @ A) > $o),Tree:(binary1291135688e_Tree @ A)]: (((P @ (binary1746293266le_Tip @ A)) => ((! [X1:(binary1291135688e_Tree @ A),X24:A,X32:(binary1291135688e_Tree @ A)]: (((P @ X1) => ((P @ X32) => (P @ ((((binary210054475elle_T @ A) @ X1) @ X24) @ X32)))))) => (P @ Tree)))))).
% 2.83/1.52 thf(fact_20_tlookup_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int),K:int]: ((((((binary1265063667lookup @ A) @ H) @ K) @ (binary1746293266le_Tip @ A)) = (none @ A))))).
% 2.83/1.52 thf(fact_21_tlookup_Osimps_I2_J, axiom, (! [A:$tType,K:int,H:(A > int),X2:A,T1:(binary1291135688e_Tree @ A),T22:(binary1291135688e_Tree @ A)]: ((((((ord_less @ int) @ K) @ (H @ X2)) => (((((binary1265063667lookup @ A) @ H) @ K) @ ((((binary210054475elle_T @ A) @ T1) @ X2) @ T22)) = ((((binary1265063667lookup @ A) @ H) @ K) @ T1))) & ((~ ((((ord_less @ int) @ K) @ (H @ X2)))) => (((((ord_less @ int) @ (H @ X2)) @ K) => (((((binary1265063667lookup @ A) @ H) @ K) @ ((((binary210054475elle_T @ A) @ T1) @ X2) @ T22)) = ((((binary1265063667lookup @ A) @ H) @ K) @ T22))) & ((~ ((((ord_less @ int) @ (H @ X2)) @ K))) => (((((binary1265063667lookup @ A) @ H) @ K) @ ((((binary210054475elle_T @ A) @ T1) @ X2) @ T22)) = ((some @ A) @ X2))))))))).
% 2.83/1.52 thf(fact_22_Tree_Odistinct_I1_J, axiom, (! [A:$tType,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: (((binary1746293266le_Tip @ A) != ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23))))).
% 2.83/1.52 thf(fact_23_not__Some__eq, axiom, (! [A:$tType,X2:(option @ A)]: (((! [Y3:A]: ((X2 != ((some @ A) @ Y3)))) = (X2 = (none @ A)))))).
% 2.83/1.52 thf(fact_24_not__None__eq, axiom, (! [A:$tType,X2:(option @ A)]: (((X2 != (none @ A)) = (? [Y3:A]: ((X2 = ((some @ A) @ Y3)))))))).
% 2.83/1.52 thf(fact_25_case__optionE, axiom, (! [A:$tType,P:$o,Q:(A > $o),X2:(option @ A)]: (((((((case_option @ $o) @ A) @ P) @ Q) @ X2) => (((X2 = (none @ A)) => (~ (P))) => (~ ((! [Y4:A]: (((X2 = ((some @ A) @ Y4)) => (~ ((Q @ Y4))))))))))))).
% 2.83/1.52 thf(fact_26_disjE__realizer2, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > $o),X2:(option @ A),R:(B > $o),F:B,G:(A > B)]: (((((((case_option @ $o) @ A) @ P) @ Q) @ X2) => ((P => (R @ F)) => ((! [Q2:A]: (((Q @ Q2) => (R @ (G @ Q2))))) => (R @ (((((case_option @ B) @ A) @ F) @ G) @ X2)))))))).
% 2.83/1.52 thf(fact_27_option_Odisc__eq__case_I1_J, axiom, (! [A:$tType,Option:(option @ A)]: (((Option = (none @ A)) = (((((case_option @ $o) @ A) @ $true) @ (^ [Uu:A]: ($false))) @ Option))))).
% 2.83/1.52 thf(fact_28_option_Odisc__eq__case_I2_J, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) = (((((case_option @ $o) @ A) @ $false) @ (^ [Uu:A]: ($true))) @ Option))))).
% 2.83/1.52 thf(fact_29_option_Oinject, axiom, (! [A:$tType,X25:A,Y24:A]: (((((some @ A) @ X25) = ((some @ A) @ Y24)) = (X25 = Y24))))).
% 2.83/1.52 thf(fact_30_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.83/1.52 thf(fact_31_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.83/1.52 thf(fact_32_all__not__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A4))))) = (A4 = (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_33_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 2.83/1.52 thf(fact_34_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.83/1.52 thf(fact_35_equals0D, axiom, (! [A:$tType,A4:(set @ A),A2:A]: (((A4 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A4))))))).
% 2.83/1.52 thf(fact_36_equals0I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A4))))) => (A4 = (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_37_ex__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A4))) = (A4 != (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_38_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X3:A]: ($false))))))).
% 2.83/1.52 thf(fact_39_option_Ocase__distrib, axiom, (! [C2:$tType,B:$tType,A:$tType,H:(B > C2),F1:B,F2:(A > B),Option:(option @ A)]: (((H @ (((((case_option @ B) @ A) @ F1) @ F2) @ Option)) = (((((case_option @ C2) @ A) @ (H @ F1)) @ (^ [X3:A]: ((H @ (F2 @ X3))))) @ Option))))).
% 2.83/1.52 thf(fact_40_option_Odistinct_I1_J, axiom, (! [A:$tType,X25:A]: (((none @ A) != ((some @ A) @ X25))))).
% 2.83/1.52 thf(fact_41_option_OdiscI, axiom, (! [A:$tType,Option:(option @ A),X25:A]: (((Option = ((some @ A) @ X25)) => (Option != (none @ A)))))).
% 2.83/1.52 thf(fact_42_option_Oexhaust, axiom, (! [A:$tType,Y:(option @ A)]: (((Y != (none @ A)) => (~ ((! [X24:A]: ((Y != ((some @ A) @ X24)))))))))).
% 2.83/1.52 thf(fact_43_option_Oinducts, axiom, (! [A:$tType,P:((option @ A) > $o),Option:(option @ A)]: (((P @ (none @ A)) => ((! [X4:A]: ((P @ ((some @ A) @ X4)))) => (P @ Option)))))).
% 2.83/1.52 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.83/1.52 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 2.83/1.52 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.83/1.52 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.83/1.52 thf(fact_48_split__option__ex, axiom, (! [A:$tType]: (((^ [P2:((option @ A) > $o)]: ((? [X5:(option @ A)]: ((P2 @ X5))))) = (^ [P3:((option @ A) > $o)]: (((P3 @ (none @ A)) | (? [X3:A]: ((P3 @ ((some @ A) @ X3))))))))))).
% 2.83/1.52 thf(fact_49_split__option__all, axiom, (! [A:$tType]: (((^ [P2:((option @ A) > $o)]: ((! [X5:(option @ A)]: ((P2 @ X5))))) = (^ [P3:((option @ A) > $o)]: (((P3 @ (none @ A)) & (! [X3:A]: ((P3 @ ((some @ A) @ X3))))))))))).
% 2.83/1.52 thf(fact_50_combine__options__cases, axiom, (! [A:$tType,B:$tType,X2:(option @ A),P:((option @ A) > ((option @ B) > $o)),Y:(option @ B)]: ((((X2 = (none @ A)) => ((P @ X2) @ Y)) => (((Y = (none @ B)) => ((P @ X2) @ Y)) => ((! [A5:A,B4:B]: (((X2 = ((some @ A) @ A5)) => ((Y = ((some @ B) @ B4)) => ((P @ X2) @ Y))))) => ((P @ X2) @ Y))))))).
% 2.83/1.52 thf(fact_51_option_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F2:(A > B)]: (((((((case_option @ B) @ A) @ F1) @ F2) @ (none @ A)) = F1)))).
% 2.83/1.52 thf(fact_52_option_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F2:(A > B),X25:A]: (((((((case_option @ B) @ A) @ F1) @ F2) @ ((some @ A) @ X25)) = (F2 @ X25))))).
% 2.83/1.52 thf(fact_53_bot__apply, axiom, (! [C2:$tType,D:$tType]: (((bot @ C2) => ((bot_bot @ (D > C2)) = (^ [X3:D]: ((bot_bot @ C2)))))))).
% 2.83/1.52 thf(fact_54_bot_Oextremum__strict, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ (bot_bot @ A)))))))))).
% 2.83/1.52 thf(fact_55_bot_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((A2 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A2)))))))).
% 2.83/1.52 thf(fact_56_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A6:(set @ A)]: ((A6 = (bot_bot @ (set @ A))))))))).
% 2.83/1.52 thf(fact_57_eqs__def, axiom, (! [A:$tType]: (((binary64540844le_eqs @ A) = (^ [H2:(A > int),X3:A]: (((collect @ A) @ (^ [Y3:A]: (((H2 @ Y3) = (H2 @ X3))))))))))).
% 2.83/1.52 thf(fact_58_option_Osplit__sel, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F2:(A > B),Option:(option @ A)]: (((P @ (((((case_option @ B) @ A) @ F1) @ F2) @ Option)) = (((Option = (none @ A)) => (P @ F1)) & ((Option = ((some @ A) @ ((the2 @ A) @ Option))) => (P @ (F2 @ ((the2 @ A) @ Option))))))))).
% 2.83/1.52 thf(fact_59_option_Osplit__sel__asm, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F2:(A > B),Option:(option @ A)]: (((P @ (((((case_option @ B) @ A) @ F1) @ F2) @ Option)) = (~ ((((Option = (none @ A)) & (~ ((P @ F1)))) | ((Option = ((some @ A) @ ((the2 @ A) @ Option))) & (~ ((P @ (F2 @ ((the2 @ A) @ Option))))))))))))).
% 2.83/1.52 thf(fact_60_set__empty__eq, axiom, (! [A:$tType,Xo:(option @ A)]: (((((set_option @ A) @ Xo) = (bot_bot @ (set @ A))) = (Xo = (none @ A)))))).
% 2.83/1.52 thf(fact_61_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X3:A]: ((bot_bot @ B)))))))).
% 2.83/1.52 thf(fact_62_elem__set, axiom, (! [A:$tType,X2:A,Xo:(option @ A)]: (((((member @ A) @ X2) @ ((set_option @ A) @ Xo)) = (Xo = ((some @ A) @ X2)))))).
% 2.83/1.52 thf(fact_63_option_Ocollapse, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) => (((some @ A) @ ((the2 @ A) @ Option)) = Option))))).
% 2.83/1.52 thf(fact_64_option_Oset__sel, axiom, (! [A:$tType,A2:(option @ A)]: (((A2 != (none @ A)) => (((member @ A) @ ((the2 @ A) @ A2)) @ ((set_option @ A) @ A2)))))).
% 2.83/1.52 thf(fact_65_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.83/1.52 thf(fact_66_not__psubset__empty, axiom, (! [A:$tType,A4:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A4) @ (bot_bot @ (set @ A)))))))).
% 2.83/1.52 thf(fact_67_option_Oexpand, axiom, (! [A:$tType,Option:(option @ A),Option2:(option @ A)]: ((((Option = (none @ A)) = (Option2 = (none @ A))) => (((Option != (none @ A)) => ((Option2 != (none @ A)) => (((the2 @ A) @ Option) = ((the2 @ A) @ Option2)))) => (Option = Option2)))))).
% 2.83/1.52 thf(fact_68_option_Osel, axiom, (! [A:$tType,X25:A]: ((((the2 @ A) @ ((some @ A) @ X25)) = X25)))).
% 2.83/1.52 thf(fact_69_option_Oset__intros, axiom, (! [A:$tType,X25:A]: ((((member @ A) @ X25) @ ((set_option @ A) @ ((some @ A) @ X25)))))).
% 2.83/1.52 thf(fact_70_option_Oset__cases, axiom, (! [A:$tType,E:A,A2:(option @ A)]: (((((member @ A) @ E) @ ((set_option @ A) @ A2)) => (A2 = ((some @ A) @ E)))))).
% 2.83/1.52 thf(fact_71_ospec, axiom, (! [A:$tType,A4:(option @ A),P:(A > $o),X2:A]: (((! [X4:A]: (((((member @ A) @ X4) @ ((set_option @ A) @ A4)) => (P @ X4)))) => ((A4 = ((some @ A) @ X2)) => (P @ X2)))))).
% 2.83/1.52 thf(fact_72_option_Oexhaust__sel, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) => (Option = ((some @ A) @ ((the2 @ A) @ Option))))))).
% 2.83/1.52 thf(fact_73_option_Osimps_I14_J, axiom, (! [A:$tType]: ((((set_option @ A) @ (none @ A)) = (bot_bot @ (set @ A)))))).
% 2.83/1.52 thf(fact_74_option_Ocase__eq__if, axiom, (! [A:$tType,B:$tType]: ((((case_option @ B) @ A) = (^ [F12:B,F22:(A > B),Option3:(option @ A)]: (((((if @ B) @ (Option3 = (none @ A))) @ F12) @ (F22 @ ((the2 @ A) @ Option3))))))))).
% 2.83/1.52 thf(fact_75_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.83/1.52 thf(fact_76_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.83/1.52 thf(fact_77_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) = ((((ord_less @ A) @ Y) @ X2) | (X2 = Y))))))))).
% 2.83/1.52 thf(fact_78_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.83/1.52 thf(fact_79_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ B4) => ((P @ A5) @ B4)))) => ((! [A5:A]: (((P @ A5) @ A5))) => ((! [A5:A,B4:A]: ((((P @ B4) @ A5) => ((P @ A5) @ B4)))) => ((P @ A2) @ B2)))))))))).
% 2.83/1.52 thf(fact_80_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X5:A]: ((P2 @ X5))))) = (^ [P3:(A > $o)]: ((? [N:A]: (((P3 @ N) & (! [M:A]: (((((ord_less @ A) @ M) @ N) => (~ ((P3 @ M)))))))))))))))).
% 2.83/1.52 thf(fact_81_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.83/1.52 thf(fact_82_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.83/1.52 thf(fact_83_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.83/1.52 thf(fact_84_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) => ((X2 != Y) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.83/1.52 thf(fact_85_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,P:$o]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ X2) => P)))))))).
% 2.83/1.52 thf(fact_86_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (Y != X2)))))))).
% 2.83/1.52 thf(fact_87_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X2:A]: (((~ ((((ord_less @ A) @ Y) @ X2))) => ((~ ((((ord_less @ A) @ X2) @ Y))) = (X2 = Y))))))))).
% 2.83/1.52 thf(fact_88_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X4) => (P @ Y5)))) => (P @ X4)))) => (P @ A2)))))))).
% 2.83/1.52 thf(fact_89_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.83/1.52 thf(fact_90_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (X2 != Y)))))))).
% 2.83/1.52 thf(fact_91_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 2.83/1.52 thf(fact_92_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.83/1.52 thf(fact_93_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.83/1.52 thf(fact_94_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((~ ((((ord_less @ A) @ X2) @ X2))))))))).
% 2.83/1.52 thf(fact_95_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) | ((X2 = Y) | (((ord_less @ A) @ Y) @ X2))))))))).
% 2.83/1.52 thf(fact_96_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z2:A]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X2) @ Z2))))))))).
% 2.83/1.52 thf(fact_97_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.83/1.52 thf(fact_98_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.83/1.52 thf(fact_99_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (X2 != Y)))))))).
% 2.83/1.52 thf(fact_100_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (? [Z3:A]: (((((ord_less @ A) @ X2) @ Z3) & (((ord_less @ A) @ Z3) @ Y))))))))))).
% 2.83/1.52 thf(fact_101_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.83/1.52 thf(fact_102_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((X2 != Y) = ((((ord_less @ A) @ X2) @ Y) | (((ord_less @ A) @ Y) @ X2))))))))).
% 2.83/1.52 thf(fact_103_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((X2 != Y) => ((~ ((((ord_less @ A) @ X2) @ Y))) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.83/1.52 thf(fact_104_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X2:A]: ((? [X_1:A]: ((((ord_less @ A) @ X2) @ X_1))))))))).
% 2.83/1.52 thf(fact_105_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X2:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X2))))))))).
% 2.83/1.52 thf(fact_106_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C2) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.83/1.52 thf(fact_107_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X4:B,Y4:B]: (((((ord_less @ B) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.83/1.52 thf(fact_108_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ B) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 2.83/1.52 thf(fact_109_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X4:B,Y4:B]: (((((ord_less @ B) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.83/1.52 thf(fact_110_option_Osimps_I15_J, axiom, (! [A:$tType,X25:A]: ((((set_option @ A) @ ((some @ A) @ X25)) = (((insert @ A) @ X25) @ (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_111_option_Othe__def, axiom, (! [A:$tType]: (((the2 @ A) = ((((case_option @ A) @ A) @ (undefined @ A)) @ (^ [X26:A]: (X26))))))).
% 2.83/1.52 thf(fact_112_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.83/1.52 thf(fact_113_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.83/1.52 thf(fact_114_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A2:A]: ((? [B4:A]: (((((ord_less @ A) @ A2) @ B4) | (((ord_less @ A) @ B4) @ A2)))))))))).
% 2.83/1.52 thf(fact_115_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X2:A,Y:A]: (((X2 != Y) => ((~ ((((ord_less @ A) @ X2) @ Y))) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.83/1.52 thf(fact_116_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: ((? [X_1:A]: ((((ord_less @ A) @ X) @ X_1))))))))).
% 2.83/1.52 thf(fact_117_insert__absorb2, axiom, (! [A:$tType,X2:A,A4:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ X2) @ A4)) = (((insert @ A) @ X2) @ A4))))).
% 2.83/1.52 thf(fact_118_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A4)) = ((A2 = B2) | (((member @ A) @ A2) @ A4)))))).
% 2.83/1.52 thf(fact_119_insertCI, axiom, (! [A:$tType,A2:A,B5:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B5))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B5)))))).
% 2.83/1.52 thf(fact_120_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_121_singleton__conv, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ (^ [X3:A]: ((X3 = A2)))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_122_singleton__conv2, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ ((^ [Y2:A,Z:A]: ((Y2 = Z))) @ A2)) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_123_psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B5) => ((((ord_less @ (set @ A)) @ B5) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 2.83/1.52 thf(fact_124_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A6)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B6)))))))))).
% 2.83/1.52 thf(fact_125_psubsetD, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A4) @ B5) => ((((member @ A) @ C) @ A4) => (((member @ A) @ C) @ B5)))))).
% 2.83/1.52 thf(fact_126_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => (? [B7:(set @ A)]: (((A4 = (((insert @ A) @ A2) @ B7)) & (~ ((((member @ A) @ A2) @ B7)))))))))).
% 2.83/1.52 thf(fact_127_insert__commute, axiom, (! [A:$tType,X2:A,Y:A,A4:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ Y) @ A4)) = (((insert @ A) @ Y) @ (((insert @ A) @ X2) @ A4)))))).
% 2.83/1.52 thf(fact_128_insert__eq__iff, axiom, (! [A:$tType,A2:A,A4:(set @ A),B2:A,B5:(set @ A)]: (((~ ((((member @ A) @ A2) @ A4))) => ((~ ((((member @ A) @ B2) @ B5))) => (((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B2) @ B5)) = (((A2 = B2) => (A4 = B5)) & ((A2 != B2) => (? [C4:(set @ A)]: (((A4 = (((insert @ A) @ B2) @ C4)) & ((~ ((((member @ A) @ B2) @ C4))) & ((B5 = (((insert @ A) @ A2) @ C4)) & (~ ((((member @ A) @ A2) @ C4)))))))))))))))).
