%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM167^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n014.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 08:41:49 AM UTC 2026
% Result : Theorem 3.06s 1.63s
% Output : Refutation 3.06s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM167^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n014.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 04:15:42 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : COM167^1 : TPTP v9.2.1. Released v7.0.0.
% 0.30/0.46 % Domain : Computing Theory
% 0.30/0.46 % Problem : Binary decision diagram 250
% 0.30/0.46 % Version : [Bla16] axioms : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [OS08] Ortner & Schirmer (2008), BDD Normalisation
% 0.30/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.30/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Bla16]
% 0.30/0.46 % Names : bindag__250.p [Bla16]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.30/0.46 % Syntax : Number of formulae : 353 ( 121 unt; 59 typ; 0 def)
% 0.30/0.46 % Number of atoms : 819 ( 290 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 9 ( 2 avg)
% 0.30/0.46 % Number of connectives : 3553 ( 95 ~; 30 |; 63 &;3042 @)
% 0.30/0.46 % ( 0 <=>; 323 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 19 ( 7 avg)
% 0.30/0.46 % Number of types : 3 ( 2 usr)
% 0.30/0.46 % Number of type conns : 312 ( 312 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 60 ( 57 usr; 6 con; 0-6 aty)
% 0.30/0.46 % Number of variables : 1079 ( 81 ^; 900 !; 50 ?;1079 :)
% 0.30/0.46 % ( 48 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.46 % 2016-07-13 14:45:46.829
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 %----Could-be-implicit typings (4)
% 0.30/0.46 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
% 0.30/0.46 binDag_Mirabelle_dag: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Simpl__Heap_Oref,type,
% 0.30/0.46 simpl_ref: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_itself,type,
% 0.30/0.46 itself: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 %----Explicit typings (55)
% 0.30/0.46 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.30/0.46 bounded_lattice:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_HOL_Otype,type,
% 0.30/0.46 type:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.30/0.46 bot:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.46 ord:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.46 order:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Lattices_Olattice,type,
% 0.30/0.46 lattice:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.30/0.46 no_bot:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.30/0.46 no_top:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.46 linorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.46 preorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.30/0.46 order_bot:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.30/0.46 wellorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Odense__order,type,
% 0.30/0.47 dense_order:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.30/0.47 linordered_idom:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Fields_Olinordered__field,type,
% 0.30/0.47 linordered_field:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.30/0.47 semilattice_sup:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 0.30/0.47 bounded_lattice_bot:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.30/0.47 bounde1808546759up_bot:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.30/0.47 condit1656338222tinuum:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
% 0.30/0.47 binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODag,type,
% 0.30/0.47 binDag_Mirabelle_Dag: simpl_ref > ( simpl_ref > simpl_ref ) > ( simpl_ref > simpl_ref ) > binDag_Mirabelle_dag > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
% 0.30/0.47 binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
% 0.30/0.47 binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
% 0.30/0.47 binDag1297733282se_dag:
% 0.30/0.47 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
% 0.30/0.47 binDag1442713106ec_dag:
% 0.30/0.47 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
% 0.30/0.47 binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
% 0.30/0.47 binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Fun_Ofun__upd,type,
% 0.30/0.47 fun_upd:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Fun_Ooverride__on,type,
% 0.30/0.47 override_on:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Groups_Ocomm__monoid,type,
% 0.30/0.47 comm_monoid:
% 0.30/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Groups_Omonoid,type,
% 0.30/0.47 monoid:
% 0.30/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_If,type,
% 0.30/0.47 if:
% 0.30/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Lattices_Osemilattice__neutr,type,
% 0.30/0.47 semilattice_neutr:
% 0.30/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.30/0.47 sup_sup:
% 0.30/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.47 bot_bot:
% 0.30/0.47 !>[A: $tType] : A ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.47 ord_less:
% 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.47 ord_less_eq:
% 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Pure_Otype,type,
% 0.30/0.47 type2:
% 0.30/0.47 !>[A: $tType] : ( itself @ A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Relation_Oinv__imagep,type,
% 0.30/0.47 inv_imagep:
% 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_OCollect,type,
% 0.30/0.47 collect:
% 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_OPow,type,
% 0.30/0.47 pow:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Obind,type,
% 0.30/0.47 bind:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Oinsert,type,
% 0.30/0.47 insert:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Ois__empty,type,
% 0.30/0.47 is_empty:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Ois__singleton,type,
% 0.30/0.47 is_singleton:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Opairwise,type,
% 0.30/0.47 pairwise:
% 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Othe__elem,type,
% 0.30/0.47 the_elem:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Simpl__Heap_ONull,type,
% 0.30/0.47 simpl_Null: simpl_ref ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Zorn_Opred__on_Ochain,type,
% 0.30/0.47 pred_chain:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Zorn_Opred__on_Omaxchain,type,
% 0.30/0.47 pred_maxchain:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_member,type,
% 0.30/0.47 member:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_l,type,
% 0.30/0.47 l: simpl_ref > simpl_ref ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_p,type,
% 0.30/0.47 p: simpl_ref ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_r,type,
% 0.30/0.47 r: simpl_ref > simpl_ref ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_t,type,
% 0.30/0.47 t: binDag_Mirabelle_dag ).