% 2.83/1.52 thf(fact_129_insert__absorb, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => ((((insert @ A) @ A2) @ A4) = A4))))).
% 2.83/1.52 thf(fact_130_insert__ident, axiom, (! [A:$tType,X2:A,A4:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X2) @ A4))) => ((~ ((((member @ A) @ X2) @ B5))) => (((((insert @ A) @ X2) @ A4) = (((insert @ A) @ X2) @ B5)) = (A4 = B5))))))).
% 2.83/1.52 thf(fact_131_Set_Oset__insert, axiom, (! [A:$tType,X2:A,A4:(set @ A)]: (((((member @ A) @ X2) @ A4) => (~ ((! [B7:(set @ A)]: (((A4 = (((insert @ A) @ X2) @ B7)) => (((member @ A) @ X2) @ B7)))))))))).
% 2.83/1.52 thf(fact_132_insertI2, axiom, (! [A:$tType,A2:A,B5:(set @ A),B2:A]: (((((member @ A) @ A2) @ B5) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B5)))))).
% 2.83/1.52 thf(fact_133_insertI1, axiom, (! [A:$tType,A2:A,B5:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B5))))).
% 2.83/1.52 thf(fact_134_insertE, axiom, (! [A:$tType,A2:A,B2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A4)) => ((A2 != B2) => (((member @ A) @ A2) @ A4)))))).
% 2.83/1.52 thf(fact_135_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A3:A,B6:(set @ A)]: (((collect @ A) @ (^ [X3:A]: (((X3 = A3) | (((member @ A) @ X3) @ B6))))))))))).
% 2.83/1.52 thf(fact_136_insert__Collect, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((insert @ A) @ A2) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U:A]: (((U != A2) => (P @ U))))))))).
% 2.83/1.52 thf(fact_137_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 2.83/1.52 thf(fact_138_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 2.83/1.52 thf(fact_139_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C:A,D2:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C) @ (((insert @ A) @ D2) @ (bot_bot @ (set @ A))))) = (((A2 = C) & (B2 = D2)) | ((A2 = D2) & (B2 = C))))))).
% 2.83/1.52 thf(fact_140_insert__not__empty, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((insert @ A) @ A2) @ A4) != (bot_bot @ (set @ A)))))).
% 2.83/1.52 thf(fact_141_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 2.83/1.52 thf(fact_142_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X3:A]: (((X3 = A2) & (P @ X3))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X3:A]: (((X3 = A2) & (P @ X3))))) = (bot_bot @ (set @ A)))))))).
% 2.83/1.52 thf(fact_143_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X3:A]: (((A2 = X3) & (P @ X3))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X3:A]: (((A2 = X3) & (P @ X3))))) = (bot_bot @ (set @ A)))))))).
% 2.83/1.52 thf(fact_144_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X))))))))).
% 2.83/1.52 thf(fact_145_the__elem__eq, axiom, (! [A:$tType,X2:A]: ((((the_elem @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = X2)))).
% 2.83/1.52 thf(fact_146_is__singletonI, axiom, (! [A:$tType,X2:A]: (((is_singleton @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_147_these__insert__Some, axiom, (! [A:$tType,X2:A,A4:(set @ (option @ A))]: ((((these @ A) @ (((insert @ (option @ A)) @ ((some @ A) @ X2)) @ A4)) = (((insert @ A) @ X2) @ ((these @ A) @ A4)))))).
% 2.83/1.52 thf(fact_148_is__singletonE, axiom, (! [A:$tType,A4:(set @ A)]: ((((is_singleton @ A) @ A4) => (~ ((! [X4:A]: ((A4 != (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))))).
% 2.83/1.52 thf(fact_149_these__empty, axiom, (! [A:$tType]: ((((these @ A) @ (bot_bot @ (set @ (option @ A)))) = (bot_bot @ (set @ A)))))).
% 2.83/1.52 thf(fact_150_these__insert__None, axiom, (! [A:$tType,A4:(set @ (option @ A))]: ((((these @ A) @ (((insert @ (option @ A)) @ (none @ A)) @ A4)) = ((these @ A) @ A4))))).
% 2.83/1.52 thf(fact_151_in__these__eq, axiom, (! [A:$tType,X2:A,A4:(set @ (option @ A))]: (((((member @ A) @ X2) @ ((these @ A) @ A4)) = (((member @ (option @ A)) @ ((some @ A) @ X2)) @ A4))))).
% 2.83/1.52 thf(fact_152_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A6:(set @ A)]: ((A6 = (((insert @ A) @ ((the_elem @ A) @ A6)) @ (bot_bot @ (set @ A)))))))))).
% 2.83/1.52 thf(fact_153_is__singletonI_H, axiom, (! [A:$tType,A4:(set @ A)]: (((A4 != (bot_bot @ (set @ A))) => ((! [X4:A,Y4:A]: (((((member @ A) @ X4) @ A4) => ((((member @ A) @ Y4) @ A4) => (X4 = Y4))))) => ((is_singleton @ A) @ A4)))))).
% 2.83/1.52 thf(fact_154_these__not__empty__eq, axiom, (! [A:$tType,B5:(set @ (option @ A))]: (((((these @ A) @ B5) != (bot_bot @ (set @ A))) = ((B5 != (bot_bot @ (set @ (option @ A)))) & (B5 != (((insert @ (option @ A)) @ (none @ A)) @ (bot_bot @ (set @ (option @ A)))))))))).
% 2.83/1.52 thf(fact_155_these__empty__eq, axiom, (! [A:$tType,B5:(set @ (option @ A))]: (((((these @ A) @ B5) = (bot_bot @ (set @ A))) = ((B5 = (bot_bot @ (set @ (option @ A)))) | (B5 = (((insert @ (option @ A)) @ (none @ A)) @ (bot_bot @ (set @ (option @ A)))))))))).
% 2.83/1.52 thf(fact_156_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A6:(set @ A)]: ((? [X3:A]: ((A6 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))).
% 2.83/1.52 thf(fact_157_the__elem__def, axiom, (! [A:$tType]: (((the_elem @ A) = (^ [X6:(set @ A)]: (((the @ A) @ (^ [X3:A]: ((X6 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))))).
% 2.83/1.52 thf(fact_158_setOf_Osimps_I2_J, axiom, (! [A:$tType,T1:(binary1291135688e_Tree @ A),X2:A,T22:(binary1291135688e_Tree @ A)]: ((((binary1653327646_setOf @ A) @ ((((binary210054475elle_T @ A) @ T1) @ X2) @ T22)) = (((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ ((binary1653327646_setOf @ A) @ T1)) @ ((binary1653327646_setOf @ A) @ T22))) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))).
% 2.83/1.52 thf(fact_159_Option_Othese__def, axiom, (! [A:$tType]: (((these @ A) = (^ [A6:(set @ (option @ A))]: (((((image @ (option @ A)) @ A) @ (the2 @ A)) @ ((collect @ (option @ A)) @ (^ [X3:(option @ A)]: (((((member @ (option @ A)) @ X3) @ A6) & (X3 != (none @ A))))))))))))).
% 2.83/1.52 thf(fact_160_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X2:B,A4:(set @ B)]: (((B2 = (F @ X2)) => ((((member @ B) @ X2) @ A4) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A4))))))).
% 2.83/1.52 thf(fact_161_Un__iff, axiom, (! [A:$tType,C:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) = ((((member @ A) @ C) @ A4) | (((member @ A) @ C) @ B5)))))).
% 2.83/1.52 thf(fact_162_UnCI, axiom, (! [A:$tType,C:A,B5:(set @ A),A4:(set @ A)]: ((((~ ((((member @ A) @ C) @ B5))) => (((member @ A) @ C) @ A4)) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B5)))))).
% 2.83/1.52 thf(fact_163_image__ident, axiom, (! [A:$tType,Y6:(set @ A)]: ((((((image @ A) @ A) @ (^ [X3:A]: (X3))) @ Y6) = Y6)))).
% 2.83/1.52 thf(fact_164_image__is__empty, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B)]: (((((((image @ B) @ A) @ F) @ A4) = (bot_bot @ (set @ A))) = (A4 = (bot_bot @ (set @ B))))))).
% 2.83/1.52 thf(fact_165_empty__is__image, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image @ B) @ A) @ F) @ A4)) = (A4 = (bot_bot @ (set @ B))))))).
% 2.83/1.52 thf(fact_166_image__empty, axiom, (! [B:$tType,A:$tType,F:(B > A)]: ((((((image @ B) @ A) @ F) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.83/1.52 thf(fact_167_image__insert, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:B,B5:(set @ B)]: ((((((image @ B) @ A) @ F) @ (((insert @ B) @ A2) @ B5)) = (((insert @ A) @ (F @ A2)) @ ((((image @ B) @ A) @ F) @ B5)))))).
% 2.83/1.52 thf(fact_168_insert__image, axiom, (! [B:$tType,A:$tType,X2:A,A4:(set @ A),F:(A > B)]: (((((member @ A) @ X2) @ A4) => ((((insert @ B) @ (F @ X2)) @ ((((image @ A) @ B) @ F) @ A4)) = ((((image @ A) @ B) @ F) @ A4)))))).
% 2.83/1.52 thf(fact_169_Un__empty, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A4) @ B5) = (bot_bot @ (set @ A))) = ((A4 = (bot_bot @ (set @ A))) & (B5 = (bot_bot @ (set @ A)))))))).
% 2.83/1.52 thf(fact_170_Un__insert__left, axiom, (! [A:$tType,A2:A,B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A2) @ B5)) @ C3) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ B5) @ C3)))))).
% 2.83/1.52 thf(fact_171_Un__insert__right, axiom, (! [A:$tType,A4:(set @ A),A2:A,B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ B5)) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ A4) @ B5)))))).
% 2.83/1.52 thf(fact_172_insert__def, axiom, (! [A:$tType]: (((insert @ A) = (^ [A3:A]: (((sup_sup @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: ((X3 = A3))))))))))).
% 2.83/1.52 thf(fact_173_Un__left__commute, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B5) @ C3)) = (((sup_sup @ (set @ A)) @ B5) @ (((sup_sup @ (set @ A)) @ A4) @ C3)))))).
% 2.83/1.52 thf(fact_174_Un__left__absorb, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) = (((sup_sup @ (set @ A)) @ A4) @ B5))))).
% 2.83/1.52 thf(fact_175_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X2:A,A4:(set @ A),B2:B,F:(A > B)]: (((((member @ A) @ X2) @ A4) => ((B2 = (F @ X2)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F) @ A4))))))).
% 2.83/1.52 thf(fact_176_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B),P:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F) @ A4)) => (P @ X4)))) => (! [X:B]: (((((member @ B) @ X) @ A4) => (P @ (F @ X))))))))).
% 2.83/1.52 thf(fact_177_image__cong, axiom, (! [B:$tType,A:$tType,M2:(set @ A),N2:(set @ A),F:(A > B),G:(A > B)]: (((M2 = N2) => ((! [X4:A]: (((((member @ A) @ X4) @ N2) => ((F @ X4) = (G @ X4))))) => (((((image @ A) @ B) @ F) @ M2) = ((((image @ A) @ B) @ G) @ N2))))))).
% 2.83/1.52 thf(fact_178_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B),P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F) @ A4)) & (P @ X)))) => (? [X4:B]: (((((member @ B) @ X4) @ A4) & (P @ (F @ X4))))))))).
% 2.83/1.52 thf(fact_179_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((sup_sup @ (set @ A)) @ B6) @ A6))))))).
% 2.83/1.52 thf(fact_180_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F:(B > A),A4:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F) @ A4)) = (? [X3:B]: (((((member @ B) @ X3) @ A4) & (Z2 = (F @ X3))))))))).
% 2.83/1.52 thf(fact_181_Un__absorb, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ A4) = A4)))).
% 2.83/1.52 thf(fact_182_image__Un, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B),B5:(set @ B)]: ((((((image @ B) @ A) @ F) @ (((sup_sup @ (set @ B)) @ A4) @ B5)) = (((sup_sup @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A4)) @ ((((image @ B) @ A) @ F) @ B5)))))).
% 2.83/1.52 thf(fact_183_Un__assoc, axiom, (! [A:$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)))))).
% 2.83/1.52 thf(fact_184_ball__Un, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) => (P @ X3)))) = ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (P @ X3)))) & (! [X3:A]: (((((member @ A) @ X3) @ B5) => (P @ X3))))))))).
% 2.83/1.52 thf(fact_185_imageI, axiom, (! [B:$tType,A:$tType,X2:A,A4:(set @ A),F:(A > B)]: (((((member @ A) @ X2) @ A4) => (((member @ B) @ (F @ X2)) @ ((((image @ A) @ B) @ F) @ A4)))))).
% 2.83/1.52 thf(fact_186_bex__Un, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),P:(A > $o)]: (((? [X3:A]: (((((member @ A) @ X3) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) & (P @ X3)))) = ((? [X3:A]: (((((member @ A) @ X3) @ A4) & (P @ X3)))) | (? [X3:A]: (((((member @ A) @ X3) @ B5) & (P @ X3))))))))).
% 2.83/1.52 thf(fact_187_UnI2, axiom, (! [A:$tType,C:A,B5:(set @ A),A4:(set @ A)]: (((((member @ A) @ C) @ B5) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B5)))))).
% 2.83/1.52 thf(fact_188_UnI1, axiom, (! [A:$tType,C:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ A4) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B5)))))).
% 2.83/1.52 thf(fact_189_UnE, axiom, (! [A:$tType,C:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B5)) => ((~ ((((member @ A) @ C) @ A4))) => (((member @ A) @ C) @ B5)))))).
% 2.83/1.52 thf(fact_190_Un__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A6) | (((member @ A) @ X3) @ B6))))))))))).
% 2.83/1.52 thf(fact_191_imageE, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),A4:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A4)) => (~ ((! [X4:B]: (((B2 = (F @ X4)) => (~ ((((member @ B) @ X4) @ A4)))))))))))).
% 2.83/1.52 thf(fact_192_image__image, axiom, (! [A:$tType,B:$tType,C2:$tType,F:(B > A),G:(C2 > B),A4:(set @ C2)]: ((((((image @ B) @ A) @ F) @ ((((image @ C2) @ B) @ G) @ A4)) = ((((image @ C2) @ A) @ (^ [X3:C2]: ((F @ (G @ X3))))) @ A4))))).
% 2.83/1.52 thf(fact_193_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B),P:(A > $o)]: ((((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F) @ A4)) & (P @ X3))))) = ((((image @ B) @ A) @ F) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ A4) & (P @ (F @ X3))))))))))).
% 2.83/1.52 thf(fact_194_Collect__disj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X3:A]: (((P @ X3) | (Q @ X3))))) = (((sup_sup @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.83/1.52 thf(fact_195_Un__empty__right, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = A4)))).
% 2.83/1.52 thf(fact_196_Un__empty__left, axiom, (! [A:$tType,B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B5) = B5)))).
% 2.83/1.52 thf(fact_197_singleton__Un__iff, axiom, (! [A:$tType,X2:A,A4:(set @ A),B5:(set @ A)]: ((((((insert @ A) @ X2) @ (bot_bot @ (set @ A))) = (((sup_sup @ (set @ A)) @ A4) @ B5)) = (((A4 = (bot_bot @ (set @ A))) & (B5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))) | (((A4 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) & (B5 = (bot_bot @ (set @ A)))) | ((A4 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) & (B5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))))))).
% 2.83/1.52 thf(fact_198_Un__singleton__iff, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),X2:A]: ((((((sup_sup @ (set @ A)) @ A4) @ B5) = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = (((A4 = (bot_bot @ (set @ A))) & (B5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))) | (((A4 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) & (B5 = (bot_bot @ (set @ A)))) | ((A4 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) & (B5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))))))).
% 2.83/1.52 thf(fact_199_insert__is__Un, axiom, (! [A:$tType]: (((insert @ A) = (^ [A3:A]: (((sup_sup @ (set @ A)) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))))))))).
% 2.83/1.52 thf(fact_200_image__constant, axiom, (! [A:$tType,B:$tType,X2:A,A4:(set @ A),C:B]: (((((member @ A) @ X2) @ A4) => (((((image @ A) @ B) @ (^ [X3:A]: (C))) @ A4) = (((insert @ B) @ C) @ (bot_bot @ (set @ B)))))))).
% 2.83/1.52 thf(fact_201_image__constant__conv, axiom, (! [B:$tType,A:$tType,A4:(set @ B),C:A]: ((((A4 = (bot_bot @ (set @ B))) => (((((image @ B) @ A) @ (^ [X3:B]: (C))) @ A4) = (bot_bot @ (set @ A)))) & ((A4 != (bot_bot @ (set @ B))) => (((((image @ B) @ A) @ (^ [X3:B]: (C))) @ A4) = (((insert @ A) @ C) @ (bot_bot @ (set @ A))))))))).
% 2.83/1.52 thf(fact_202_the__elem__image__unique, axiom, (! [B:$tType,A:$tType,A4:(set @ A),F:(A > B),X2:A]: (((A4 != (bot_bot @ (set @ A))) => ((! [Y4:A]: (((((member @ A) @ Y4) @ A4) => ((F @ Y4) = (F @ X2))))) => (((the_elem @ B) @ ((((image @ A) @ B) @ F) @ A4)) = (F @ X2))))))).
% 2.83/1.52 thf(fact_203_sup__bot__left, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X2:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X2) = X2))))))).
% 2.83/1.52 thf(fact_204_sup__bot__right, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X2:A]: (((((sup_sup @ A) @ X2) @ (bot_bot @ A)) = X2))))))).
% 2.83/1.52 thf(fact_205_bot__eq__sup__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X2:A,Y:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X2) @ Y)) = ((X2 = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.83/1.52 thf(fact_206_sup__bot_Oright__neutral, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ (bot_bot @ A)) = A2))))))).
% 2.83/1.52 thf(fact_207_sup__bot_Oneutr__eq__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A,B2:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ A2) @ B2)) = ((A2 = (bot_bot @ A)) & (B2 = (bot_bot @ A)))))))))).
% 2.83/1.52 thf(fact_208_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A2) = A2))))))).
% 2.83/1.52 thf(fact_209_sup__bot_Oeq__neutr__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A,B2:A]: ((((((sup_sup @ A) @ A2) @ B2) = (bot_bot @ A)) = ((A2 = (bot_bot @ A)) & (B2 = (bot_bot @ A)))))))))).
% 2.83/1.52 thf(fact_210_sup__eq__bot__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X2:A,Y:A]: ((((((sup_sup @ A) @ X2) @ Y) = (bot_bot @ A)) = ((X2 = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.83/1.52 thf(fact_211_these__image__Some__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((these @ A) @ ((((image @ A) @ (option @ A)) @ (some @ A)) @ A4)) = A4)))).
% 2.83/1.52 thf(fact_212_sup__set__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((sup_sup @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A6)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B6))))))))))).