% 0.30/0.47
% 0.30/0.47 %----Relevant facts (256)
% 0.30/0.47 thf(fact_0_Dag__Null,axiom,
% 0.30/0.47 ! [L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag_Mirabelle_Dag @ simpl_Null @ L @ R @ T )
% 0.30/0.47 = ( T = binDag_Mirabelle_Tip ) ) ).
% 0.30/0.47
% 0.30/0.47 % Dag_Null
% 0.30/0.47 thf(fact_1_Dag_Osimps_I1_J,axiom,
% 0.30/0.47 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref] :
% 0.30/0.47 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ binDag_Mirabelle_Tip )
% 0.30/0.47 = ( P = simpl_Null ) ) ).
% 0.30/0.47
% 0.30/0.47 % Dag.simps(1)
% 0.30/0.47 thf(fact_2_Dag__upd__same__l__lemma,axiom,
% 0.30/0.47 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( P != simpl_Null )
% 0.30/0.47 => ~ ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ P @ P ) @ R @ T ) ) ).
% 0.30/0.47
% 0.30/0.47 % Dag_upd_same_l_lemma
% 0.30/0.47 thf(fact_3_fun__upd__upd,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z: B] :
% 0.30/0.47 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z )
% 0.30/0.47 = ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_upd
% 0.30/0.47 thf(fact_4_fun__upd__triv,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.30/0.47 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 0.30/0.47 = F ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_triv
% 0.30/0.47 thf(fact_5_fun__upd__apply,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( fun_upd @ B @ A )
% 0.30/0.47 = ( ^ [F2: B > A,X2: B,Y2: A,Z2: B] : ( if @ A @ ( Z2 = X2 ) @ Y2 @ ( F2 @ Z2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_apply
% 0.30/0.47 thf(fact_6_notin__Dag__update__l,axiom,
% 0.30/0.47 ! [Q: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,Y: simpl_ref,R: simpl_ref > simpl_ref] :
% 0.30/0.47 ( ~ ( member @ simpl_ref @ Q @ ( binDag1380252983set_of @ T ) )
% 0.30/0.47 => ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q @ Y ) @ R @ T )
% 0.30/0.47 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % notin_Dag_update_l
% 0.30/0.47 thf(fact_7_notin__Dag__update__r,axiom,
% 0.30/0.47 ! [Q: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Y: simpl_ref] :
% 0.30/0.47 ( ~ ( member @ simpl_ref @ Q @ ( binDag1380252983set_of @ T ) )
% 0.30/0.47 => ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q @ Y ) @ T )
% 0.30/0.47 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % notin_Dag_update_r
% 0.30/0.47 thf(fact_8_Null__notin__Dag,axiom,
% 0.30/0.47 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 0.30/0.47 => ~ ( member @ simpl_ref @ simpl_Null @ ( binDag1380252983set_of @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Null_notin_Dag
% 0.30/0.47 thf(fact_9_Dag__Ref,axiom,
% 0.30/0.47 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( P != simpl_Null )
% 0.30/0.47 => ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 0.30/0.47 = ( ? [Lt: binDag_Mirabelle_dag,Rt: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( T
% 0.30/0.47 = ( binDag476092410e_Node @ Lt @ P @ Rt ) )
% 0.30/0.47 & ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt )
% 0.30/0.47 & ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Dag_Ref
% 0.30/0.47 thf(fact_10_Dag_Osimps_I2_J,axiom,
% 0.30/0.47 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 0.30/0.47 = ( ( P = A2 )
% 0.30/0.47 & ( P != simpl_Null )
% 0.30/0.47 & ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt2 )
% 0.30/0.47 & ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Dag.simps(2)
% 0.30/0.47 thf(fact_11_fun__upd__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( fun_upd @ A @ B )
% 0.30/0.47 = ( ^ [F2: A > B,A3: A,B2: B,X2: A] : ( if @ B @ ( X2 = A3 ) @ B2 @ ( F2 @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_def
% 0.30/0.47 thf(fact_12_fun__upd__eqD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z: B] :
% 0.30/0.47 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.30/0.47 = ( fun_upd @ A @ B @ G @ X @ Z ) )
% 0.30/0.47 => ( Y = Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_eqD
% 0.30/0.47 thf(fact_13_fun__upd__idem,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 0.30/0.47 ( ( ( F @ X )
% 0.30/0.47 = Y )
% 0.30/0.47 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 0.30/0.47 = F ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_idem
% 0.30/0.47 thf(fact_14_fun__upd__same,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 0.30/0.47 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_same
% 0.30/0.47 thf(fact_15_dag_Oinject,axiom,
% 0.30/0.47 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
% 0.30/0.47 = ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
% 0.30/0.47 = ( ( X21 = Y21 )
% 0.30/0.47 & ( X22 = Y22 )
% 0.30/0.47 & ( X23 = Y23 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dag.inject