% 2.83/1.52 thf(fact_213_sup__Un__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((sup_sup @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S)))) = (^ [X3:A]: ((((member @ A) @ X3) @ (((sup_sup @ (set @ A)) @ R) @ S)))))))).
% 2.83/1.52 thf(fact_214_None__notin__image__Some, axiom, (! [A:$tType,A4:(set @ A)]: ((~ ((((member @ (option @ A)) @ (none @ A)) @ ((((image @ A) @ (option @ A)) @ (some @ A)) @ A4))))))).
% 2.83/1.52 thf(fact_215_Some__image__these__eq, axiom, (! [A:$tType,A4:(set @ (option @ A))]: ((((((image @ A) @ (option @ A)) @ (some @ A)) @ ((these @ A) @ A4)) = ((collect @ (option @ A)) @ (^ [X3:(option @ A)]: (((((member @ (option @ A)) @ X3) @ A4) & (X3 != (none @ A)))))))))).
% 2.83/1.52 thf(fact_216_sup_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C:A,B2:A,A2:A]: (((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.83/1.52 thf(fact_217_sup_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ C) @ A2) => (((ord_less @ A) @ C) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.83/1.52 thf(fact_218_sup_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less @ A) = (^ [B3:A,A3:A]: (((A3 = (((sup_sup @ A) @ A3) @ B3)) & (A3 != B3))))))))).
% 2.83/1.52 thf(fact_219_sup_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C:A,A2:A]: (((((ord_less @ A) @ (((sup_sup @ A) @ B2) @ C)) @ A2) => (~ (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ C) @ A2))))))))))))).
% 2.83/1.52 thf(fact_220_less__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,B2:A,A2:A]: (((((ord_less @ A) @ X2) @ B2) => (((ord_less @ A) @ X2) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.83/1.52 thf(fact_221_less__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,A2:A,B2:A]: (((((ord_less @ A) @ X2) @ A2) => (((ord_less @ A) @ X2) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 2.83/1.52 thf(fact_222_the__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X4:A]: (((P @ X4) => (X4 = A2)))) => (((the @ A) @ P) = A2)))))).
% 2.83/1.52 thf(fact_223_the__eq__trivial, axiom, (! [A:$tType,A2:A]: ((((the @ A) @ (^ [X3:A]: ((X3 = A2)))) = A2)))).
% 2.83/1.52 thf(fact_224_the__sym__eq__trivial, axiom, (! [A:$tType,X2:A]: ((((the @ A) @ ((^ [Y2:A,Z:A]: ((Y2 = Z))) @ X2)) = X2)))).
% 2.83/1.52 thf(fact_225_the1__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((? [X:A]: (((P @ X) & (! [Y4:A]: (((P @ Y4) => (Y4 = X))))))) => ((P @ A2) => (((the @ A) @ P) = A2)))))).
% 2.83/1.52 thf(fact_226_the1I2, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((? [X:A]: (((P @ X) & (! [Y4:A]: (((P @ Y4) => (Y4 = X))))))) => ((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (Q @ ((the @ A) @ P))))))).
% 2.83/1.52 thf(fact_227_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P3:$o,X3:A,Y3:A]: (((the @ A) @ (^ [Z4:A]: (((P3 => (Z4 = X3)) & ((~ (P3)) => (Z4 = Y3)))))))))))).
% 2.83/1.52 thf(fact_228_theI2, axiom, (! [A:$tType,P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X4:A]: (((P @ X4) => (X4 = A2)))) => ((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (Q @ ((the @ A) @ P)))))))).
% 2.83/1.52 thf(fact_229_theI_H, axiom, (! [A:$tType,P:(A > $o)]: (((? [X:A]: (((P @ X) & (! [Y4:A]: (((P @ Y4) => (Y4 = X))))))) => (P @ ((the @ A) @ P)))))).
% 2.83/1.52 thf(fact_230_theI, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X4:A]: (((P @ X4) => (X4 = A2)))) => (P @ ((the @ A) @ P))))))).
% 2.83/1.52 thf(fact_231_Tree_Osimps_I15_J, axiom, (! [A:$tType,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: ((((binary2130109271t_Tree @ A) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23)) = (((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ ((binary2130109271t_Tree @ A) @ X21)) @ (((insert @ A) @ X22) @ (bot_bot @ (set @ A))))) @ ((binary2130109271t_Tree @ A) @ X23)))))).
% 2.83/1.52 thf(fact_232_bind__singleton__conv__image, axiom, (! [A:$tType,B:$tType,A4:(set @ B),F:(B > A)]: ((((((bind @ B) @ A) @ A4) @ (^ [X3:B]: ((((insert @ A) @ (F @ X3)) @ (bot_bot @ (set @ A)))))) = ((((image @ B) @ A) @ F) @ A4))))).
% 2.83/1.52 thf(fact_233_empty__bind, axiom, (! [B:$tType,A:$tType,F:(B > (set @ A))]: ((((((bind @ B) @ A) @ (bot_bot @ (set @ B))) @ F) = (bot_bot @ (set @ A)))))).
% 2.83/1.52 thf(fact_234_Set_Obind__bind, axiom, (! [C2:$tType,B:$tType,A:$tType,A4:(set @ A),B5:(A > (set @ C2)),C3:(C2 > (set @ B))]: ((((((bind @ C2) @ B) @ ((((bind @ A) @ C2) @ A4) @ B5)) @ C3) = ((((bind @ A) @ B) @ A4) @ (^ [X3:A]: (((((bind @ C2) @ B) @ (B5 @ X3)) @ C3)))))))).
% 2.83/1.52 thf(fact_235_Tree_Oset__intros_I3_J, axiom, (! [A:$tType,Ya:A,X23:(binary1291135688e_Tree @ A),X21:(binary1291135688e_Tree @ A),X22:A]: (((((member @ A) @ Ya) @ ((binary2130109271t_Tree @ A) @ X23)) => (((member @ A) @ Ya) @ ((binary2130109271t_Tree @ A) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23))))))).
% 2.83/1.52 thf(fact_236_Tree_Oset__intros_I2_J, axiom, (! [A:$tType,X22:A,X21:(binary1291135688e_Tree @ A),X23:(binary1291135688e_Tree @ A)]: ((((member @ A) @ X22) @ ((binary2130109271t_Tree @ A) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23)))))).
% 2.83/1.52 thf(fact_237_Tree_Oset__intros_I1_J, axiom, (! [A:$tType,Y:A,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: (((((member @ A) @ Y) @ ((binary2130109271t_Tree @ A) @ X21)) => (((member @ A) @ Y) @ ((binary2130109271t_Tree @ A) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23))))))).
% 2.83/1.52 thf(fact_238_Tree_Oset__cases, axiom, (! [A:$tType,E:A,A2:(binary1291135688e_Tree @ A)]: (((((member @ A) @ E) @ ((binary2130109271t_Tree @ A) @ A2)) => ((! [Z1:(binary1291135688e_Tree @ A)]: (((? [Z22:A,Z32:(binary1291135688e_Tree @ A)]: ((A2 = ((((binary210054475elle_T @ A) @ Z1) @ Z22) @ Z32)))) => (~ ((((member @ A) @ E) @ ((binary2130109271t_Tree @ A) @ Z1))))))) => ((! [Z1:(binary1291135688e_Tree @ A),Z32:(binary1291135688e_Tree @ A)]: ((A2 != ((((binary210054475elle_T @ A) @ Z1) @ E) @ Z32)))) => (~ ((! [Z1:(binary1291135688e_Tree @ A),Z22:A,Z32:(binary1291135688e_Tree @ A)]: (((A2 = ((((binary210054475elle_T @ A) @ Z1) @ Z22) @ Z32)) => (~ ((((member @ A) @ E) @ ((binary2130109271t_Tree @ A) @ Z32))))))))))))))).
% 2.83/1.52 thf(fact_239_bind__const, axiom, (! [B:$tType,A:$tType,A4:(set @ B),B5:(set @ A)]: ((((A4 = (bot_bot @ (set @ B))) => (((((bind @ B) @ A) @ A4) @ (^ [Uu:B]: (B5))) = (bot_bot @ (set @ A)))) & ((A4 != (bot_bot @ (set @ B))) => (((((bind @ B) @ A) @ A4) @ (^ [Uu:B]: (B5))) = B5)))))).
% 2.83/1.52 thf(fact_240_nonempty__bind__const, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B5:(set @ B)]: (((A4 != (bot_bot @ (set @ A))) => (((((bind @ A) @ B) @ A4) @ (^ [Uu:A]: (B5))) = B5))))).
% 2.83/1.52 thf(fact_241_Tree_Osimps_I14_J, axiom, (! [A:$tType]: ((((binary2130109271t_Tree @ A) @ (binary1746293266le_Tip @ A)) = (bot_bot @ (set @ A)))))).
% 2.83/1.52 thf(fact_242_Inf_OINF__identity__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A4:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (^ [X3:A]: (X3))) @ A4)) = (Inf @ A4))))).
% 2.83/1.52 thf(fact_243_Sup_OSUP__identity__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A4:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (^ [X3:A]: (X3))) @ A4)) = (Sup @ A4))))).
% 2.83/1.52 thf(fact_244_range__constant, axiom, (! [B:$tType,A:$tType,X2:A]: ((((((image @ B) @ A) @ (^ [Uu:B]: (X2))) @ (top_top @ (set @ B))) = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))).
% 2.83/1.52 thf(fact_245_INF__eq__bot__iff, axiom, (! [B:$tType,A:$tType]: (((comple1035589618norder @ A) => (! [F:(B > A),A4:(set @ B)]: (((((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A4)) = (bot_bot @ A)) = (! [X3:A]: (((((ord_less @ A) @ (bot_bot @ A)) @ X3) => (? [Y3:B]: (((((member @ B) @ Y3) @ A4) & (((ord_less @ A) @ (F @ Y3)) @ X3)))))))))))))).
% 2.83/1.52 thf(fact_246_top__apply, axiom, (! [C2:$tType,D:$tType]: (((top @ C2) => ((top_top @ (D > C2)) = (^ [X3:D]: ((top_top @ C2)))))))).
% 2.83/1.52 thf(fact_247_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.83/1.52 thf(fact_248_Inf__apply, axiom, (! [B:$tType,A:$tType]: (((complete_Inf @ B) => ((complete_Inf_Inf @ (A > B)) = (^ [A6:(set @ (A > B)),X3:A]: (((complete_Inf_Inf @ B) @ ((((image @ (A > B)) @ B) @ (^ [F3:(A > B)]: ((F3 @ X3)))) @ A6))))))))).
% 2.83/1.52 thf(fact_249_INT__constant, axiom, (! [B:$tType,A:$tType,A4:(set @ B),C:(set @ A)]: ((((A4 = (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [Y3:B]: (C))) @ A4)) = (top_top @ (set @ A)))) & ((A4 != (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [Y3:B]: (C))) @ A4)) = C)))))).
% 2.83/1.52 thf(fact_250_Collect__const, axiom, (! [A:$tType,P:$o]: (((P => (((collect @ A) @ (^ [S2:A]: (P))) = (top_top @ (set @ A)))) & ((~ (P)) => (((collect @ A) @ (^ [S2:A]: (P))) = (bot_bot @ (set @ A)))))))).
% 2.83/1.52 thf(fact_251_INF__apply, axiom, (! [A:$tType,B:$tType,C2:$tType]: (((complete_Inf @ A) => (! [F:(C2 > (B > A)),A4:(set @ C2),X2:B]: (((((complete_Inf_Inf @ (B > A)) @ ((((image @ C2) @ (B > A)) @ F) @ A4)) @ X2) = ((complete_Inf_Inf @ A) @ ((((image @ C2) @ A) @ (^ [Y3:C2]: (((F @ Y3) @ X2)))) @ A4))))))))).
% 2.83/1.52 thf(fact_252_INF__identity__eq, axiom, (! [A:$tType]: (((complete_Inf @ A) => (! [A4:(set @ A)]: ((((complete_Inf_Inf @ A) @ ((((image @ A) @ A) @ (^ [X3:A]: (X3))) @ A4)) = ((complete_Inf_Inf @ A) @ A4)))))))).
% 2.83/1.52 thf(fact_253_Inf__eq__bot__iff, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (! [A4:(set @ A)]: (((((complete_Inf_Inf @ A) @ A4) = (bot_bot @ A)) = (! [X3:A]: (((((ord_less @ A) @ (bot_bot @ A)) @ X3) => (? [Y3:A]: (((((member @ A) @ Y3) @ A4) & (((ord_less @ A) @ Y3) @ X3)))))))))))))).
% 2.83/1.52 thf(fact_254_Inf__empty, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (((complete_Inf_Inf @ A) @ (bot_bot @ (set @ A))) = (top_top @ A)))))).
% 2.83/1.52 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, (bounded_lattice @ $o)).
% 2.83/1.52 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A7:$tType]: ((bounded_lattice @ (set @ A7))))).
% 2.83/1.52 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A7:$tType,A8:$tType]: (((bounded_lattice @ A8) => (bounded_lattice @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A7:$tType,A8:$tType]: (((bounded_lattice @ A8) => (bounde1808546759up_bot @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (comple187826305attice @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A7:$tType,A8:$tType]: (((semilattice_sup @ A8) => (semilattice_sup @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Complete__Lattices_OInf, axiom, (! [A7:$tType,A8:$tType]: (((complete_Inf @ A8) => (complete_Inf @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A8:$tType]: (((order_bot @ A8) => (order_bot @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Orderings_Otop, axiom, (! [A7:$tType,A8:$tType]: (((top @ A8) => (top @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A8:$tType]: (((bot @ A8) => (bot @ (A7 > A8)))))).
% 2.83/1.52 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3, axiom, (semilattice_sup @ int)).
% 2.83/1.52 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 2.83/1.52 thf(tcon_Int_Oint___Complete__Lattices_OInf_4, axiom, (complete_Inf @ int)).
% 2.83/1.52 thf(tcon_Int_Oint___Orderings_Opreorder_5, axiom, (preorder @ int)).
% 2.83/1.52 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 2.83/1.52 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, (no_top @ int)).
% 2.83/1.52 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, (no_bot @ int)).
% 2.83/1.52 thf(tcon_Int_Oint___Orderings_Oorder_6, axiom, (order @ int)).
% 2.83/1.52 thf(tcon_Int_Oint___Orderings_Oord_7, axiom, (ord @ int)).
% 2.83/1.52 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_8, axiom, (! [A7:$tType]: ((bounde1808546759up_bot @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_9, axiom, (! [A7:$tType]: ((comple187826305attice @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10, axiom, (! [A7:$tType]: ((semilattice_sup @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Complete__Lattices_OInf_11, axiom, (! [A7:$tType]: ((complete_Inf @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Orderings_Oorder__bot_12, axiom, (! [A7:$tType]: ((order_bot @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Orderings_Opreorder_13, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Orderings_Oorder_14, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Orderings_Otop_15, axiom, (! [A7:$tType]: ((top @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Orderings_Oord_16, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.83/1.52 thf(tcon_Set_Oset___Orderings_Obot_17, axiom, (! [A7:$tType]: ((bot @ (set @ A7))))).
% 2.83/1.52 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_18, axiom, (bounde1808546759up_bot @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_19, axiom, (comple187826305attice @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_20, axiom, (semilattice_sup @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Complete__Lattices_OInf_21, axiom, (complete_Inf @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Orderings_Oorder__bot_22, axiom, (order_bot @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_23, axiom, (preorder @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder_24, axiom, (linorder @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_25, axiom, (order @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Orderings_Otop_26, axiom, (top @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Orderings_Oord_27, axiom, (ord @ $o)).
% 2.83/1.52 thf(tcon_HOL_Obool___Orderings_Obot_28, axiom, (bot @ $o)).
% 2.83/1.52 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.83/1.52 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $false) @ X2) @ Y) = Y)))).
% 2.83/1.52 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $true) @ X2) @ Y) = X2)))).
% 2.83/1.52 thf(conj_0, conjecture, $false).
% 2.83/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.UyQQfMpErQ/DTF2DTF25161.p
% 2.83/1.52 ------------------------
% 2.83/1.57 ---- Cleaned THF ---
% 2.83/1.57 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
% 2.83/1.57 binary1291135688e_Tree: $tType > $tType ).
% 2.83/1.57
% 2.83/1.57 thf(ty_t_Option_Ooption,type,
% 2.83/1.57 option: $tType > $tType ).
% 2.83/1.57
% 2.83/1.57 thf(ty_t_Set_Oset,type,
% 2.83/1.57 set: $tType > $tType ).
% 2.83/1.57
% 2.83/1.57 thf(ty_t_Int_Oint,type,
% 2.83/1.57 int: $tType ).
% 2.83/1.57
% 2.83/1.57 thf(ty_tf_a,type,
% 2.83/1.57 a: $tType ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.83/1.57 bounded_lattice:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_HOL_Otype,type,
% 2.83/1.57 type:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Obot,type,
% 2.83/1.57 bot:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Oord,type,
% 2.83/1.57 ord:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Otop,type,
% 2.83/1.57 top:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Oorder,type,
% 2.83/1.57 order:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Ono__bot,type,
% 2.83/1.57 no_bot:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Ono__top,type,
% 2.83/1.57 no_top:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Olinorder,type,
% 2.83/1.57 linorder:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Opreorder,type,
% 2.83/1.57 preorder:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.83/1.57 order_bot:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Owellorder,type,
% 2.83/1.57 wellorder:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Complete__Lattices_OInf,type,
% 2.83/1.57 complete_Inf:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Orderings_Odense__order,type,
% 2.83/1.57 dense_order:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.83/1.57 linordered_idom:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Fields_Olinordered__field,type,
% 2.83/1.57 linordered_field:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.83/1.57 semilattice_sup:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.83/1.57 comple187826305attice:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 2.83/1.57 comple1035589618norder:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 2.83/1.57 bounde1808546759up_bot:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.83/1.57 condit1656338222tinuum:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
% 2.83/1.57 binary210054475elle_T:
% 2.83/1.57 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
% 2.83/1.57 binary1746293266le_Tip:
% 2.83/1.57 !>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Oset__Tree,type,
% 2.83/1.57 binary2130109271t_Tree:
% 2.83/1.57 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oeqs,type,
% 2.83/1.57 binary64540844le_eqs:
% 2.83/1.57 !>[A: $tType] : ( ( A > int ) > A > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
% 2.83/1.57 binary827270440e_memb:
% 2.83/1.57 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
% 2.83/1.57 binary1653327646_setOf:
% 2.83/1.57 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
% 2.83/1.57 binary1610619414edTree:
% 2.83/1.57 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
% 2.83/1.57 binary231205461t_pred:
% 2.83/1.57 !>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Otlookup,type,
% 2.83/1.57 binary1265063667lookup:
% 2.83/1.57 !>[A: $tType] : ( ( A > int ) > int > ( binary1291135688e_Tree @ A ) > ( option @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
% 2.83/1.57 complete_Inf_Inf:
% 2.83/1.57 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_HOL_OThe,type,
% 2.83/1.57 the:
% 2.83/1.57 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_HOL_Oundefined,type,
% 2.83/1.57 undefined:
% 2.83/1.57 !>[A: $tType] : A ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_If,type,
% 2.83/1.57 if:
% 2.83/1.57 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.83/1.57 sup_sup:
% 2.83/1.57 !>[A: $tType] : ( A > A > A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Option_Ooption_ONone,type,
% 2.83/1.57 none:
% 2.83/1.57 !>[A: $tType] : ( option @ A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Option_Ooption_OSome,type,
% 2.83/1.57 some:
% 2.83/1.57 !>[A: $tType] : ( A > ( option @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Option_Ooption_Ocase__option,type,
% 2.83/1.57 case_option:
% 2.83/1.57 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Option_Ooption_Oset__option,type,
% 2.83/1.57 set_option:
% 2.83/1.57 !>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Option_Ooption_Othe,type,
% 2.83/1.57 the2:
% 2.83/1.57 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Option_Othese,type,
% 2.83/1.57 these:
% 2.83/1.57 !>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.83/1.57 bot_bot:
% 2.83/1.57 !>[A: $tType] : A ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.83/1.57 ord_less:
% 2.83/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.83/1.57 top_top:
% 2.83/1.57 !>[A: $tType] : A ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Set_OCollect,type,
% 2.83/1.57 collect:
% 2.83/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Set_Obind,type,
% 2.83/1.57 bind:
% 2.83/1.57 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Set_Oimage,type,
% 2.83/1.57 image:
% 2.83/1.57 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Set_Oinsert,type,
% 2.83/1.57 insert:
% 2.83/1.57 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Set_Ois__empty,type,
% 2.83/1.57 is_empty:
% 2.83/1.57 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Set_Ois__singleton,type,
% 2.83/1.57 is_singleton:
% 2.83/1.57 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Set_Othe__elem,type,
% 2.83/1.57 the_elem:
% 2.83/1.57 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_member,type,
% 2.83/1.57 member:
% 2.83/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_h,type,
% 2.83/1.57 h: a > int ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_t,type,
% 2.83/1.57 t: binary1291135688e_Tree @ a ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_x,type,
% 2.83/1.57 x: a ).
% 2.83/1.57
% 2.83/1.57 thf(fact_0_h,axiom,
% 2.83/1.57 member @ a @ x @ ( binary1653327646_setOf @ a @ t ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_1_sorted__distinct,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int,A2: A,B2: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A2 @ B2 @ T ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_2_s,axiom,
% 2.83/1.57 binary1610619414edTree @ a @ h @ t ).
% 2.83/1.57
% 2.83/1.57 thf(fact_3_res,axiom,
% 2.83/1.57 ~ ( binary827270440e_memb @ a @ h @ x @ t ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_4__092_060open_062h_Ax_A_092_060noteq_062_Ah_Ax_092_060close_062,axiom,
% 2.83/1.57 ( ( h @ x )
% 2.83/1.57 != ( h @ x ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_5_o1,axiom,
% 2.83/1.57 ! [X: a] :
% 2.83/1.57 ( ( member @ a @ X @ ( binary1653327646_setOf @ a @ t ) )
% 2.83/1.57 => ( ( h @ X )
% 2.83/1.57 != ( h @ x ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_6_sorted__distinct__pred__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( binary231205461t_pred @ A )
% 2.83/1.57 = ( ^ [H2: A > int,A3: A,B3: A,T2: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( ( binary1610619414edTree @ A @ H2 @ T2 )
% 2.83/1.57 & ( member @ A @ A3 @ ( binary1653327646_setOf @ A @ T2 ) )
% 2.83/1.57 & ( member @ A @ B3 @ ( binary1653327646_setOf @ A @ T2 ) )
% 2.83/1.57 & ( ( H2 @ A3 )
% 2.83/1.57 = ( H2 @ B3 ) ) )
% 2.83/1.57 => ( A3 = B3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_7_tNone,axiom,
% 2.83/1.57 ( ( binary1265063667lookup @ a @ h @ ( h @ x ) @ t )
% 2.83/1.57 = ( none @ a ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_8_sortedTree_Osimps_I1_J,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_9_sortLemmaL,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 2.83/1.57 => ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_10_sortLemmaR,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 2.83/1.57 => ( binary1610619414edTree @ A @ H @ T22 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_11_tlookup__none,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,K: int] :
% 2.83/1.57 ( ( ( binary1610619414edTree @ A @ H @ T )
% 2.83/1.57 & ( ( binary1265063667lookup @ A @ H @ K @ T )
% 2.83/1.57 = ( none @ A ) ) )
% 2.83/1.57 => ! [X: A] :
% 2.83/1.57 ( ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) )
% 2.83/1.57 => ( ( H @ X )
% 2.83/1.57 != K ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_12_tlookup__some,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,K: int,X2: A] :
% 2.83/1.57 ( ( ( binary1610619414edTree @ A @ H @ T )
% 2.83/1.57 & ( ( binary1265063667lookup @ A @ H @ K @ T )
% 2.83/1.57 = ( some @ A @ X2 ) ) )
% 2.83/1.57 => ( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T ) )
% 2.83/1.57 & ( ( H @ X2 )
% 2.83/1.57 = K ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_13_tlookup__finds,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,Y: A] :
% 2.83/1.57 ( ( ( binary1610619414edTree @ A @ H @ T )
% 2.83/1.57 & ( member @ A @ Y @ ( binary1653327646_setOf @ A @ T ) ) )
% 2.83/1.57 => ( ( binary1265063667lookup @ A @ H @ ( H @ Y ) @ T )
% 2.83/1.57 = ( some @ A @ Y ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_14_sortedTree_Osimps_I2_J,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 2.83/1.57 = ( ( binary1610619414edTree @ A @ H @ T1 )
% 2.83/1.57 & ! [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T1 ) )
% 2.83/1.57 => ( ord_less @ int @ ( H @ X3 ) @ ( H @ X2 ) ) )
% 2.83/1.57 & ! [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T22 ) )
% 2.83/1.57 => ( ord_less @ int @ ( H @ X2 ) @ ( H @ X3 ) ) )
% 2.83/1.57 & ( binary1610619414edTree @ A @ H @ T22 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_15_setOf_Osimps_I1_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_16_memb__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( binary827270440e_memb @ A )
% 2.83/1.57 = ( ^ [H2: A > int,X3: A,T2: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( case_option @ $o @ A @ $false
% 2.83/1.57 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.83/1.57 @ X3 )
% 2.83/1.57 @ ( binary1265063667lookup @ A @ H2 @ ( H2 @ X3 ) @ T2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_17_Tree_Oinject,axiom,
% 2.83/1.57 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
% 2.83/1.57 = ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
% 2.83/1.57 = ( ( X21 = Y21 )
% 2.83/1.57 & ( X22 = Y22 )
% 2.83/1.57 & ( X23 = Y23 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_18_Tree_Oexhaust,axiom,
% 2.83/1.57 ! [A: $tType,Y: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( Y
% 2.83/1.57 != ( binary1746293266le_Tip @ A ) )
% 2.83/1.57 => ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( Y
% 2.83/1.57 != ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_19_Tree_Oinduct,axiom,
% 2.83/1.57 ! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( P @ ( binary1746293266le_Tip @ A ) )
% 2.83/1.57 => ( ! [X1: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( P @ X1 )
% 2.83/1.57 => ( ( P @ X32 )
% 2.83/1.57 => ( P @ ( binary210054475elle_T @ A @ X1 @ X24 @ X32 ) ) ) )
% 2.83/1.57 => ( P @ Tree ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_20_tlookup_Osimps_I1_J,axiom,
% 2.83/1.57 ! [A: $tType,H: A > int,K: int] :
% 2.83/1.57 ( ( binary1265063667lookup @ A @ H @ K @ ( binary1746293266le_Tip @ A ) )
% 2.83/1.57 = ( none @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_21_tlookup_Osimps_I2_J,axiom,
% 2.83/1.57 ! [A: $tType,K: int,H: A > int,X2: A,T1: binary1291135688e_Tree @ A,T22: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( ( ord_less @ int @ K @ ( H @ X2 ) )
% 2.83/1.57 => ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 2.83/1.57 = ( binary1265063667lookup @ A @ H @ K @ T1 ) ) )
% 2.83/1.57 & ( ~ ( ord_less @ int @ K @ ( H @ X2 ) )
% 2.83/1.57 => ( ( ( ord_less @ int @ ( H @ X2 ) @ K )
% 2.83/1.57 => ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 2.83/1.57 = ( binary1265063667lookup @ A @ H @ K @ T22 ) ) )
% 2.83/1.57 & ( ~ ( ord_less @ int @ ( H @ X2 ) @ K )
% 2.83/1.57 => ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 2.83/1.57 = ( some @ A @ X2 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_22_Tree_Odistinct_I1_J,axiom,
% 2.83/1.57 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( binary1746293266le_Tip @ A )
% 2.83/1.57 != ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_23_not__Some__eq,axiom,
% 2.83/1.57 ! [A: $tType,X2: option @ A] :
% 2.83/1.57 ( ( ! [Y3: A] :
% 2.83/1.57 ( X2
% 2.83/1.57 != ( some @ A @ Y3 ) ) )
% 2.83/1.57 = ( X2
% 2.83/1.57 = ( none @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_24_not__None__eq,axiom,
% 2.83/1.57 ! [A: $tType,X2: option @ A] :
% 2.83/1.57 ( ( X2
% 2.83/1.57 != ( none @ A ) )
% 2.83/1.57 = ( ? [Y3: A] :
% 2.83/1.57 ( X2
% 2.83/1.57 = ( some @ A @ Y3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_25_case__optionE,axiom,
% 2.83/1.57 ! [A: $tType,P: $o,Q: A > $o,X2: option @ A] :
% 2.83/1.57 ( ( case_option @ $o @ A @ P @ Q @ X2 )
% 2.83/1.57 => ( ( ( X2
% 2.83/1.57 = ( none @ A ) )
% 2.83/1.57 => ~ P )
% 2.83/1.57 => ~ ! [Y4: A] :
% 2.83/1.57 ( ( X2
% 2.83/1.57 = ( some @ A @ Y4 ) )
% 2.83/1.57 => ~ ( Q @ Y4 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_26_disjE__realizer2,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,P: $o,Q: A > $o,X2: option @ A,R: B > $o,F: B,G: A > B] :
% 2.83/1.57 ( ( case_option @ $o @ A @ P @ Q @ X2 )
% 2.83/1.57 => ( ( P
% 2.83/1.57 => ( R @ F ) )
% 2.83/1.57 => ( ! [Q2: A] :
% 2.83/1.57 ( ( Q @ Q2 )
% 2.83/1.57 => ( R @ ( G @ Q2 ) ) )
% 2.83/1.57 => ( R @ ( case_option @ B @ A @ F @ G @ X2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_27_option_Odisc__eq__case_I1_J,axiom,
% 2.83/1.57 ! [A: $tType,Option: option @ A] :
% 2.83/1.57 ( ( Option
% 2.83/1.57 = ( none @ A ) )
% 2.83/1.57 = ( case_option @ $o @ A @ $true
% 2.83/1.57 @ ^ [Uu: A] : $false
% 2.83/1.57 @ Option ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_28_option_Odisc__eq__case_I2_J,axiom,
% 2.83/1.57 ! [A: $tType,Option: option @ A] :
% 2.83/1.57 ( ( Option
% 2.83/1.57 != ( none @ A ) )
% 2.83/1.57 = ( case_option @ $o @ A @ $false
% 2.83/1.57 @ ^ [Uu: A] : $true
% 2.83/1.57 @ Option ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_29_option_Oinject,axiom,
% 2.83/1.57 ! [A: $tType,X25: A,Y24: A] :
% 2.83/1.57 ( ( ( some @ A @ X25 )
% 2.83/1.57 = ( some @ A @ Y24 ) )
% 2.83/1.57 = ( X25 = Y24 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_30_empty__Collect__eq,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o] :
% 2.83/1.57 ( ( ( bot_bot @ ( set @ A ) )
% 2.83/1.57 = ( collect @ A @ P ) )
% 2.83/1.57 = ( ! [X3: A] :
% 2.83/1.57 ~ ( P @ X3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_31_Collect__empty__eq,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o] :
% 2.83/1.57 ( ( ( collect @ A @ P )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( ! [X3: A] :
% 2.83/1.57 ~ ( P @ X3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_32_all__not__in__conv,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( ! [X3: A] :
% 2.83/1.57 ~ ( member @ A @ X3 @ A4 ) )
% 2.83/1.57 = ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_33_empty__iff,axiom,
% 2.83/1.57 ! [A: $tType,C: A] :
% 2.83/1.57 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_34_emptyE,axiom,
% 2.83/1.57 ! [A: $tType,A2: A] :
% 2.83/1.57 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_35_equals0D,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,A2: A] :
% 2.83/1.57 ( ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_36_equals0I,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ! [Y4: A] :
% 2.83/1.57 ~ ( member @ A @ Y4 @ A4 )
% 2.83/1.57 => ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_37_ex__in__conv,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
% 2.83/1.57 = ( A4
% 2.83/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_38_empty__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bot_bot @ ( set @ A ) )
% 2.83/1.57 = ( collect @ A
% 2.83/1.57 @ ^ [X3: A] : $false ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_39_option_Ocase__distrib,axiom,
% 2.83/1.57 ! [C2: $tType,B: $tType,A: $tType,H: B > C2,F1: B,F2: A > B,Option: option @ A] :
% 2.83/1.57 ( ( H @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
% 2.83/1.57 = ( case_option @ C2 @ A @ ( H @ F1 )
% 2.83/1.57 @ ^ [X3: A] : ( H @ ( F2 @ X3 ) )
% 2.83/1.57 @ Option ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_40_option_Odistinct_I1_J,axiom,
% 2.83/1.57 ! [A: $tType,X25: A] :
% 2.83/1.57 ( ( none @ A )
% 2.83/1.57 != ( some @ A @ X25 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_41_option_OdiscI,axiom,
% 2.83/1.57 ! [A: $tType,Option: option @ A,X25: A] :
% 2.83/1.57 ( ( Option
% 2.83/1.57 = ( some @ A @ X25 ) )
% 2.83/1.57 => ( Option
% 2.83/1.57 != ( none @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_42_option_Oexhaust,axiom,
% 2.83/1.57 ! [A: $tType,Y: option @ A] :
% 2.83/1.57 ( ( Y
% 2.83/1.57 != ( none @ A ) )
% 2.83/1.57 => ~ ! [X24: A] :
% 2.83/1.57 ( Y
% 2.83/1.57 != ( some @ A @ X24 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_43_option_Oinducts,axiom,
% 2.83/1.57 ! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
% 2.83/1.57 ( ( P @ ( none @ A ) )
% 2.83/1.57 => ( ! [X4: A] : ( P @ ( some @ A @ X4 ) )
% 2.83/1.57 => ( P @ Option ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_44_mem__Collect__eq,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,P: A > $o] :
% 2.83/1.57 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.83/1.57 = ( P @ A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_45_Collect__mem__eq,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( collect @ A
% 2.83/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 2.83/1.57 = A4 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_46_Collect__cong,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.83/1.57 ( ! [X4: A] :
% 2.83/1.57 ( ( P @ X4 )
% 2.83/1.57 = ( Q @ X4 ) )
% 2.83/1.57 => ( ( collect @ A @ P )
% 2.83/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_47_ext,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.83/1.57 ( ! [X4: A] :
% 2.83/1.57 ( ( F @ X4 )
% 2.83/1.57 = ( G @ X4 ) )
% 2.83/1.57 => ( F = G ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_48_split__option__ex,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ^ [P2: ( option @ A ) > $o] :
% 2.83/1.57 ? [X5: option @ A] : ( P2 @ X5 ) )
% 2.83/1.57 = ( ^ [P3: ( option @ A ) > $o] :
% 2.83/1.57 ( ( P3 @ ( none @ A ) )
% 2.83/1.57 | ? [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_49_split__option__all,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ^ [P2: ( option @ A ) > $o] :
% 2.83/1.57 ! [X5: option @ A] : ( P2 @ X5 ) )
% 2.83/1.57 = ( ^ [P3: ( option @ A ) > $o] :
% 2.83/1.57 ( ( P3 @ ( none @ A ) )
% 2.83/1.57 & ! [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_50_combine__options__cases,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,X2: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
% 2.83/1.57 ( ( ( X2
% 2.83/1.57 = ( none @ A ) )
% 2.83/1.57 => ( P @ X2 @ Y ) )
% 2.83/1.57 => ( ( ( Y
% 2.83/1.57 = ( none @ B ) )
% 2.83/1.57 => ( P @ X2 @ Y ) )
% 2.83/1.57 => ( ! [A5: A,B4: B] :
% 2.83/1.57 ( ( X2
% 2.83/1.57 = ( some @ A @ A5 ) )
% 2.83/1.57 => ( ( Y
% 2.83/1.57 = ( some @ B @ B4 ) )
% 2.83/1.57 => ( P @ X2 @ Y ) ) )
% 2.83/1.57 => ( P @ X2 @ Y ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_51_option_Osimps_I4_J,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,F1: B,F2: A > B] :
% 2.83/1.57 ( ( case_option @ B @ A @ F1 @ F2 @ ( none @ A ) )
% 2.83/1.57 = F1 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_52_option_Osimps_I5_J,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,F1: B,F2: A > B,X25: A] :
% 2.83/1.57 ( ( case_option @ B @ A @ F1 @ F2 @ ( some @ A @ X25 ) )
% 2.83/1.57 = ( F2 @ X25 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_53_bot__apply,axiom,
% 2.83/1.57 ! [C2: $tType,D: $tType] :
% 2.83/1.57 ( ( bot @ C2 )
% 2.83/1.57 => ( ( bot_bot @ ( D > C2 ) )
% 2.83/1.57 = ( ^ [X3: D] : ( bot_bot @ C2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_54_bot_Oextremum__strict,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order_bot @ A )
% 2.83/1.57 => ! [A2: A] :
% 2.83/1.57 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_55_bot_Onot__eq__extremum,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order_bot @ A )
% 2.83/1.57 => ! [A2: A] :
% 2.83/1.57 ( ( A2
% 2.83/1.57 != ( bot_bot @ A ) )
% 2.83/1.57 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_56_Set_Ois__empty__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( is_empty @ A )
% 2.83/1.57 = ( ^ [A6: set @ A] :
% 2.83/1.57 ( A6
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_57_eqs__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( binary64540844le_eqs @ A )
% 2.83/1.57 = ( ^ [H2: A > int,X3: A] :
% 2.83/1.57 ( collect @ A
% 2.83/1.57 @ ^ [Y3: A] :
% 2.83/1.57 ( ( H2 @ Y3 )
% 2.83/1.57 = ( H2 @ X3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_58_option_Osplit__sel,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > B,Option: option @ A] :
% 2.83/1.57 ( ( P @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
% 2.83/1.57 = ( ( ( Option
% 2.83/1.57 = ( none @ A ) )
% 2.83/1.57 => ( P @ F1 ) )
% 2.83/1.57 & ( ( Option
% 2.83/1.57 = ( some @ A @ ( the2 @ A @ Option ) ) )
% 2.83/1.57 => ( P @ ( F2 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_59_option_Osplit__sel__asm,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > B,Option: option @ A] :
% 2.83/1.57 ( ( P @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
% 2.83/1.57 = ( ~ ( ( ( Option
% 2.83/1.57 = ( none @ A ) )
% 2.83/1.57 & ~ ( P @ F1 ) )
% 2.83/1.57 | ( ( Option
% 2.83/1.57 = ( some @ A @ ( the2 @ A @ Option ) ) )
% 2.83/1.57 & ~ ( P @ ( F2 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_60_set__empty__eq,axiom,
% 2.83/1.57 ! [A: $tType,Xo: option @ A] :
% 2.83/1.57 ( ( ( set_option @ A @ Xo )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( Xo
% 2.83/1.57 = ( none @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_61_bot__fun__def,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( bot @ B )
% 2.83/1.57 => ( ( bot_bot @ ( A > B ) )
% 2.83/1.57 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_62_elem__set,axiom,
% 2.83/1.57 ! [A: $tType,X2: A,Xo: option @ A] :
% 2.83/1.57 ( ( member @ A @ X2 @ ( set_option @ A @ Xo ) )
% 2.83/1.57 = ( Xo
% 2.83/1.57 = ( some @ A @ X2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_63_option_Ocollapse,axiom,
% 2.83/1.57 ! [A: $tType,Option: option @ A] :
% 2.83/1.57 ( ( Option
% 2.83/1.57 != ( none @ A ) )
% 2.83/1.57 => ( ( some @ A @ ( the2 @ A @ Option ) )
% 2.83/1.57 = Option ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_64_option_Oset__sel,axiom,
% 2.83/1.57 ! [A: $tType,A2: option @ A] :
% 2.83/1.57 ( ( A2
% 2.83/1.57 != ( none @ A ) )
% 2.83/1.57 => ( member @ A @ ( the2 @ A @ A2 ) @ ( set_option @ A @ A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_65_bot__set__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bot_bot @ ( set @ A ) )
% 2.83/1.57 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_66_not__psubset__empty,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_67_option_Oexpand,axiom,
% 2.83/1.57 ! [A: $tType,Option: option @ A,Option2: option @ A] :
% 2.83/1.57 ( ( ( Option
% 2.83/1.57 = ( none @ A ) )
% 2.83/1.57 = ( Option2
% 2.83/1.57 = ( none @ A ) ) )
% 2.83/1.57 => ( ( ( Option
% 2.83/1.57 != ( none @ A ) )
% 2.83/1.57 => ( ( Option2
% 2.83/1.57 != ( none @ A ) )
% 2.83/1.57 => ( ( the2 @ A @ Option )
% 2.83/1.57 = ( the2 @ A @ Option2 ) ) ) )
% 2.83/1.57 => ( Option = Option2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_68_option_Osel,axiom,
% 2.83/1.57 ! [A: $tType,X25: A] :
% 2.83/1.57 ( ( the2 @ A @ ( some @ A @ X25 ) )
% 2.83/1.57 = X25 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_69_option_Oset__intros,axiom,
% 2.83/1.57 ! [A: $tType,X25: A] : ( member @ A @ X25 @ ( set_option @ A @ ( some @ A @ X25 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_70_option_Oset__cases,axiom,
% 2.83/1.57 ! [A: $tType,E: A,A2: option @ A] :
% 2.83/1.57 ( ( member @ A @ E @ ( set_option @ A @ A2 ) )
% 2.83/1.57 => ( A2
% 2.83/1.57 = ( some @ A @ E ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_71_ospec,axiom,
% 2.83/1.57 ! [A: $tType,A4: option @ A,P: A > $o,X2: A] :
% 2.83/1.57 ( ! [X4: A] :
% 2.83/1.57 ( ( member @ A @ X4 @ ( set_option @ A @ A4 ) )
% 2.83/1.57 => ( P @ X4 ) )
% 2.83/1.57 => ( ( A4