% 0.30/0.47 thf(fact_16_dag_Oexhaust,axiom,
% 0.30/0.47 ! [Y: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( Y != binDag_Mirabelle_Tip )
% 0.30/0.47 => ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dag.exhaust
% 0.30/0.47 thf(fact_17_dag_Oinduct,axiom,
% 0.30/0.47 ! [P2: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( P2 @ binDag_Mirabelle_Tip )
% 0.30/0.47 => ( ! [X1: binDag_Mirabelle_dag,X24: simpl_ref,X3: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( P2 @ X1 )
% 0.30/0.47 => ( ( P2 @ X3 )
% 0.30/0.47 => ( P2 @ ( binDag476092410e_Node @ X1 @ X24 @ X3 ) ) ) )
% 0.30/0.47 => ( P2 @ Dag ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dag.induct
% 0.30/0.47 thf(fact_18_dag_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.30/0.47 ( binDag_Mirabelle_Tip
% 0.30/0.47 != ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dag.distinct(1)
% 0.30/0.47 thf(fact_19_fun__upd__idem__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 0.30/0.47 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.30/0.47 = F )
% 0.30/0.47 = ( ( F @ X )
% 0.30/0.47 = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_idem_iff
% 0.30/0.47 thf(fact_20_fun__upd__twist,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,C: A,M: A > B,B3: B,D: B] :
% 0.30/0.47 ( ( A2 != C )
% 0.30/0.47 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A2 @ B3 ) @ C @ D )
% 0.30/0.47 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C @ D ) @ A2 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_twist
% 0.30/0.47 thf(fact_21_fun__upd__other,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y: B] :
% 0.30/0.47 ( ( Z != X )
% 0.30/0.47 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z )
% 0.30/0.47 = ( F @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_other
% 0.30/0.47 thf(fact_22_DAG_Osimps_I2_J,axiom,
% 0.30/0.47 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.30/0.47 = ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
% 0.30/0.47 & ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
% 0.30/0.47 & ( binDag_Mirabelle_DAG @ L )
% 0.30/0.47 & ( binDag_Mirabelle_DAG @ R ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % DAG.simps(2)
% 0.30/0.47 thf(fact_23_in__set__of__decomp,axiom,
% 0.30/0.47 ! [P: simpl_ref,T: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( member @ simpl_ref @ P @ ( binDag1380252983set_of @ T ) )
% 0.30/0.47 = ( ? [L2: binDag_Mirabelle_dag,R2: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( T
% 0.30/0.47 = ( binDag476092410e_Node @ L2 @ P @ R2 ) )
% 0.30/0.47 | ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ L2 @ P @ R2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_set_of_decomp
% 0.30/0.47 thf(fact_24_dag_Osimps_I4_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
% 0.30/0.47 ( ( binDag1297733282se_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
% 0.30/0.47 = F1 ) ).
% 0.30/0.47
% 0.30/0.47 % dag.simps(4)
% 0.30/0.47 thf(fact_25_dag_Osimps_I6_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
% 0.30/0.47 ( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
% 0.30/0.47 = F1 ) ).
% 0.30/0.47
% 0.30/0.47 % dag.simps(6)
% 0.30/0.47 thf(fact_26_DAG_Osimps_I1_J,axiom,
% 0.30/0.47 binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% 0.30/0.47
% 0.30/0.47 % DAG.simps(1)
% 0.30/0.47 thf(fact_27_dag_Osimps_I7_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 0.30/0.47 = ( F22 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X23 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dag.simps(7)
% 0.30/0.47 thf(fact_28_dag_Osimps_I5_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag1297733282se_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 0.30/0.47 = ( F22 @ X21 @ X22 @ X23 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dag.simps(5)
% 0.30/0.47 thf(fact_29_set__of__Tip,axiom,
% 0.30/0.47 ( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
% 0.30/0.47 = ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % set_of_Tip
% 0.30/0.47 thf(fact_30_subdag_Osimps_I1_J,axiom,
% 0.30/0.47 ! [T: binDag_Mirabelle_dag] :
% 0.30/0.47 ~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% 0.30/0.47
% 0.30/0.47 % subdag.simps(1)
% 0.30/0.47 thf(fact_31_subdag_Osimps_I2_J,axiom,
% 0.30/0.47 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T )
% 0.30/0.47 = ( ( T = L )
% 0.30/0.47 | ( T = R )
% 0.30/0.47 | ( binDag786255756subdag @ L @ T )
% 0.30/0.47 | ( binDag786255756subdag @ R @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subdag.simps(2)
% 0.30/0.47 thf(fact_32_subdag__not__sym,axiom,
% 0.30/0.47 ! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag786255756subdag @ S @ T )
% 0.30/0.47 => ~ ( binDag786255756subdag @ T @ S ) ) ).