% 2.83/1.57 = ( some @ A @ X2 ) )
% 2.83/1.57 => ( P @ X2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_72_option_Oexhaust__sel,axiom,
% 2.83/1.57 ! [A: $tType,Option: option @ A] :
% 2.83/1.57 ( ( Option
% 2.83/1.57 != ( none @ A ) )
% 2.83/1.57 => ( Option
% 2.83/1.57 = ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_73_option_Osimps_I14_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( set_option @ A @ ( none @ A ) )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_74_option_Ocase__eq__if,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType] :
% 2.83/1.57 ( ( case_option @ B @ A )
% 2.83/1.57 = ( ^ [F12: B,F22: A > B,Option3: option @ A] :
% 2.83/1.57 ( if @ B
% 2.83/1.57 @ ( Option3
% 2.83/1.57 = ( none @ A ) )
% 2.83/1.57 @ F12
% 2.83/1.57 @ ( F22 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_75_dual__order_Ostrict__implies__not__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [B2: A,A2: A] :
% 2.83/1.57 ( ( ord_less @ A @ B2 @ A2 )
% 2.83/1.57 => ( A2 != B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_76_order_Ostrict__implies__not__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( A2 != B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_77_not__less__iff__gr__or__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.83/1.57 = ( ( ord_less @ A @ Y @ X2 )
% 2.83/1.57 | ( X2 = Y ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_78_dual__order_Ostrict__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [B2: A,A2: A,C: A] :
% 2.83/1.57 ( ( ord_less @ A @ B2 @ A2 )
% 2.83/1.57 => ( ( ord_less @ A @ C @ B2 )
% 2.83/1.57 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_79_linorder__less__wlog,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.83/1.57 ( ! [A5: A,B4: A] :
% 2.83/1.57 ( ( ord_less @ A @ A5 @ B4 )
% 2.83/1.57 => ( P @ A5 @ B4 ) )
% 2.83/1.57 => ( ! [A5: A] : ( P @ A5 @ A5 )
% 2.83/1.57 => ( ! [A5: A,B4: A] :
% 2.83/1.57 ( ( P @ B4 @ A5 )
% 2.83/1.57 => ( P @ A5 @ B4 ) )
% 2.83/1.57 => ( P @ A2 @ B2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_80_exists__least__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( wellorder @ A )
% 2.83/1.57 => ( ( ^ [P2: A > $o] :
% 2.83/1.57 ? [X5: A] : ( P2 @ X5 ) )
% 2.83/1.57 = ( ^ [P3: A > $o] :
% 2.83/1.57 ? [N: A] :
% 2.83/1.57 ( ( P3 @ N )
% 2.83/1.57 & ! [M: A] :
% 2.83/1.57 ( ( ord_less @ A @ M @ N )
% 2.83/1.57 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_81_less__imp__not__less,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_82_order_Ostrict__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less @ A @ B2 @ C )
% 2.83/1.57 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_83_dual__order_Oirrefl,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A] :
% 2.83/1.57 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_84_linorder__cases,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ~ ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ( ( X2 != Y )
% 2.83/1.57 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_85_less__imp__triv,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A,P: $o] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ( ( ord_less @ A @ Y @ X2 )
% 2.83/1.57 => P ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_86_less__imp__not__eq2,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ( Y != X2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_87_antisym__conv3,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [Y: A,X2: A] :
% 2.83/1.57 ( ~ ( ord_less @ A @ Y @ X2 )
% 2.83/1.57 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.83/1.57 = ( X2 = Y ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_88_less__induct,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( wellorder @ A )
% 2.83/1.57 => ! [P: A > $o,A2: A] :
% 2.83/1.57 ( ! [X4: A] :
% 2.83/1.57 ( ! [Y5: A] :
% 2.83/1.57 ( ( ord_less @ A @ Y5 @ X4 )
% 2.83/1.57 => ( P @ Y5 ) )
% 2.83/1.57 => ( P @ X4 ) )
% 2.83/1.57 => ( P @ A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_89_less__not__sym,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_90_less__imp__not__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ( X2 != Y ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_91_dual__order_Oasym,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [B2: A,A2: A] :
% 2.83/1.57 ( ( ord_less @ A @ B2 @ A2 )
% 2.83/1.57 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_92_ord__less__eq__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ord @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( B2 = C )
% 2.83/1.57 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_93_ord__eq__less__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ord @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( A2 = B2 )
% 2.83/1.57 => ( ( ord_less @ A @ B2 @ C )
% 2.83/1.57 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_94_less__irrefl,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X2: A] :
% 2.83/1.57 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_95_less__linear,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 | ( X2 = Y )
% 2.83/1.57 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_96_less__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A,Z2: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ( ( ord_less @ A @ Y @ Z2 )
% 2.83/1.57 => ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_97_less__asym_H,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_98_less__asym,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_99_less__imp__neq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ( X2 != Y ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_100_dense,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( dense_order @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ? [Z3: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ Z3 )
% 2.83/1.57 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_101_order_Oasym,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_102_neq__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( X2 != Y )
% 2.83/1.57 = ( ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 | ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_103_neqE,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( X2 != Y )
% 2.83/1.57 => ( ~ ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_104_gt__ex,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( no_top @ A )
% 2.83/1.57 => ! [X2: A] :
% 2.83/1.57 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_105_lt__ex,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( no_bot @ A )
% 2.83/1.57 => ! [X2: A] :
% 2.83/1.57 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_106_order__less__subst2,axiom,
% 2.83/1.57 ! [A: $tType,C2: $tType] :
% 2.83/1.57 ( ( ( order @ C2 )
% 2.83/1.57 & ( order @ A ) )
% 2.83/1.57 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 2.83/1.57 => ( ! [X4: A,Y4: A] :
% 2.83/1.57 ( ( ord_less @ A @ X4 @ Y4 )
% 2.83/1.57 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.83/1.57 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_107_order__less__subst1,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType] :
% 2.83/1.57 ( ( ( order @ B )
% 2.83/1.57 & ( order @ A ) )
% 2.83/1.57 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.83/1.57 => ( ( ord_less @ B @ B2 @ C )
% 2.83/1.57 => ( ! [X4: B,Y4: B] :
% 2.83/1.57 ( ( ord_less @ B @ X4 @ Y4 )
% 2.83/1.57 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.83/1.57 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_108_ord__less__eq__subst,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType] :
% 2.83/1.57 ( ( ( ord @ B )
% 2.83/1.57 & ( ord @ A ) )
% 2.83/1.57 => ! [A2: A,B2: A,F: A > B,C: B] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ( F @ B2 )
% 2.83/1.57 = C )
% 2.83/1.57 => ( ! [X4: A,Y4: A] :
% 2.83/1.57 ( ( ord_less @ A @ X4 @ Y4 )
% 2.83/1.57 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.83/1.57 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_109_ord__eq__less__subst,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType] :
% 2.83/1.57 ( ( ( ord @ B )
% 2.83/1.57 & ( ord @ A ) )
% 2.83/1.57 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.83/1.57 ( ( A2
% 2.83/1.57 = ( F @ B2 ) )
% 2.83/1.57 => ( ( ord_less @ B @ B2 @ C )
% 2.83/1.57 => ( ! [X4: B,Y4: B] :
% 2.83/1.57 ( ( ord_less @ B @ X4 @ Y4 )
% 2.83/1.57 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.83/1.57 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_110_option_Osimps_I15_J,axiom,
% 2.83/1.57 ! [A: $tType,X25: A] :
% 2.83/1.57 ( ( set_option @ A @ ( some @ A @ X25 ) )
% 2.83/1.57 = ( insert @ A @ X25 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_111_option_Othe__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( the2 @ A )
% 2.83/1.57 = ( case_option @ A @ A @ ( undefined @ A )
% 2.83/1.57 @ ^ [X26: A] : X26 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_112_bot__empty__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bot_bot @ ( A > $o ) )
% 2.83/1.57 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_113_Collect__empty__eq__bot,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o] :
% 2.83/1.57 ( ( ( collect @ A @ P )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( P
% 2.83/1.57 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_114_ex__gt__or__lt,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( condit1656338222tinuum @ A )
% 2.83/1.57 => ! [A2: A] :
% 2.83/1.57 ? [B4: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B4 )
% 2.83/1.57 | ( ord_less @ A @ B4 @ A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_115_linorder__neqE__linordered__idom,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( X2 != Y )
% 2.83/1.57 => ( ~ ( ord_less @ A @ X2 @ Y )
% 2.83/1.57 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_116_linordered__field__no__ub,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_field @ A )
% 2.83/1.57 => ! [X: A] :
% 2.83/1.57 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_117_insert__absorb2,axiom,
% 2.83/1.57 ! [A: $tType,X2: A,A4: set @ A] :
% 2.83/1.57 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A4 ) )
% 2.83/1.57 = ( insert @ A @ X2 @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_118_insert__iff,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 2.83/1.57 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 2.83/1.57 = ( ( A2 = B2 )
% 2.83/1.57 | ( member @ A @ A2 @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_119_insertCI,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 2.83/1.57 ( ( ~ ( member @ A @ A2 @ B5 )
% 2.83/1.57 => ( A2 = B2 ) )
% 2.83/1.57 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_120_singletonI,axiom,
% 2.83/1.57 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_121_singleton__conv,axiom,
% 2.83/1.57 ! [A: $tType,A2: A] :
% 2.83/1.57 ( ( collect @ A
% 2.83/1.57 @ ^ [X3: A] : ( X3 = A2 ) )
% 2.83/1.57 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_122_singleton__conv2,axiom,
% 2.83/1.57 ! [A: $tType,A2: A] :
% 2.83/1.57 ( ( collect @ A
% 2.83/1.57 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.83/1.57 @ A2 ) )
% 2.83/1.57 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_123_psubset__trans,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 2.83/1.57 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 2.83/1.57 => ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
% 2.83/1.57 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_124_less__set__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ord_less @ ( set @ A ) )
% 2.83/1.57 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.83/1.57 ( ord_less @ ( A > $o )
% 2.83/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ A6 )
% 2.83/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_125_psubsetD,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,C: A] :
% 2.83/1.57 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 2.83/1.57 => ( ( member @ A @ C @ A4 )
% 2.83/1.57 => ( member @ A @ C @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_126_mk__disjoint__insert,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,A4: set @ A] :
% 2.83/1.57 ( ( member @ A @ A2 @ A4 )
% 2.83/1.57 => ? [B7: set @ A] :
% 2.83/1.57 ( ( A4
% 2.83/1.57 = ( insert @ A @ A2 @ B7 ) )
% 2.83/1.57 & ~ ( member @ A @ A2 @ B7 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_127_insert__commute,axiom,
% 2.83/1.57 ! [A: $tType,X2: A,Y: A,A4: set @ A] :
% 2.83/1.57 ( ( insert @ A @ X2 @ ( insert @ A @ Y @ A4 ) )
% 2.83/1.57 = ( insert @ A @ Y @ ( insert @ A @ X2 @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_128_insert__eq__iff,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,A4: set @ A,B2: A,B5: set @ A] :
% 2.83/1.57 ( ~ ( member @ A @ A2 @ A4 )
% 2.83/1.57 => ( ~ ( member @ A @ B2 @ B5 )
% 2.83/1.57 => ( ( ( insert @ A @ A2 @ A4 )
% 2.83/1.57 = ( insert @ A @ B2 @ B5 ) )
% 2.83/1.57 = ( ( ( A2 = B2 )
% 2.83/1.57 => ( A4 = B5 ) )
% 2.83/1.57 & ( ( A2 != B2 )
% 2.83/1.57 => ? [C4: set @ A] :
% 2.83/1.57 ( ( A4
% 2.83/1.57 = ( insert @ A @ B2 @ C4 ) )
% 2.83/1.57 & ~ ( member @ A @ B2 @ C4 )
% 2.83/1.57 & ( B5
% 2.83/1.57 = ( insert @ A @ A2 @ C4 ) )
% 2.83/1.57 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_129_insert__absorb,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,A4: set @ A] :
% 2.83/1.57 ( ( member @ A @ A2 @ A4 )
% 2.83/1.57 => ( ( insert @ A @ A2 @ A4 )
% 2.83/1.57 = A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_130_insert__ident,axiom,
% 2.83/1.57 ! [A: $tType,X2: A,A4: set @ A,B5: set @ A] :
% 2.83/1.57 ( ~ ( member @ A @ X2 @ A4 )
% 2.83/1.57 => ( ~ ( member @ A @ X2 @ B5 )
% 2.83/1.57 => ( ( ( insert @ A @ X2 @ A4 )
% 2.83/1.57 = ( insert @ A @ X2 @ B5 ) )
% 2.83/1.57 = ( A4 = B5 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_131_Set_Oset__insert,axiom,
% 2.83/1.57 ! [A: $tType,X2: A,A4: set @ A] :
% 2.83/1.57 ( ( member @ A @ X2 @ A4 )
% 2.83/1.57 => ~ ! [B7: set @ A] :
% 2.83/1.57 ( ( A4
% 2.83/1.57 = ( insert @ A @ X2 @ B7 ) )
% 2.83/1.57 => ( member @ A @ X2 @ B7 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_132_insertI2,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 2.83/1.57 ( ( member @ A @ A2 @ B5 )
% 2.83/1.57 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_133_insertI1,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_134_insertE,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 2.83/1.57 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 2.83/1.57 => ( ( A2 != B2 )
% 2.83/1.57 => ( member @ A @ A2 @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_135_insert__compr,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( insert @ A )
% 2.83/1.57 = ( ^ [A3: A,B6: set @ A] :
% 2.83/1.57 ( collect @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( ( X3 = A3 )
% 2.83/1.57 | ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_136_insert__Collect,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,P: A > $o] :
% 2.83/1.57 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 2.83/1.57 = ( collect @ A
% 2.83/1.57 @ ^ [U: A] :
% 2.83/1.57 ( ( U != A2 )
% 2.83/1.57 => ( P @ U ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_137_singletonD,axiom,
% 2.83/1.57 ! [A: $tType,B2: A,A2: A] :
% 2.83/1.57 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 => ( B2 = A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_138_singleton__iff,axiom,
% 2.83/1.57 ! [A: $tType,B2: A,A2: A] :
% 2.83/1.57 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 = ( B2 = A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_139_doubleton__eq__iff,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,B2: A,C: A,D2: A] :
% 2.83/1.57 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 = ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.83/1.57 = ( ( ( A2 = C )
% 2.83/1.57 & ( B2 = D2 ) )
% 2.83/1.57 | ( ( A2 = D2 )
% 2.83/1.57 & ( B2 = C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_140_insert__not__empty,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,A4: set @ A] :
% 2.83/1.57 ( ( insert @ A @ A2 @ A4 )
% 2.83/1.57 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_141_singleton__inject,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,B2: A] :
% 2.83/1.57 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 => ( A2 = B2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_142_Collect__conv__if,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,A2: A] :
% 2.83/1.57 ( ( ( P @ A2 )
% 2.83/1.57 => ( ( collect @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( ( X3 = A2 )
% 2.83/1.57 & ( P @ X3 ) ) )
% 2.83/1.57 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.83/1.57 & ( ~ ( P @ A2 )
% 2.83/1.57 => ( ( collect @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( ( X3 = A2 )
% 2.83/1.57 & ( P @ X3 ) ) )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_143_Collect__conv__if2,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,A2: A] :
% 2.83/1.57 ( ( ( P @ A2 )
% 2.83/1.57 => ( ( collect @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( ( A2 = X3 )
% 2.83/1.57 & ( P @ X3 ) ) )
% 2.83/1.57 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.83/1.57 & ( ~ ( P @ A2 )
% 2.83/1.57 => ( ( collect @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( ( A2 = X3 )
% 2.83/1.57 & ( P @ X3 ) ) )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_144_linordered__field__no__lb,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_field @ A )
% 2.83/1.57 => ! [X: A] :
% 2.83/1.57 ? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_145_the__elem__eq,axiom,
% 2.83/1.57 ! [A: $tType,X2: A] :
% 2.83/1.57 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 = X2 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_146_is__singletonI,axiom,
% 2.83/1.57 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_147_these__insert__Some,axiom,
% 2.83/1.57 ! [A: $tType,X2: A,A4: set @ ( option @ A )] :
% 2.83/1.57 ( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X2 ) @ A4 ) )
% 2.83/1.57 = ( insert @ A @ X2 @ ( these @ A @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_148_is__singletonE,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( is_singleton @ A @ A4 )
% 2.83/1.57 => ~ ! [X4: A] :
% 2.83/1.57 ( A4
% 2.83/1.57 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_149_these__empty,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_150_these__insert__None,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ ( option @ A )] :
% 2.83/1.57 ( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A4 ) )
% 2.83/1.57 = ( these @ A @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_151_in__these__eq,axiom,
% 2.83/1.57 ! [A: $tType,X2: A,A4: set @ ( option @ A )] :
% 2.83/1.57 ( ( member @ A @ X2 @ ( these @ A @ A4 ) )
% 2.83/1.57 = ( member @ ( option @ A ) @ ( some @ A @ X2 ) @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_152_is__singleton__the__elem,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( is_singleton @ A )
% 2.83/1.57 = ( ^ [A6: set @ A] :
% 2.83/1.57 ( A6
% 2.83/1.57 = ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_153_is__singletonI_H,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( A4
% 2.83/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 => ( ! [X4: A,Y4: A] :
% 2.83/1.57 ( ( member @ A @ X4 @ A4 )
% 2.83/1.57 => ( ( member @ A @ Y4 @ A4 )
% 2.83/1.57 => ( X4 = Y4 ) ) )
% 2.83/1.57 => ( is_singleton @ A @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_154_these__not__empty__eq,axiom,
% 2.83/1.57 ! [A: $tType,B5: set @ ( option @ A )] :
% 2.83/1.57 ( ( ( these @ A @ B5 )
% 2.83/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( ( B5
% 2.83/1.57 != ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 2.83/1.57 & ( B5
% 2.83/1.57 != ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_155_these__empty__eq,axiom,
% 2.83/1.57 ! [A: $tType,B5: set @ ( option @ A )] :
% 2.83/1.57 ( ( ( these @ A @ B5 )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( ( B5
% 2.83/1.57 = ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 2.83/1.57 | ( B5
% 2.83/1.57 = ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_156_is__singleton__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( is_singleton @ A )
% 2.83/1.57 = ( ^ [A6: set @ A] :
% 2.83/1.57 ? [X3: A] :
% 2.83/1.57 ( A6
% 2.83/1.57 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_157_the__elem__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( the_elem @ A )
% 2.83/1.57 = ( ^ [X6: set @ A] :
% 2.83/1.57 ( the @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( X6
% 2.83/1.57 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_158_setOf_Osimps_I2_J,axiom,
% 2.83/1.57 ! [A: $tType,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
% 2.83/1.57 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T22 ) ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_159_Option_Othese__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( these @ A )
% 2.83/1.57 = ( ^ [A6: set @ ( option @ A )] :
% 2.83/1.57 ( image @ ( option @ A ) @ A @ ( the2 @ A )
% 2.83/1.57 @ ( collect @ ( option @ A )
% 2.83/1.57 @ ^ [X3: option @ A] :
% 2.83/1.57 ( ( member @ ( option @ A ) @ X3 @ A6 )
% 2.83/1.57 & ( X3
% 2.83/1.57 != ( none @ A ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_160_image__eqI,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A4: set @ B] :
% 2.83/1.57 ( ( B2
% 2.83/1.57 = ( F @ X2 ) )
% 2.83/1.57 => ( ( member @ B @ X2 @ A4 )
% 2.83/1.57 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_161_Un__iff,axiom,
% 2.83/1.57 ! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
% 2.83/1.57 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 2.83/1.57 = ( ( member @ A @ C @ A4 )
% 2.83/1.57 | ( member @ A @ C @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_162_UnCI,axiom,
% 2.83/1.57 ! [A: $tType,C: A,B5: set @ A,A4: set @ A] :
% 2.83/1.57 ( ( ~ ( member @ A @ C @ B5 )
% 2.83/1.57 => ( member @ A @ C @ A4 ) )
% 2.83/1.57 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_163_image__ident,axiom,
% 2.83/1.57 ! [A: $tType,Y6: set @ A] :
% 2.83/1.57 ( ( image @ A @ A
% 2.83/1.57 @ ^ [X3: A] : X3
% 2.83/1.57 @ Y6 )
% 2.83/1.57 = Y6 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_164_image__is__empty,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
% 2.83/1.57 ( ( ( image @ B @ A @ F @ A4 )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_165_empty__is__image,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
% 2.83/1.57 ( ( ( bot_bot @ ( set @ A ) )
% 2.83/1.57 = ( image @ B @ A @ F @ A4 ) )
% 2.83/1.57 = ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_166_image__empty,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,F: B > A] :
% 2.83/1.57 ( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_167_image__insert,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,F: B > A,A2: B,B5: set @ B] :
% 2.83/1.57 ( ( image @ B @ A @ F @ ( insert @ B @ A2 @ B5 ) )
% 2.83/1.57 = ( insert @ A @ ( F @ A2 ) @ ( image @ B @ A @ F @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_168_insert__image,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
% 2.83/1.57 ( ( member @ A @ X2 @ A4 )
% 2.83/1.57 => ( ( insert @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) )
% 2.83/1.57 = ( image @ A @ B @ F @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_169_Un__empty,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 2.83/1.57 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 & ( B5
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_170_Un__insert__left,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,B5: set @ A,C3: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B5 ) @ C3 )
% 2.83/1.57 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_171_Un__insert__right,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
% 2.83/1.57 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_172_insert__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( insert @ A )
% 2.83/1.57 = ( ^ [A3: A] :
% 2.83/1.57 ( sup_sup @ ( set @ A )
% 2.83/1.57 @ ( collect @ A
% 2.83/1.57 @ ^ [X3: A] : ( X3 = A3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_173_Un__left__commute,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
% 2.83/1.57 = ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_174_Un__left__absorb,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 2.83/1.57 = ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_175_rev__image__eqI,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,X2: A,A4: set @ A,B2: B,F: A > B] :
% 2.83/1.57 ( ( member @ A @ X2 @ A4 )
% 2.83/1.57 => ( ( B2
% 2.83/1.57 = ( F @ X2 ) )
% 2.83/1.57 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A4 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_176_ball__imageD,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 2.83/1.57 ( ! [X4: A] :
% 2.83/1.57 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A4 ) )
% 2.83/1.57 => ( P @ X4 ) )
% 2.83/1.57 => ! [X: B] :
% 2.83/1.57 ( ( member @ B @ X @ A4 )
% 2.83/1.57 => ( P @ ( F @ X ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_177_image__cong,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,M2: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 2.83/1.57 ( ( M2 = N2 )
% 2.83/1.57 => ( ! [X4: A] :
% 2.83/1.57 ( ( member @ A @ X4 @ N2 )
% 2.83/1.57 => ( ( F @ X4 )
% 2.83/1.57 = ( G @ X4 ) ) )
% 2.83/1.57 => ( ( image @ A @ B @ F @ M2 )
% 2.83/1.57 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_178_bex__imageD,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 2.83/1.57 ( ? [X: A] :
% 2.83/1.57 ( ( member @ A @ X @ ( image @ B @ A @ F @ A4 ) )
% 2.83/1.57 & ( P @ X ) )
% 2.83/1.57 => ? [X4: B] :
% 2.83/1.57 ( ( member @ B @ X4 @ A4 )
% 2.83/1.57 & ( P @ ( F @ X4 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_179_Un__commute,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) )
% 2.83/1.57 = ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_180_image__iff,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,Z2: A,F: B > A,A4: set @ B] :
% 2.83/1.57 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A4 ) )
% 2.83/1.57 = ( ? [X3: B] :
% 2.83/1.57 ( ( member @ B @ X3 @ A4 )
% 2.83/1.57 & ( Z2
% 2.83/1.57 = ( F @ X3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_181_Un__absorb,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 2.83/1.57 = A4 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_182_image__Un,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,B5: set @ B] :
% 2.83/1.57 ( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A4 @ B5 ) )
% 2.83/1.57 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A4 ) @ ( image @ B @ A @ F @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_183_Un__assoc,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
% 2.83/1.57 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_184_ball__Un,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
% 2.83/1.57 ( ( ! [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 2.83/1.57 => ( P @ X3 ) ) )
% 2.83/1.57 = ( ! [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ A4 )
% 2.83/1.57 => ( P @ X3 ) )
% 2.83/1.57 & ! [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ B5 )
% 2.83/1.57 => ( P @ X3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_185_imageI,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
% 2.83/1.57 ( ( member @ A @ X2 @ A4 )
% 2.83/1.57 => ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_186_bex__Un,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
% 2.83/1.57 ( ( ? [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 2.83/1.57 & ( P @ X3 ) ) )
% 2.83/1.57 = ( ? [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ A4 )
% 2.83/1.57 & ( P @ X3 ) )
% 2.83/1.57 | ? [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ B5 )
% 2.83/1.57 & ( P @ X3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_187_UnI2,axiom,
% 2.83/1.57 ! [A: $tType,C: A,B5: set @ A,A4: set @ A] :
% 2.83/1.57 ( ( member @ A @ C @ B5 )
% 2.83/1.57 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_188_UnI1,axiom,
% 2.83/1.57 ! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
% 2.83/1.57 ( ( member @ A @ C @ A4 )
% 2.83/1.57 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_189_UnE,axiom,
% 2.83/1.57 ! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
% 2.83/1.57 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 2.83/1.57 => ( ~ ( member @ A @ C @ A4 )
% 2.83/1.57 => ( member @ A @ C @ B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_190_Un__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) )
% 2.83/1.57 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.83/1.57 ( collect @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ A6 )
% 2.83/1.57 | ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_191_imageE,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,B2: A,F: B > A,A4: set @ B] :
% 2.83/1.57 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) )
% 2.83/1.57 => ~ ! [X4: B] :
% 2.83/1.57 ( ( B2
% 2.83/1.57 = ( F @ X4 ) )
% 2.83/1.57 => ~ ( member @ B @ X4 @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_192_image__image,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,C2: $tType,F: B > A,G: C2 > B,A4: set @ C2] :
% 2.83/1.57 ( ( image @ B @ A @ F @ ( image @ C2 @ B @ G @ A4 ) )
% 2.83/1.57 = ( image @ C2 @ A
% 2.83/1.57 @ ^ [X3: C2] : ( F @ ( G @ X3 ) )
% 2.83/1.57 @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_193_Compr__image__eq,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
% 2.83/1.57 ( ( collect @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A4 ) )
% 2.83/1.57 & ( P @ X3 ) ) )
% 2.83/1.57 = ( image @ B @ A @ F
% 2.83/1.57 @ ( collect @ B
% 2.83/1.57 @ ^ [X3: B] :
% 2.83/1.57 ( ( member @ B @ X3 @ A4 )
% 2.83/1.57 & ( P @ ( F @ X3 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_194_Collect__disj__eq,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.83/1.57 ( ( collect @ A
% 2.83/1.57 @ ^ [X3: A] :
% 2.83/1.57 ( ( P @ X3 )
% 2.83/1.57 | ( Q @ X3 ) ) )
% 2.83/1.57 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_195_Un__empty__right,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = A4 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_196_Un__empty__left,axiom,
% 2.83/1.57 ! [A: $tType,B5: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
% 2.83/1.57 = B5 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_197_singleton__Un__iff,axiom,
% 2.83/1.57 ! [A: $tType,X2: A,A4: set @ A,B5: set @ A] :
% 2.83/1.57 ( ( ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 = ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
% 2.83/1.57 = ( ( ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 & ( B5
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.83/1.57 | ( ( A4
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 & ( B5
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 | ( ( A4
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 & ( B5
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_198_Un__singleton__iff,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,X2: A] :
% 2.83/1.57 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 = ( ( ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 & ( B5
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.83/1.57 | ( ( A4
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 & ( B5
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 | ( ( A4
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 & ( B5
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_199_insert__is__Un,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( insert @ A )
% 2.83/1.57 = ( ^ [A3: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_200_image__constant,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,X2: A,A4: set @ A,C: B] :
% 2.83/1.57 ( ( member @ A @ X2 @ A4 )
% 2.83/1.57 => ( ( image @ A @ B
% 2.83/1.57 @ ^ [X3: A] : C
% 2.83/1.57 @ A4 )
% 2.83/1.57 = ( insert @ B @ C @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_201_image__constant__conv,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,A4: set @ B,C: A] :
% 2.83/1.57 ( ( ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ B ) ) )
% 2.83/1.57 => ( ( image @ B @ A
% 2.83/1.57 @ ^ [X3: B] : C
% 2.83/1.57 @ A4 )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 & ( ( A4
% 2.83/1.57 != ( bot_bot @ ( set @ B ) ) )
% 2.83/1.57 => ( ( image @ B @ A
% 2.83/1.57 @ ^ [X3: B] : C
% 2.83/1.57 @ A4 )
% 2.83/1.57 = ( insert @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_202_the__elem__image__unique,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,A4: set @ A,F: A > B,X2: A] :
% 2.83/1.57 ( ( A4
% 2.83/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 => ( ! [Y4: A] :
% 2.83/1.57 ( ( member @ A @ Y4 @ A4 )
% 2.83/1.57 => ( ( F @ Y4 )
% 2.83/1.57 = ( F @ X2 ) ) )
% 2.83/1.57 => ( ( the_elem @ B @ ( image @ A @ B @ F @ A4 ) )
% 2.83/1.57 = ( F @ X2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_203_sup__bot__left,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bounde1808546759up_bot @ A )
% 2.83/1.57 => ! [X2: A] :
% 2.83/1.57 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X2 )
% 2.83/1.57 = X2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_204_sup__bot__right,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bounde1808546759up_bot @ A )
% 2.83/1.57 => ! [X2: A] :
% 2.83/1.57 ( ( sup_sup @ A @ X2 @ ( bot_bot @ A ) )
% 2.83/1.57 = X2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_205_bot__eq__sup__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bounde1808546759up_bot @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ( bot_bot @ A )
% 2.83/1.57 = ( sup_sup @ A @ X2 @ Y ) )
% 2.83/1.57 = ( ( X2
% 2.83/1.57 = ( bot_bot @ A ) )
% 2.83/1.57 & ( Y
% 2.83/1.57 = ( bot_bot @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_206_sup__bot_Oright__neutral,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bounde1808546759up_bot @ A )
% 2.83/1.57 => ! [A2: A] :
% 2.83/1.57 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 2.83/1.57 = A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_207_sup__bot_Oneutr__eq__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bounde1808546759up_bot @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ( bot_bot @ A )
% 2.83/1.57 = ( sup_sup @ A @ A2 @ B2 ) )
% 2.83/1.57 = ( ( A2
% 2.83/1.57 = ( bot_bot @ A ) )
% 2.83/1.57 & ( B2
% 2.83/1.57 = ( bot_bot @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_208_sup__bot_Oleft__neutral,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bounde1808546759up_bot @ A )
% 2.83/1.57 => ! [A2: A] :
% 2.83/1.57 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 2.83/1.57 = A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_209_sup__bot_Oeq__neutr__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bounde1808546759up_bot @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ( sup_sup @ A @ A2 @ B2 )
% 2.83/1.57 = ( bot_bot @ A ) )
% 2.83/1.57 = ( ( A2
% 2.83/1.57 = ( bot_bot @ A ) )
% 2.83/1.57 & ( B2
% 2.83/1.57 = ( bot_bot @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_210_sup__eq__bot__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( bounde1808546759up_bot @ A )
% 2.83/1.57 => ! [X2: A,Y: A] :
% 2.83/1.57 ( ( ( sup_sup @ A @ X2 @ Y )
% 2.83/1.57 = ( bot_bot @ A ) )
% 2.83/1.57 = ( ( X2
% 2.83/1.57 = ( bot_bot @ A ) )
% 2.83/1.57 & ( Y
% 2.83/1.57 = ( bot_bot @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_211_these__image__Some__eq,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) )
% 2.83/1.57 = A4 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_212_sup__set__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( sup_sup @ ( set @ A ) )
% 2.83/1.57 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.83/1.57 ( collect @ A
% 2.83/1.57 @ ( sup_sup @ ( A > $o )
% 2.83/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ A6 )
% 2.83/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_213_sup__Un__eq,axiom,
% 2.83/1.57 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.83/1.57 ( ( sup_sup @ ( A > $o )
% 2.83/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 2.83/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 2.83/1.57 = ( ^ [X3: A] : ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_214_None__notin__image__Some,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_215_Some__image__these__eq,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ ( option @ A )] :
% 2.83/1.57 ( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
% 2.83/1.57 = ( collect @ ( option @ A )
% 2.83/1.57 @ ^ [X3: option @ A] :
% 2.83/1.57 ( ( member @ ( option @ A ) @ X3 @ A4 )
% 2.83/1.57 & ( X3
% 2.83/1.57 != ( none @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_216_sup_Ostrict__coboundedI2,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( semilattice_sup @ A )
% 2.83/1.57 => ! [C: A,B2: A,A2: A] :
% 2.83/1.57 ( ( ord_less @ A @ C @ B2 )
% 2.83/1.57 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_217_sup_Ostrict__coboundedI1,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( semilattice_sup @ A )
% 2.83/1.57 => ! [C: A,A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ C @ A2 )