% 0.30/0.47
% 0.30/0.47 % subdag_not_sym
% 0.30/0.47 thf(fact_33_subdag__trans,axiom,
% 0.30/0.47 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag786255756subdag @ T @ S )
% 0.30/0.47 => ( ( binDag786255756subdag @ S @ R )
% 0.30/0.47 => ( binDag786255756subdag @ T @ R ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subdag_trans
% 0.30/0.47 thf(fact_34_subdag__neq,axiom,
% 0.30/0.47 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag786255756subdag @ T @ S )
% 0.30/0.47 => ( T != S ) ) ).
% 0.30/0.47
% 0.30/0.47 % subdag_neq
% 0.30/0.47 thf(fact_35_subdag__NodeD,axiom,
% 0.30/0.47 ! [T: binDag_Mirabelle_dag,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 0.30/0.47 => ( ( binDag786255756subdag @ T @ Lt2 )
% 0.30/0.47 & ( binDag786255756subdag @ T @ Rt2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subdag_NodeD
% 0.30/0.47 thf(fact_36_empty__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A @ P2 ) )
% 0.30/0.47 = ( ! [X2: A] :
% 0.30/0.47 ~ ( P2 @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_Collect_eq
% 0.30/0.47 thf(fact_37_Collect__empty__eq,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ( ( collect @ A @ P2 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( ! [X2: A] :
% 0.30/0.47 ~ ( P2 @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_empty_eq
% 0.30/0.47 thf(fact_38_all__not__in__conv,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( ! [X2: A] :
% 0.30/0.47 ~ ( member @ A @ X2 @ A4 ) )
% 0.30/0.47 = ( A4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % all_not_in_conv
% 0.30/0.47 thf(fact_39_empty__iff,axiom,
% 0.30/0.47 ! [A: $tType,C: A] :
% 0.30/0.47 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_iff
% 0.30/0.47 thf(fact_40_bot__apply,axiom,
% 0.30/0.47 ! [C2: $tType,D2: $tType] :
% 0.30/0.47 ( ( bot @ C2 @ ( type2 @ C2 ) )
% 0.30/0.47 => ( ( bot_bot @ ( D2 > C2 ) )
% 0.30/0.47 = ( ^ [X2: D2] : ( bot_bot @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_apply
% 0.30/0.47 thf(fact_41_override__on__emptyset,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.47 ( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = F ) ).
% 0.30/0.47
% 0.30/0.47 % override_on_emptyset
% 0.30/0.47 thf(fact_42_ex__in__conv,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( ? [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.30/0.47 = ( A4
% 0.30/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ex_in_conv
% 0.30/0.47 thf(fact_43_equals0I,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ! [Y3: A] :
% 0.30/0.47 ~ ( member @ A @ Y3 @ A4 )
% 0.30/0.47 => ( A4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % equals0I
% 0.30/0.47 thf(fact_44_equals0D,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,A2: A] :
% 0.30/0.47 ( ( A4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % equals0D
% 0.30/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,P2: A > $o] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.30/0.47 = ( P2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Collect_eq
% 0.30/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.30/0.47 = A4 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mem_eq
% 0.30/0.47 thf(fact_47_Collect__cong,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ! [X4: A] :
% 0.30/0.47 ( ( P2 @ X4 )
% 0.30/0.47 = ( Q2 @ X4 ) )
% 0.30/0.47 => ( ( collect @ A @ P2 )
% 0.30/0.47 = ( collect @ A @ Q2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_cong
% 0.30/0.47 thf(fact_48_ext,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.47 ( ! [X4: A] :
% 0.30/0.47 ( ( F @ X4 )
% 0.30/0.47 = ( G @ X4 ) )
% 0.30/0.47 => ( F = G ) ) ).