% 2.83/1.57 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_218_sup_Ostrict__order__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( semilattice_sup @ A )
% 2.83/1.57 => ( ( ord_less @ A )
% 2.83/1.57 = ( ^ [B3: A,A3: A] :
% 2.83/1.57 ( ( A3
% 2.83/1.57 = ( sup_sup @ A @ A3 @ B3 ) )
% 2.83/1.57 & ( A3 != B3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_219_sup_Ostrict__boundedE,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( semilattice_sup @ A )
% 2.83/1.57 => ! [B2: A,C: A,A2: A] :
% 2.83/1.57 ( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
% 2.83/1.57 => ~ ( ( ord_less @ A @ B2 @ A2 )
% 2.83/1.57 => ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_220_less__supI2,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( semilattice_sup @ A )
% 2.83/1.57 => ! [X2: A,B2: A,A2: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ B2 )
% 2.83/1.57 => ( ord_less @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_221_less__supI1,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( semilattice_sup @ A )
% 2.83/1.57 => ! [X2: A,A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ X2 @ A2 )
% 2.83/1.57 => ( ord_less @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_222_the__equality,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,A2: A] :
% 2.83/1.57 ( ( P @ A2 )
% 2.83/1.57 => ( ! [X4: A] :
% 2.83/1.57 ( ( P @ X4 )
% 2.83/1.57 => ( X4 = A2 ) )
% 2.83/1.57 => ( ( the @ A @ P )
% 2.83/1.57 = A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_223_the__eq__trivial,axiom,
% 2.83/1.57 ! [A: $tType,A2: A] :
% 2.83/1.57 ( ( the @ A
% 2.83/1.57 @ ^ [X3: A] : ( X3 = A2 ) )
% 2.83/1.57 = A2 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_224_the__sym__eq__trivial,axiom,
% 2.83/1.57 ! [A: $tType,X2: A] :
% 2.83/1.57 ( ( the @ A
% 2.83/1.57 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.83/1.57 @ X2 ) )
% 2.83/1.57 = X2 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_225_the1__equality,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,A2: A] :
% 2.83/1.57 ( ? [X: A] :
% 2.83/1.57 ( ( P @ X )
% 2.83/1.57 & ! [Y4: A] :
% 2.83/1.57 ( ( P @ Y4 )
% 2.83/1.57 => ( Y4 = X ) ) )
% 2.83/1.57 => ( ( P @ A2 )
% 2.83/1.57 => ( ( the @ A @ P )
% 2.83/1.57 = A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_226_the1I2,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.83/1.57 ( ? [X: A] :
% 2.83/1.57 ( ( P @ X )
% 2.83/1.57 & ! [Y4: A] :
% 2.83/1.57 ( ( P @ Y4 )
% 2.83/1.57 => ( Y4 = X ) ) )
% 2.83/1.57 => ( ! [X4: A] :
% 2.83/1.57 ( ( P @ X4 )
% 2.83/1.57 => ( Q @ X4 ) )
% 2.83/1.57 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_227_If__def,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( if @ A )
% 2.83/1.57 = ( ^ [P3: $o,X3: A,Y3: A] :
% 2.83/1.57 ( the @ A
% 2.83/1.57 @ ^ [Z4: A] :
% 2.83/1.57 ( ( P3
% 2.83/1.57 => ( Z4 = X3 ) )
% 2.83/1.57 & ( ~ P3
% 2.83/1.57 => ( Z4 = Y3 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_228_theI2,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 2.83/1.57 ( ( P @ A2 )
% 2.83/1.57 => ( ! [X4: A] :
% 2.83/1.57 ( ( P @ X4 )
% 2.83/1.57 => ( X4 = A2 ) )
% 2.83/1.57 => ( ! [X4: A] :
% 2.83/1.57 ( ( P @ X4 )
% 2.83/1.57 => ( Q @ X4 ) )
% 2.83/1.57 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_229_theI_H,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o] :
% 2.83/1.57 ( ? [X: A] :
% 2.83/1.57 ( ( P @ X )
% 2.83/1.57 & ! [Y4: A] :
% 2.83/1.57 ( ( P @ Y4 )
% 2.83/1.57 => ( Y4 = X ) ) )
% 2.83/1.57 => ( P @ ( the @ A @ P ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_230_theI,axiom,
% 2.83/1.57 ! [A: $tType,P: A > $o,A2: A] :
% 2.83/1.57 ( ( P @ A2 )
% 2.83/1.57 => ( ! [X4: A] :
% 2.83/1.57 ( ( P @ X4 )
% 2.83/1.57 => ( X4 = A2 ) )
% 2.83/1.57 => ( P @ ( the @ A @ P ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_231_Tree_Osimps_I15_J,axiom,
% 2.83/1.57 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
% 2.83/1.57 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary2130109271t_Tree @ A @ X21 ) @ ( insert @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( binary2130109271t_Tree @ A @ X23 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_232_bind__singleton__conv__image,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,A4: set @ B,F: B > A] :
% 2.83/1.57 ( ( bind @ B @ A @ A4
% 2.83/1.57 @ ^ [X3: B] : ( insert @ A @ ( F @ X3 ) @ ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 = ( image @ B @ A @ F @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_233_empty__bind,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,F: B > ( set @ A )] :
% 2.83/1.57 ( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_234_Set_Obind__bind,axiom,
% 2.83/1.57 ! [C2: $tType,B: $tType,A: $tType,A4: set @ A,B5: A > ( set @ C2 ),C3: C2 > ( set @ B )] :
% 2.83/1.57 ( ( bind @ C2 @ B @ ( bind @ A @ C2 @ A4 @ B5 ) @ C3 )
% 2.83/1.57 = ( bind @ A @ B @ A4
% 2.83/1.57 @ ^ [X3: A] : ( bind @ C2 @ B @ ( B5 @ X3 ) @ C3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_235_Tree_Oset__intros_I3_J,axiom,
% 2.83/1.57 ! [A: $tType,Ya: A,X23: binary1291135688e_Tree @ A,X21: binary1291135688e_Tree @ A,X22: A] :
% 2.83/1.57 ( ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ X23 ) )
% 2.83/1.57 => ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_236_Tree_Oset__intros_I2_J,axiom,
% 2.83/1.57 ! [A: $tType,X22: A,X21: binary1291135688e_Tree @ A,X23: binary1291135688e_Tree @ A] : ( member @ A @ X22 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_237_Tree_Oset__intros_I1_J,axiom,
% 2.83/1.57 ! [A: $tType,Y: A,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( member @ A @ Y @ ( binary2130109271t_Tree @ A @ X21 ) )
% 2.83/1.57 => ( member @ A @ Y @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_238_Tree_Oset__cases,axiom,
% 2.83/1.57 ! [A: $tType,E: A,A2: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( member @ A @ E @ ( binary2130109271t_Tree @ A @ A2 ) )
% 2.83/1.57 => ( ! [Z1: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ? [Z22: A,Z32: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( A2
% 2.83/1.57 = ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
% 2.83/1.57 => ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z1 ) ) )
% 2.83/1.57 => ( ! [Z1: binary1291135688e_Tree @ A,Z32: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( A2
% 2.83/1.57 != ( binary210054475elle_T @ A @ Z1 @ E @ Z32 ) )
% 2.83/1.57 => ~ ! [Z1: binary1291135688e_Tree @ A,Z22: A,Z32: binary1291135688e_Tree @ A] :
% 2.83/1.57 ( ( A2
% 2.83/1.57 = ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
% 2.83/1.57 => ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z32 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_239_bind__const,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,A4: set @ B,B5: set @ A] :
% 2.83/1.57 ( ( ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ B ) ) )
% 2.83/1.57 => ( ( bind @ B @ A @ A4
% 2.83/1.57 @ ^ [Uu: B] : B5 )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) )
% 2.83/1.57 & ( ( A4
% 2.83/1.57 != ( bot_bot @ ( set @ B ) ) )
% 2.83/1.57 => ( ( bind @ B @ A @ A4
% 2.83/1.57 @ ^ [Uu: B] : B5 )
% 2.83/1.57 = B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_240_nonempty__bind__const,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,A4: set @ A,B5: set @ B] :
% 2.83/1.57 ( ( A4
% 2.83/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.83/1.57 => ( ( bind @ A @ B @ A4
% 2.83/1.57 @ ^ [Uu: A] : B5 )
% 2.83/1.57 = B5 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_241_Tree_Osimps_I14_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( binary2130109271t_Tree @ A @ ( binary1746293266le_Tip @ A ) )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_242_Inf_OINF__identity__eq,axiom,
% 2.83/1.57 ! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
% 2.83/1.57 ( ( Inf
% 2.83/1.57 @ ( image @ A @ A
% 2.83/1.57 @ ^ [X3: A] : X3
% 2.83/1.57 @ A4 ) )
% 2.83/1.57 = ( Inf @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_243_Sup_OSUP__identity__eq,axiom,
% 2.83/1.57 ! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
% 2.83/1.57 ( ( Sup
% 2.83/1.57 @ ( image @ A @ A
% 2.83/1.57 @ ^ [X3: A] : X3
% 2.83/1.57 @ A4 ) )
% 2.83/1.57 = ( Sup @ A4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_244_range__constant,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,X2: A] :
% 2.83/1.57 ( ( image @ B @ A
% 2.83/1.57 @ ^ [Uu: B] : X2
% 2.83/1.57 @ ( top_top @ ( set @ B ) ) )
% 2.83/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_245_INF__eq__bot__iff,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( comple1035589618norder @ A )
% 2.83/1.57 => ! [F: B > A,A4: set @ B] :
% 2.83/1.57 ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A4 ) )
% 2.83/1.57 = ( bot_bot @ A ) )
% 2.83/1.57 = ( ! [X3: A] :
% 2.83/1.57 ( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
% 2.83/1.57 => ? [Y3: B] :
% 2.83/1.57 ( ( member @ B @ Y3 @ A4 )
% 2.83/1.57 & ( ord_less @ A @ ( F @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_246_top__apply,axiom,
% 2.83/1.57 ! [C2: $tType,D: $tType] :
% 2.83/1.57 ( ( top @ C2 )
% 2.83/1.57 => ( ( top_top @ ( D > C2 ) )
% 2.83/1.57 = ( ^ [X3: D] : ( top_top @ C2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_247_UNIV__I,axiom,
% 2.83/1.57 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_248_Inf__apply,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( complete_Inf @ B )
% 2.83/1.57 => ( ( complete_Inf_Inf @ ( A > B ) )
% 2.83/1.57 = ( ^ [A6: set @ ( A > B ),X3: A] :
% 2.83/1.57 ( complete_Inf_Inf @ B
% 2.83/1.57 @ ( image @ ( A > B ) @ B
% 2.83/1.57 @ ^ [F3: A > B] : ( F3 @ X3 )
% 2.83/1.57 @ A6 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_249_INT__constant,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,A4: set @ B,C: set @ A] :
% 2.83/1.57 ( ( ( A4
% 2.83/1.57 = ( bot_bot @ ( set @ B ) ) )
% 2.83/1.57 => ( ( complete_Inf_Inf @ ( set @ A )
% 2.83/1.57 @ ( image @ B @ ( set @ A )
% 2.83/1.57 @ ^ [Y3: B] : C
% 2.83/1.57 @ A4 ) )
% 2.83/1.57 = ( top_top @ ( set @ A ) ) ) )
% 2.83/1.57 & ( ( A4
% 2.83/1.57 != ( bot_bot @ ( set @ B ) ) )
% 2.83/1.57 => ( ( complete_Inf_Inf @ ( set @ A )
% 2.83/1.57 @ ( image @ B @ ( set @ A )
% 2.83/1.57 @ ^ [Y3: B] : C
% 2.83/1.57 @ A4 ) )
% 2.83/1.57 = C ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_250_Collect__const,axiom,
% 2.83/1.57 ! [A: $tType,P: $o] :
% 2.83/1.57 ( ( P
% 2.83/1.57 => ( ( collect @ A
% 2.83/1.57 @ ^ [S2: A] : P )
% 2.83/1.57 = ( top_top @ ( set @ A ) ) ) )
% 2.83/1.57 & ( ~ P
% 2.83/1.57 => ( ( collect @ A
% 2.83/1.57 @ ^ [S2: A] : P )
% 2.83/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_251_INF__apply,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,C2: $tType] :
% 2.83/1.57 ( ( complete_Inf @ A )
% 2.83/1.57 => ! [F: C2 > B > A,A4: set @ C2,X2: B] :
% 2.83/1.57 ( ( complete_Inf_Inf @ ( B > A ) @ ( image @ C2 @ ( B > A ) @ F @ A4 ) @ X2 )
% 2.83/1.57 = ( complete_Inf_Inf @ A
% 2.83/1.57 @ ( image @ C2 @ A
% 2.83/1.57 @ ^ [Y3: C2] : ( F @ Y3 @ X2 )
% 2.83/1.57 @ A4 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_252_INF__identity__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( complete_Inf @ A )
% 2.83/1.57 => ! [A4: set @ A] :
% 2.83/1.57 ( ( complete_Inf_Inf @ A
% 2.83/1.57 @ ( image @ A @ A
% 2.83/1.57 @ ^ [X3: A] : X3
% 2.83/1.57 @ A4 ) )
% 2.83/1.57 = ( complete_Inf_Inf @ A @ A4 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_253_Inf__eq__bot__iff,axiom,
% 2.83/1.58 ! [A: $tType] :
% 2.83/1.58 ( ( comple1035589618norder @ A )
% 2.83/1.58 => ! [A4: set @ A] :
% 2.83/1.58 ( ( ( complete_Inf_Inf @ A @ A4 )
% 2.83/1.58 = ( bot_bot @ A ) )
% 2.83/1.58 = ( ! [X3: A] :
% 2.83/1.58 ( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
% 2.83/1.58 => ? [Y3: A] :
% 2.83/1.58 ( ( member @ A @ Y3 @ A4 )
% 2.83/1.58 & ( ord_less @ A @ Y3 @ X3 ) ) ) ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(fact_254_Inf__empty,axiom,
% 2.83/1.58 ! [A: $tType] :
% 2.83/1.58 ( ( comple187826305attice @ A )
% 2.83/1.58 => ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.83/1.58 = ( top_top @ A ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.83/1.58 bounded_lattice @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.83/1.58 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( bounded_lattice @ A8 )
% 2.83/1.58 => ( bounded_lattice @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( bounded_lattice @ A8 )
% 2.83/1.58 => ( bounde1808546759up_bot @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( comple187826305attice @ A8 )
% 2.83/1.58 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( semilattice_sup @ A8 )
% 2.83/1.58 => ( semilattice_sup @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Complete__Lattices_OInf,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( complete_Inf @ A8 )
% 2.83/1.58 => ( complete_Inf @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( order_bot @ A8 )
% 2.83/1.58 => ( order_bot @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( preorder @ A8 )
% 2.83/1.58 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( order @ A8 )
% 2.83/1.58 => ( order @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Orderings_Otop,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( top @ A8 )
% 2.83/1.58 => ( top @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Orderings_Oord,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( ord @ A8 )
% 2.83/1.58 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_fun___Orderings_Obot,axiom,
% 2.83/1.58 ! [A7: $tType,A8: $tType] :
% 2.83/1.58 ( ( bot @ A8 )
% 2.83/1.58 => ( bot @ ( A7 > A8 ) ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
% 2.83/1.58 semilattice_sup @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.83/1.58 linordered_idom @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Complete__Lattices_OInf_4,axiom,
% 2.83/1.58 complete_Inf @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Orderings_Opreorder_5,axiom,
% 2.83/1.58 preorder @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.83/1.58 linorder @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 2.83/1.58 no_top @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 2.83/1.58 no_bot @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Orderings_Oorder_6,axiom,
% 2.83/1.58 order @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Int_Oint___Orderings_Oord_7,axiom,
% 2.83/1.58 ord @ int ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_8,axiom,
% 2.83/1.58 ! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_9,axiom,
% 2.83/1.58 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10,axiom,
% 2.83/1.58 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Complete__Lattices_OInf_11,axiom,
% 2.83/1.58 ! [A7: $tType] : ( complete_Inf @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Orderings_Oorder__bot_12,axiom,
% 2.83/1.58 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Orderings_Opreorder_13,axiom,
% 2.83/1.58 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
% 2.83/1.58 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Orderings_Otop_15,axiom,
% 2.83/1.58 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
% 2.83/1.58 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_Set_Oset___Orderings_Obot_17,axiom,
% 2.83/1.58 ! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_18,axiom,
% 2.83/1.58 bounde1808546759up_bot @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_19,axiom,
% 2.83/1.58 comple187826305attice @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_20,axiom,
% 2.83/1.58 semilattice_sup @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Complete__Lattices_OInf_21,axiom,
% 2.83/1.58 complete_Inf @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Orderings_Oorder__bot_22,axiom,
% 2.83/1.58 order_bot @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
% 2.83/1.58 preorder @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Orderings_Olinorder_24,axiom,
% 2.83/1.58 linorder @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Orderings_Oorder_25,axiom,
% 2.83/1.58 order @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Orderings_Otop_26,axiom,
% 2.83/1.58 top @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
% 2.83/1.58 ord @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(tcon_HOL_Obool___Orderings_Obot_28,axiom,
% 2.83/1.58 bot @ $o ).
% 2.83/1.58
% 2.83/1.58 thf(help_If_3_1_T,axiom,
% 2.83/1.58 ! [P: $o] :
% 2.83/1.58 ( ( P = $true )
% 2.83/1.58 | ( P = $false ) ) ).
% 2.83/1.58
% 2.83/1.58 thf(help_If_2_1_T,axiom,
% 2.83/1.58 ! [A: $tType,X2: A,Y: A] :
% 2.83/1.58 ( ( if @ A @ $false @ X2 @ Y )
% 2.83/1.58 = Y ) ).
% 2.83/1.58
% 2.83/1.58 thf(help_If_1_1_T,axiom,
% 2.83/1.58 ! [A: $tType,X2: A,Y: A] :
% 2.83/1.58 ( ( if @ A @ $true @ X2 @ Y )
% 2.83/1.58 = X2 ) ).
% 2.83/1.58
% 2.83/1.58 thf(conj_0,conjecture,
% 2.83/1.58 $false ).
% 2.83/1.58 ------------------------
% 2.83/1.59 % (25442)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_25161 for (2999ds/275Mi)
% 2.83/1.59 % (25443)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25161 for (2999ds/18Mi)
% 2.83/1.59 % (25438)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_25161 for (2999ds/4Mi)
% 2.92/1.60 % (25438)Instruction limit reached!
% 2.92/1.60 % (25438)------------------------------
% 2.92/1.60 % (25438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.60 % (25438)Termination reason: Unknown
% 2.92/1.60 % (25438)Termination phase: shuffling
% 2.92/1.60
% 2.92/1.60 % (25438)Memory used [KB]: 1407
% 2.92/1.60 % (25438)Time elapsed: 0.004 s
% 2.92/1.60 % (25438)Instructions burned: 4 (million)
% 2.92/1.60 % (25438)------------------------------
% 2.92/1.60 % (25438)------------------------------
% 2.92/1.60 % (25437)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25161 for (2999ds/183Mi)
% 2.92/1.60 % (25440)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25161 for (2999ds/2Mi)
% 2.92/1.60 % (25439)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_25161 for (2999ds/27Mi)
% 2.92/1.60 % (25441)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_25161 for (2999ds/2Mi)
% 2.92/1.60 % (25443)Instruction limit reached!
% 2.92/1.60 % (25443)------------------------------
% 2.92/1.60 % (25443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.60 % (25443)Termination reason: Unknown
% 2.92/1.60 % (25443)Termination phase: Property scanning
% 2.92/1.60
% 2.92/1.60 % (25443)Memory used [KB]: 1791
% 2.92/1.60 % (25443)Time elapsed: 0.009 s
% 2.92/1.60 % (25443)Instructions burned: 20 (million)
% 2.92/1.60 % (25443)------------------------------
% 2.92/1.60 % (25443)------------------------------
% 2.92/1.60 % (25440)Instruction limit reached!