% 0.30/0.47
% 0.30/0.47 % ext
% 0.30/0.47 thf(fact_49_override__on__apply__in,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,A4: set @ A,F: A > B,G: A > B] :
% 0.30/0.47 ( ( member @ A @ A2 @ A4 )
% 0.30/0.47 => ( ( override_on @ A @ B @ F @ G @ A4 @ A2 )
% 0.30/0.47 = ( G @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % override_on_apply_in
% 0.30/0.47 thf(fact_50_override__on__apply__notin,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,A4: set @ A,F: A > B,G: A > B] :
% 0.30/0.47 ( ~ ( member @ A @ A2 @ A4 )
% 0.30/0.47 => ( ( override_on @ A @ B @ F @ G @ A4 @ A2 )
% 0.30/0.47 = ( F @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % override_on_apply_notin
% 0.30/0.47 thf(fact_51_bot__set__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_set_def
% 0.30/0.47 thf(fact_52_override__on__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( override_on @ A @ B )
% 0.30/0.47 = ( ^ [F2: A > B,G2: A > B,A5: set @ A,A3: A] : ( if @ B @ ( member @ A @ A3 @ A5 ) @ ( G2 @ A3 ) @ ( F2 @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % override_on_def
% 0.30/0.47 thf(fact_53_bot__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( bot @ B @ ( type2 @ B ) )
% 0.30/0.47 => ( ( bot_bot @ ( A > B ) )
% 0.30/0.47 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_fun_def
% 0.30/0.47 thf(fact_54_emptyE,axiom,
% 0.30/0.47 ! [A: $tType,A2: A] :
% 0.30/0.47 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % emptyE
% 0.30/0.47 thf(fact_55_Set_Ois__empty__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( is_empty @ A )
% 0.30/0.47 = ( ^ [A5: set @ A] :
% 0.30/0.47 ( A5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Set.is_empty_def
% 0.30/0.47 thf(fact_56_Collect__empty__eq__bot,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ( ( collect @ A @ P2 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( P2
% 0.30/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_empty_eq_bot
% 0.30/0.47 thf(fact_57_bot__empty__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.30/0.47 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_empty_eq
% 0.30/0.47 thf(fact_58_is__singletonI_H,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( A4
% 0.30/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 => ( ! [X4: A,Y3: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ A4 )
% 0.30/0.47 => ( ( member @ A @ Y3 @ A4 )
% 0.30/0.47 => ( X4 = Y3 ) ) )
% 0.30/0.47 => ( is_singleton @ A @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singletonI'
% 0.30/0.47 thf(fact_59_empty__bind,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: B > ( set @ A )] :
% 0.30/0.47 ( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_bind
% 0.30/0.47 thf(fact_60_set__of__Node,axiom,
% 0.30/0.47 ! [Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag1380252983set_of @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 0.30/0.47 = ( sup_sup @ ( set @ simpl_ref ) @ ( sup_sup @ ( set @ simpl_ref ) @ ( insert @ simpl_ref @ A2 @ ( bot_bot @ ( set @ simpl_ref ) ) ) @ ( binDag1380252983set_of @ Lt2 ) ) @ ( binDag1380252983set_of @ Rt2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % set_of_Node
% 0.30/0.47 thf(fact_61_pairwise__empty,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > A > $o] : ( pairwise @ A @ P2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pairwise_empty
% 0.30/0.47 thf(fact_62_DAG__less,axiom,
% 0.30/0.47 ! [Y: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( binDag_Mirabelle_DAG @ Y )
% 0.30/0.47 => ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
% 0.30/0.47 => ( binDag_Mirabelle_DAG @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % DAG_less
% 0.30/0.47 thf(fact_63_insertCI,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B4: set @ A,B3: A] :
% 0.30/0.47 ( ( ~ ( member @ A @ A2 @ B4 )
% 0.30/0.47 => ( A2 = B3 ) )
% 0.30/0.47 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insertCI
% 0.30/0.47 thf(fact_64_insert__iff,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B3: A,A4: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
% 0.30/0.47 = ( ( A2 = B3 )
% 0.30/0.47 | ( member @ A @ A2 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_iff
% 0.30/0.47 thf(fact_65_insert__absorb2,axiom,
% 0.30/0.47 ! [A: $tType,X: A,A4: set @ A] :
% 0.30/0.47 ( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
% 0.30/0.47 = ( insert @ A @ X @ A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_absorb2
% 0.30/0.47 thf(fact_66_UnCI,axiom,
% 0.30/0.47 ! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
% 0.30/0.47 ( ( ~ ( member @ A @ C @ B4 )
% 0.30/0.47 => ( member @ A @ C @ A4 ) )
% 0.30/0.47 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnCI
% 0.30/0.47 thf(fact_67_Un__iff,axiom,
% 0.30/0.47 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 0.30/0.47 = ( ( member @ A @ C @ A4 )
% 0.30/0.47 | ( member @ A @ C @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_iff
% 0.30/0.47 thf(fact_68_singletonI,axiom,
% 0.30/0.47 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % singletonI
% 0.30/0.47 thf(fact_69_Un__empty,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( ( A4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 & ( B4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_empty
% 0.30/0.47 thf(fact_70_Un__insert__left,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B4: set @ A,C3: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B4 ) @ C3 )
% 0.30/0.47 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_insert_left
% 0.30/0.47 thf(fact_71_Un__insert__right,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,A2: A,B4: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B4 ) )
% 0.30/0.47 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_insert_right
% 0.30/0.47 thf(fact_72_is__singletonI,axiom,
% 0.30/0.47 ! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singletonI
% 0.30/0.47 thf(fact_73_pairwise__singleton,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > A > $o,A4: A] : ( pairwise @ A @ P2 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pairwise_singleton
% 0.30/0.47 thf(fact_74_singleton__Un__iff,axiom,
% 0.30/0.47 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 0.30/0.47 = ( ( ( A4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 & ( B4
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.47 | ( ( A4
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( B4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 | ( ( A4
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( B4
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_Un_iff
% 0.30/0.47 thf(fact_75_Un__singleton__iff,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.30/0.47 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = ( ( ( A4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 & ( B4
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.47 | ( ( A4
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( B4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 | ( ( A4
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( B4
% 0.30/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_singleton_iff
% 0.30/0.47 thf(fact_76_insert__is__Un,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( insert @ A )
% 0.30/0.47 = ( ^ [A3: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_is_Un
% 0.30/0.47 thf(fact_77_UnE,axiom,
% 0.30/0.47 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 0.30/0.47 => ( ~ ( member @ A @ C @ A4 )
% 0.30/0.47 => ( member @ A @ C @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnE
% 0.30/0.47 thf(fact_78_UnI1,axiom,
% 0.30/0.47 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( member @ A @ C @ A4 )
% 0.30/0.47 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnI1
% 0.30/0.47 thf(fact_79_UnI2,axiom,
% 0.30/0.47 ! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
% 0.30/0.47 ( ( member @ A @ C @ B4 )
% 0.30/0.47 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnI2
% 0.30/0.47 thf(fact_80_bex__Un,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,P2: A > $o] :
% 0.30/0.47 ( ( ? [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 0.30/0.47 & ( P2 @ X2 ) ) )
% 0.30/0.47 = ( ? [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ A4 )
% 0.30/0.47 & ( P2 @ X2 ) )
% 0.30/0.47 | ? [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ B4 )
% 0.30/0.47 & ( P2 @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_Un
% 0.30/0.47 thf(fact_81_ball__Un,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,P2: A > $o] :
% 0.30/0.47 ( ( ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 0.30/0.47 => ( P2 @ X2 ) ) )
% 0.30/0.47 = ( ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ A4 )
% 0.30/0.47 => ( P2 @ X2 ) )
% 0.30/0.47 & ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ B4 )
% 0.30/0.47 => ( P2 @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_Un
% 0.30/0.47 thf(fact_82_insertE,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B3: A,A4: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
% 0.30/0.47 => ( ( A2 != B3 )
% 0.30/0.47 => ( member @ A @ A2 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insertE
% 0.30/0.47 thf(fact_83_Un__assoc,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_assoc
% 0.30/0.47 thf(fact_84_insertI1,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % insertI1
% 0.30/0.47 thf(fact_85_insertI2,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B4: set @ A,B3: A] :
% 0.30/0.47 ( ( member @ A @ A2 @ B4 )
% 0.30/0.47 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insertI2
% 0.30/0.47 thf(fact_86_Un__absorb,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 0.30/0.47 = A4 ) ).
% 0.30/0.47
% 0.30/0.47 % Un_absorb
% 0.30/0.47 thf(fact_87_Un__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_commute
% 0.30/0.47 thf(fact_88_Set_Oset__insert,axiom,
% 0.30/0.47 ! [A: $tType,X: A,A4: set @ A] :
% 0.30/0.47 ( ( member @ A @ X @ A4 )
% 0.30/0.47 => ~ ! [B6: set @ A] :
% 0.30/0.47 ( ( A4
% 0.30/0.47 = ( insert @ A @ X @ B6 ) )
% 0.30/0.47 => ( member @ A @ X @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Set.set_insert
% 0.30/0.47 thf(fact_89_insert__ident,axiom,
% 0.30/0.47 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ~ ( member @ A @ X @ A4 )
% 0.30/0.47 => ( ~ ( member @ A @ X @ B4 )
% 0.30/0.47 => ( ( ( insert @ A @ X @ A4 )
% 0.30/0.47 = ( insert @ A @ X @ B4 ) )
% 0.30/0.47 = ( A4 = B4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_ident
% 0.30/0.47 thf(fact_90_pairwise__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( pairwise @ A )
% 0.30/0.47 = ( ^ [R3: A > A > $o,S2: set @ A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ S2 )
% 0.30/0.47 => ! [Y2: A] :
% 0.30/0.47 ( ( member @ A @ Y2 @ S2 )
% 0.30/0.47 => ( ( X2 != Y2 )
% 0.30/0.47 => ( R3 @ X2 @ Y2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pairwise_def
% 0.30/0.47 thf(fact_91_insert__absorb,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A4: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ A4 )
% 0.30/0.47 => ( ( insert @ A @ A2 @ A4 )
% 0.30/0.47 = A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_absorb
% 0.30/0.47 thf(fact_92_insert__eq__iff,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A4: set @ A,B3: A,B4: set @ A] :
% 0.30/0.47 ( ~ ( member @ A @ A2 @ A4 )