% 2.92/1.60 % (25440)------------------------------
% 2.92/1.60 % (25440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.60 % (25440)Termination reason: Unknown
% 2.92/1.60 % (25440)Termination phase: shuffling
% 2.92/1.60
% 2.92/1.60 % (25440)Memory used [KB]: 1407
% 2.92/1.60 % (25440)Time elapsed: 0.004 s
% 2.92/1.60 % (25440)Instructions burned: 2 (million)
% 2.92/1.60 % (25440)------------------------------
% 2.92/1.60 % (25440)------------------------------
% 2.92/1.60 % (25441)Instruction limit reached!
% 2.92/1.60 % (25441)------------------------------
% 2.92/1.60 % (25441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.60 % (25441)Termination reason: Unknown
% 2.92/1.60 % (25441)Termination phase: shuffling
% 2.92/1.60
% 2.92/1.60 % (25441)Memory used [KB]: 1407
% 2.92/1.60 % (25441)Time elapsed: 0.004 s
% 2.92/1.60 % (25441)Instructions burned: 3 (million)
% 2.92/1.60 % (25441)------------------------------
% 2.92/1.60 % (25441)------------------------------
% 2.92/1.60 % (25442)Refutation not found, incomplete strategy
% 2.92/1.60 % (25442)------------------------------
% 2.92/1.60 % (25442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.60 % (25442)Termination reason: Refutation not found, incomplete strategy
% 2.92/1.60
% 2.92/1.60
% 2.92/1.60 % (25442)Memory used [KB]: 6268
% 2.92/1.60 % (25442)Time elapsed: 0.011 s
% 2.92/1.60 % (25442)Instructions burned: 30 (million)
% 2.92/1.60 % (25442)------------------------------
% 2.92/1.60 % (25442)------------------------------
% 2.92/1.61 % (25445)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_25161 for (2999ds/37Mi)
% 2.92/1.61 % (25444)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25161 for (2999ds/3Mi)
% 2.92/1.61 % (25444)Instruction limit reached!
% 2.92/1.61 % (25444)------------------------------
% 2.92/1.61 % (25444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.61 % (25444)Termination reason: Unknown
% 2.92/1.61 % (25444)Termination phase: shuffling
% 2.92/1.61
% 2.92/1.61 % (25444)Memory used [KB]: 1407
% 2.92/1.61 % (25444)Time elapsed: 0.003 s
% 2.92/1.61 % (25444)Instructions burned: 3 (million)
% 2.92/1.61 % (25444)------------------------------
% 2.92/1.61 % (25444)------------------------------
% 2.92/1.61 % (25448)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_25161 for (2999ds/1041Mi)
% 2.92/1.61 % (25439)Instruction limit reached!
% 2.92/1.61 % (25439)------------------------------
% 2.92/1.61 % (25439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.61 % (25439)Termination reason: Unknown
% 2.92/1.61 % (25439)Termination phase: Property scanning
% 2.92/1.61
% 2.92/1.61 % (25439)Memory used [KB]: 1918
% 2.92/1.61 % (25439)Time elapsed: 0.019 s
% 2.92/1.61 % (25439)Instructions burned: 28 (million)
% 2.92/1.61 % (25439)------------------------------
% 2.92/1.61 % (25439)------------------------------
% 2.92/1.62 % (25446)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_25161 for (2999ds/15Mi)
% 2.92/1.62 % (25447)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_25161 for (2999ds/3Mi)
% 2.92/1.62 % (25447)Instruction limit reached!
% 2.92/1.62 % (25447)------------------------------
% 2.92/1.62 % (25447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.62 % (25447)Termination reason: Unknown
% 2.92/1.62 % (25447)Termination phase: shuffling
% 2.92/1.62
% 2.92/1.62 % (25447)Memory used [KB]: 1407
% 2.92/1.62 % (25447)Time elapsed: 0.004 s
% 2.92/1.62 % (25447)Instructions burned: 3 (million)
% 2.92/1.62 % (25447)------------------------------
% 2.92/1.62 % (25447)------------------------------
% 2.92/1.62 % (25449)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25161 for (2999ds/7Mi)
% 2.92/1.62 % (25445)Instruction limit reached!
% 2.92/1.62 % (25445)------------------------------
% 2.92/1.62 % (25445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.62 % (25445)Termination reason: Unknown
% 2.92/1.62 % (25445)Termination phase: Property scanning
% 2.92/1.62
% 2.92/1.62 % (25445)Memory used [KB]: 2174
% 2.92/1.62 % (25445)Time elapsed: 0.014 s
% 2.92/1.62 % (25445)Instructions burned: 40 (million)
% 2.92/1.62 % (25445)------------------------------
% 2.92/1.62 % (25445)------------------------------
% 2.92/1.62 % (25449)Instruction limit reached!
% 2.92/1.62 % (25449)------------------------------
% 2.92/1.62 % (25449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.62 % (25449)Termination reason: Unknown
% 2.92/1.62 % (25449)Termination phase: shuffling
% 2.92/1.62
% 2.92/1.62 % (25449)Memory used [KB]: 1535
% 2.92/1.62 % (25449)Time elapsed: 0.005 s
% 2.92/1.62 % (25449)Instructions burned: 9 (million)
% 2.92/1.62 % (25449)------------------------------
% 2.92/1.62 % (25449)------------------------------
% 2.92/1.63 % (25446)Instruction limit reached!
% 2.92/1.63 % (25446)------------------------------
% 2.92/1.63 % (25446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.63 % (25446)Termination reason: Unknown
% 2.92/1.63 % (25446)Termination phase: shuffling
% 2.92/1.63
% 2.92/1.63 % (25446)Memory used [KB]: 1663
% 2.92/1.63 % (25446)Time elapsed: 0.011 s
% 2.92/1.63 % (25446)Instructions burned: 15 (million)
% 2.92/1.63 % (25446)------------------------------
% 2.92/1.63 % (25446)------------------------------
% 2.92/1.63 % (25452)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_25161 for (2999ds/3Mi)
% 2.92/1.63 % (25450)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25161 for (2999ds/16Mi)
% 2.92/1.63 % (25452)Instruction limit reached!
% 2.92/1.63 % (25452)------------------------------
% 2.92/1.63 % (25452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.63 % (25452)Termination reason: Unknown
% 2.92/1.63 % (25452)Termination phase: shuffling
% 2.92/1.63
% 2.92/1.63 % (25452)Memory used [KB]: 1407
% 2.92/1.63 % (25452)Time elapsed: 0.003 s
% 2.92/1.63 % (25452)Instructions burned: 3 (million)
% 2.92/1.63 % (25452)------------------------------
% 2.92/1.63 % (25452)------------------------------
% 2.92/1.63 % (25453)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_25161 for (2999ds/7Mi)
% 2.92/1.64 % (25451)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_25161 for (2999ds/3Mi)
% 2.92/1.64 % (25453)Instruction limit reached!
% 2.92/1.64 % (25453)------------------------------
% 2.92/1.64 % (25453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.64 % (25453)Termination reason: Unknown
% 2.92/1.64 % (25453)Termination phase: shuffling
% 2.92/1.64
% 2.92/1.64 % (25453)Memory used [KB]: 1535
% 2.92/1.64 % (25453)Time elapsed: 0.005 s
% 2.92/1.64 % (25453)Instructions burned: 9 (million)
% 2.92/1.64 % (25453)------------------------------
% 2.92/1.64 % (25453)------------------------------
% 2.92/1.64 % (25437)First to succeed.
% 2.92/1.64 % (25451)Instruction limit reached!
% 2.92/1.64 % (25451)------------------------------
% 2.92/1.64 % (25451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.64 % (25451)Termination reason: Unknown
% 2.92/1.64 % (25451)Termination phase: shuffling
% 2.92/1.64
% 2.92/1.64 % (25451)Memory used [KB]: 1407
% 2.92/1.64 % (25451)Time elapsed: 0.004 s
% 2.92/1.64 % (25451)Instructions burned: 3 (million)
% 2.92/1.64 % (25451)------------------------------
% 2.92/1.64 % (25451)------------------------------
% 2.92/1.64 % (25437)Refutation found. Thanks to Tanya!
% 2.92/1.64 % SZS status ContradictoryAxioms for DTF2THF_25161
% 2.92/1.64 % SZS output start Proof for DTF2THF_25161
% 2.92/1.64 thf(type_def_5, type, binary1291135688e_Tree: ($tType) > $tType).
% 2.92/1.64 thf(type_def_7, type, set: ($tType) > $tType).
% 2.92/1.64 thf(type_def_8, type, int: $tType).
% 2.92/1.64 thf(type_def_9, type, option: ($tType) > $tType).
% 2.92/1.64 thf(type_def_10, type, a: $tType).
% 2.92/1.64 thf(func_def_0, type, binary1291135688e_Tree: ($tType) > $tType).
% 2.92/1.64 thf(func_def_1, type, option: ($tType) > $tType).
% 2.92/1.64 thf(func_def_2, type, set: ($tType) > $tType).
% 2.92/1.64 thf(func_def_3, type, int: $tType).
% 2.92/1.64 thf(func_def_4, type, a: $tType).
% 2.92/1.64 thf(func_def_5, type, bounded_lattice: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_7, type, bot: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_9, type, top: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_11, type, no_bot: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_12, type, no_top: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_13, type, linorder: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_14, type, preorder: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_15, type, order_bot: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_16, type, wellorder: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_17, type, complete_Inf: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_18, type, dense_order: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_19, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_20, type, linordered_field: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_21, type, semilattice_sup: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_22, type, comple187826305attice: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_23, type, comple1035589618norder: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_24, type, bounde1808546759up_bot: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_25, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.92/1.64 thf(func_def_26, type, binary210054475elle_T: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_27, type, binary1746293266le_Tip: !>[X0: $tType]:(binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_28, type, binary2130109271t_Tree: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > set @ X0)).
% 2.92/1.64 thf(func_def_29, type, binary64540844le_eqs: !>[X0: $tType]:((X0 > int) > X0 > set @ X0)).
% 2.92/1.64 thf(func_def_30, type, binary827270440e_memb: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > $o)).
% 2.92/1.64 thf(func_def_31, type, binary1653327646_setOf: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > set @ X0)).
% 2.92/1.64 thf(func_def_32, type, binary1610619414edTree: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > $o)).
% 2.92/1.64 thf(func_def_33, type, binary231205461t_pred: !>[X0: $tType]:((X0 > int) > X0 > X0 > binary1291135688e_Tree @ X0 > $o)).
% 2.92/1.64 thf(func_def_34, type, binary1265063667lookup: !>[X0: $tType]:((X0 > int) > int > binary1291135688e_Tree @ X0 > option @ X0)).
% 2.92/1.64 thf(func_def_35, type, complete_Inf_Inf: !>[X0: $tType]:(set @ X0 > X0)).
% 2.92/1.64 thf(func_def_36, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_37, type, undefined: !>[X0: $tType]:(X0)).
% 2.92/1.64 thf(func_def_38, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.92/1.64 thf(func_def_39, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.92/1.64 thf(func_def_40, type, none: !>[X0: $tType]:(option @ X0)).
% 2.92/1.64 thf(func_def_41, type, some: !>[X0: $tType]:(X0 > option @ X0)).
% 2.92/1.64 thf(func_def_42, type, case_option: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > option @ X1 > X0)).
% 2.92/1.64 thf(func_def_43, type, set_option: !>[X0: $tType]:(option @ X0 > set @ X0)).
% 2.92/1.64 thf(func_def_44, type, the2: !>[X0: $tType]:(option @ X0 > X0)).
% 2.92/1.64 thf(func_def_45, type, these: !>[X0: $tType]:(set @ option @ X0 > set @ X0)).
% 2.92/1.64 thf(func_def_46, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.92/1.64 thf(func_def_47, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.92/1.64 thf(func_def_48, type, top_top: !>[X0: $tType]:(X0)).
% 2.92/1.64 thf(func_def_49, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.92/1.64 thf(func_def_50, type, bind: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ X1)).
% 2.92/1.64 thf(func_def_51, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.92/1.64 thf(func_def_52, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.92/1.64 thf(func_def_53, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.92/1.64 thf(func_def_54, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 2.92/1.64 thf(func_def_55, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.92/1.64 thf(func_def_56, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.92/1.64 thf(func_def_57, type, h: a > int).
% 2.92/1.64 thf(func_def_58, type, t: binary1291135688e_Tree @ a).
% 2.92/1.64 thf(func_def_59, type, x: a).
% 2.92/1.64 thf(func_def_75, type, sP0: !>[X0: $tType, X1: $tType]:((option @ X0 > option @ X1 > $o) > option @ X1 > option @ X0 > $o)).
% 2.92/1.64 thf(func_def_76, type, sK1: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.92/1.64 thf(func_def_77, type, sK2: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.92/1.64 thf(func_def_78, type, sK3: !>[X0: $tType, X1: $tType]:((option @ X0 > option @ X1 > $o) > option @ X1 > option @ X0 > X0)).
% 2.92/1.64 thf(func_def_79, type, sK4: !>[X0: $tType, X1: $tType]:((option @ X0 > option @ X1 > $o) > option @ X1 > option @ X0 > X1)).
% 2.92/1.64 thf(func_def_80, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.92/1.64 thf(func_def_81, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.92/1.64 thf(func_def_82, type, sK7: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_83, type, sK8: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0)).
% 2.92/1.64 thf(func_def_84, type, sK9: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_85, type, sK10: !>[X0: $tType]:(X0 > X0)).
% 2.92/1.64 thf(func_def_86, type, sK11: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_87, type, sK12: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_88, type, sK13: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_89, type, sK14: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_90, type, sK15: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 2.92/1.64 thf(func_def_91, type, sK16: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_92, type, sK17: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > X0)).
% 2.92/1.64 thf(func_def_93, type, sK18: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_94, type, sK19: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_95, type, sK20: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > X0)).
% 2.92/1.64 thf(func_def_96, type, sK21: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_97, type, sK22: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_98, type, sK23: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_99, type, sK24: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.92/1.64 thf(func_def_100, type, sK25: !>[X0: $tType]:(option @ X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_101, type, sK26: !>[X0: $tType]:(X0 > X0)).
% 2.92/1.64 thf(func_def_102, type, sK27: !>[X0: $tType]:((X0 > $o) > option @ X0 > X0)).
% 2.92/1.64 thf(func_def_103, type, sK28: !>[X0: $tType]:(X0 > X0)).
% 2.92/1.64 thf(func_def_104, type, sK29: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_105, type, sK30: !>[X0: $tType]:(option @ X0 > X0)).
% 2.92/1.64 thf(func_def_106, type, sK31: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.92/1.64 thf(func_def_107, type, sK32: !>[X0: $tType]:(set @ X0 > X0)).
% 2.92/1.64 thf(func_def_108, type, sK33: !>[X0: $tType]:(set @ X0 > X0)).
% 2.92/1.64 thf(func_def_109, type, sK34: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.92/1.64 thf(func_def_110, type, sK35: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.92/1.64 thf(func_def_111, type, sK36: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 2.92/1.64 thf(func_def_112, type, sK37: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.92/1.64 thf(func_def_113, type, sK38: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.92/1.64 thf(func_def_114, type, sK39: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.92/1.64 thf(func_def_115, type, sK40: !>[X0: $tType]:(set @ X0 > X0)).
% 2.92/1.64 thf(func_def_116, type, sK41: !>[X0: $tType]:(set @ X0 > X0)).
% 2.92/1.64 thf(func_def_117, type, sK42: !>[X0: $tType, X1: $tType]:(X1 > (X1 > X0) > set @ X1 > X1)).
% 2.92/1.64 thf(func_def_118, type, sK43: !>[X0: $tType]:(X0 > X0)).
% 2.92/1.64 thf(func_def_119, type, sK44: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_120, type, sK45: !>[X0: $tType]:((option @ X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_121, type, sK46: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_122, type, sK47: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > $o) > (X1 > $o) > X1)).
% 2.92/1.64 thf(func_def_123, type, sK48: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_124, type, sK49: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.92/1.64 thf(func_def_125, type, sK50: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_126, type, sK51: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_127, type, sK52: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_128, type, sK53: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_129, type, sK54: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_130, type, sK55: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_131, type, sK56: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_132, type, sK57: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_133, type, sK58: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > binary1291135688e_Tree @ X0)).
% 2.92/1.64 thf(func_def_134, type, sK59: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_135, type, sK60: !>[X0: $tType]:(X0 > X0)).
% 2.92/1.64 thf(func_def_136, type, sK61: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 2.92/1.64 thf(func_def_137, type, sK62: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.92/1.64 thf(func_def_139, type, sK64: !>[X0: $tType]:(set @ X0 > X0)).
% 2.92/1.64 thf(func_def_140, type, sK65: !>[X0: $tType]:(set @ X0 > X0 > X0)).
% 2.92/1.64 thf(func_def_141, type, sK66: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X0 > X1)).
% 2.92/1.64 thf(func_def_142, type, sK67: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1)).
% 2.92/1.64 thf(func_def_143, type, sK68: !>[X0: $tType, X1: $tType]:(set @ X0 > X1 > (X0 > X1) > X0)).
% 2.92/1.64 thf(func_def_144, type, sK69: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_145, type, sK70: !>[X0: $tType]:(set @ X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_146, type, sK71: !>[X0: $tType]:(set @ X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_147, type, sK72: !>[X0: $tType]:(set @ X0 > (X0 > $o) > X0)).
% 2.92/1.64 thf(func_def_148, type, sK73: !>[X0: $tType]:(set @ X0 > (X0 > $o) > set @ X0 > X0)).
% 2.92/1.64 thf(func_def_149, type, sK74: !>[X0: $tType]:(set @ X0 > X0)).
% 2.92/1.64 thf(f1967,plain,(
% 2.92/1.64 $false),
% 2.92/1.64 inference(trivial_inequality_removal,[status(thm)],[f1246])).
% 2.92/1.64 thf(f1246,plain,(
% 2.92/1.64 ((h @ x) != (h @ x))),
% 2.92/1.64 inference(cnf_transformation,[status(thm)],[f5])).
% 2.92/1.64 thf(f5,axiom,(
% 2.92/1.64 ((h @ x) != (h @ x))),
% 2.92/1.64 file('/export/starexec/sandbox2/tmp/tmp.UyQQfMpErQ/DTF2THF_25161.p',fact_4__092_060open_062h_Ax_A_092_060noteq_062_Ah_Ax_092_060close_062)).
% 2.92/1.64 % SZS output end Proof for DTF2THF_25161
% 2.92/1.64 % (25437)------------------------------
% 2.92/1.64 % (25437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.64 % (25437)Termination reason: Refutation
% 2.92/1.64
% 2.92/1.64 % (25437)Memory used [KB]: 6780
% 2.92/1.64 % (25437)Time elapsed: 0.042 s
% 2.92/1.64 % (25437)Instructions burned: 101 (million)
% 2.92/1.64 % (25437)------------------------------
% 2.92/1.64 % (25437)------------------------------
% 2.92/1.64 % (25436)Success in time 0.059 s
%------------------------------------------------------------------------------