% 0.30/0.47 => ( ~ ( member @ A @ B3 @ B4 )
% 0.30/0.47 => ( ( ( insert @ A @ A2 @ A4 )
% 0.30/0.47 = ( insert @ A @ B3 @ B4 ) )
% 0.30/0.47 = ( ( ( A2 = B3 )
% 0.30/0.47 => ( A4 = B4 ) )
% 0.30/0.47 & ( ( A2 != B3 )
% 0.30/0.47 => ? [C4: set @ A] :
% 0.30/0.47 ( ( A4
% 0.30/0.47 = ( insert @ A @ B3 @ C4 ) )
% 0.30/0.47 & ~ ( member @ A @ B3 @ C4 )
% 0.30/0.47 & ( B4
% 0.30/0.47 = ( insert @ A @ A2 @ C4 ) )
% 0.30/0.47 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_eq_iff
% 0.30/0.47 thf(fact_93_Un__left__absorb,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_left_absorb
% 0.30/0.47 thf(fact_94_insert__commute,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 0.30/0.47 ( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
% 0.30/0.47 = ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_commute
% 0.30/0.47 thf(fact_95_Un__left__commute,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
% 0.30/0.47 = ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_left_commute
% 0.30/0.47 thf(fact_96_pairwise__insert,axiom,
% 0.30/0.47 ! [A: $tType,R: A > A > $o,X: A,S: set @ A] :
% 0.30/0.47 ( ( pairwise @ A @ R @ ( insert @ A @ X @ S ) )
% 0.30/0.47 = ( ! [Y2: A] :
% 0.30/0.47 ( ( ( member @ A @ Y2 @ S )
% 0.30/0.47 & ( Y2 != X ) )
% 0.30/0.47 => ( ( R @ X @ Y2 )
% 0.30/0.47 & ( R @ Y2 @ X ) ) )
% 0.30/0.47 & ( pairwise @ A @ R @ S ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pairwise_insert
% 0.30/0.47 thf(fact_97_mk__disjoint__insert,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A4: set @ A] :
% 0.30/0.47 ( ( member @ A @ A2 @ A4 )
% 0.30/0.47 => ? [B6: set @ A] :
% 0.30/0.47 ( ( A4
% 0.30/0.47 = ( insert @ A @ A2 @ B6 ) )
% 0.30/0.47 & ~ ( member @ A @ A2 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mk_disjoint_insert
% 0.30/0.47 thf(fact_98_ord__eq__less__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less @ B @ B3 @ C )
% 0.30/0.47 => ( ! [X4: B,Y3: B] :
% 0.30/0.47 ( ( ord_less @ B @ X4 @ Y3 )
% 0.30/0.47 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_less_subst
% 0.30/0.47 thf(fact_99_ord__less__eq__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A2: A,B3: A,F: A > B,C: B] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.47 => ( ( ( F @ B3 )
% 0.30/0.47 = C )
% 0.30/0.47 => ( ! [X4: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Y3 )
% 0.30/0.47 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.47 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_less_eq_subst
% 0.30/0.47 thf(fact_100_order__less__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less @ B @ B3 @ C )
% 0.30/0.47 => ( ! [X4: B,Y3: B] :
% 0.30/0.47 ( ( ord_less @ B @ X4 @ Y3 )
% 0.30/0.47 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_subst1
% 0.30/0.47 thf(fact_101_order__less__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C2: $tType] :
% 0.30/0.47 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.47 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 0.30/0.47 => ( ! [X4: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Y3 )
% 0.30/0.47 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_subst2
% 0.30/0.47 thf(fact_102_lt__ex,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % lt_ex
% 0.30/0.47 thf(fact_103_gt__ex,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( no_top @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % gt_ex
% 0.30/0.47 thf(fact_104_neqE,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 => ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neqE
% 0.30/0.47 thf(fact_105_neq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 = ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neq_iff
% 0.30/0.47 thf(fact_106_order_Oasym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.47 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.asym
% 0.30/0.47 thf(fact_107_dense,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Z3 )
% 0.30/0.47 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense
% 0.30/0.47 thf(fact_108_less__imp__neq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( X != Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_neq
% 0.30/0.47 thf(fact_109_less__asym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_asym
% 0.30/0.47 thf(fact_110_less__asym_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.47 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_asym'
% 0.30/0.47 thf(fact_111_less__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less @ A @ Y @ Z )
% 0.30/0.47 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_trans
% 0.30/0.47 thf(fact_112_less__linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 | ( X = Y )
% 0.30/0.47 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_linear
% 0.30/0.47 thf(fact_113_less__irrefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_irrefl
% 0.30/0.47 thf(fact_114_ord__eq__less__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A,C: A] :
% 0.30/0.47 ( ( A2 = B3 )
% 0.30/0.47 => ( ( ord_less @ A @ B3 @ C )
% 0.30/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_less_trans
% 0.30/0.47 thf(fact_115_ord__less__eq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A,C: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.47 => ( ( B3 = C )
% 0.30/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_less_eq_trans
% 0.30/0.47 thf(fact_116_dual__order_Oasym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,A2: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.30/0.47 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.asym
% 0.30/0.47 thf(fact_117_less__imp__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( X != Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq
% 0.30/0.47 thf(fact_118_less__not__sym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_not_sym
% 0.30/0.47 thf(fact_119_less__induct,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P2: A > $o,A2: A] :
% 0.30/0.47 ( ! [X4: A] :
% 0.30/0.47 ( ! [Y4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Y4 @ X4 )
% 0.30/0.47 => ( P2 @ Y4 ) )
% 0.30/0.47 => ( P2 @ X4 ) )
% 0.30/0.47 => ( P2 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_induct
% 0.30/0.47 thf(fact_120_antisym__conv3,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [Y: A,X: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ Y @ X )
% 0.30/0.47 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv3
% 0.30/0.47 thf(fact_121_less__imp__not__eq2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( Y != X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq2
% 0.30/0.47 thf(fact_122_less__imp__triv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,P2: $o] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less @ A @ Y @ X )
% 0.30/0.47 => P2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_triv
% 0.30/0.47 thf(fact_123_linorder__cases,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( X != Y )
% 0.30/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_cases
% 0.30/0.47 thf(fact_124_dual__order_Oirrefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.irrefl
% 0.30/0.47 thf(fact_125_order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A,C: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.47 => ( ( ord_less @ A @ B3 @ C )
% 0.30/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans
% 0.30/0.47 thf(fact_126_less__imp__not__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_less
% 0.30/0.47 thf(fact_127_dual__order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,A2: A,C: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.30/0.47 => ( ( ord_less @ A @ C @ B3 )
% 0.30/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans
% 0.30/0.47 thf(fact_128_not__less__iff__gr__or__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.47 = ( ( ord_less @ A @ Y @ X )
% 0.30/0.47 | ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_less_iff_gr_or_eq
% 0.30/0.47 thf(fact_129_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.47 => ( A2 != B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_implies_not_eq
% 0.30/0.47 thf(fact_130_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,A2: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.30/0.47 => ( A2 != B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_implies_not_eq
% 0.30/0.47 thf(fact_131_is__singletonE,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( is_singleton @ A @ A4 )
% 0.30/0.47 => ~ ! [X4: A] :
% 0.30/0.47 ( A4
% 0.30/0.47 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singletonE
% 0.30/0.47 thf(fact_132_is__singleton__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( is_singleton @ A )
% 0.30/0.47 = ( ^ [A5: set @ A] :
% 0.30/0.47 ? [X2: A] :
% 0.30/0.47 ( A5
% 0.30/0.47 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singleton_def
% 0.30/0.47 thf(fact_133_Un__empty__left,axiom,
% 0.30/0.47 ! [A: $tType,B4: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
% 0.30/0.47 = B4 ) ).
% 0.30/0.47
% 0.30/0.47 % Un_empty_left
% 0.30/0.47 thf(fact_134_Un__empty__right,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = A4 ) ).
% 0.30/0.47
% 0.30/0.47 % Un_empty_right
% 0.30/0.47 thf(fact_135_singletonD,axiom,
% 0.30/0.47 ! [A: $tType,B3: A,A2: A] :
% 0.30/0.47 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 => ( B3 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % singletonD
% 0.30/0.47 thf(fact_136_singleton__iff,axiom,
% 0.30/0.47 ! [A: $tType,B3: A,A2: A] :
% 0.30/0.47 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = ( B3 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_iff
% 0.30/0.47 thf(fact_137_doubleton__eq__iff,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B3: A,C: A,D: A] :
% 0.30/0.47 ( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.47 = ( ( ( A2 = C )
% 0.30/0.47 & ( B3 = D ) )
% 0.30/0.47 | ( ( A2 = D )
% 0.30/0.47 & ( B3 = C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % doubleton_eq_iff
% 0.30/0.47 thf(fact_138_insert__not__empty,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A4: set @ A] :
% 0.30/0.47 ( ( insert @ A @ A2 @ A4 )
% 0.30/0.47 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_not_empty
% 0.30/0.47 thf(fact_139_singleton__inject,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,B3: A] :
% 0.30/0.47 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 => ( A2 = B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_inject
% 0.30/0.47 thf(fact_140_bot_Oextremum__strict,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot.extremum_strict
% 0.30/0.47 thf(fact_141_bot_Onot__eq__extremum,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 != ( bot_bot @ A ) )
% 0.30/0.47 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot.not_eq_extremum
% 0.30/0.47 thf(fact_142_less__dag__Node_H,axiom,
% 0.30/0.47 ! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.30/0.47 = ( ( X = L )
% 0.30/0.47 | ( X = R )
% 0.30/0.47 | ( ord_less @ binDag_Mirabelle_dag @ X @ L )
% 0.30/0.47 | ( ord_less @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_dag_Node'
% 0.30/0.47 thf(fact_143_less__Node__dag,axiom,
% 0.30/0.47 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ X )
% 0.30/0.47 => ( ( ord_less @ binDag_Mirabelle_dag @ L @ X )
% 0.30/0.47 & ( ord_less @ binDag_Mirabelle_dag @ R @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_Node_dag
% 0.30/0.47 thf(fact_144_less__dag__def,axiom,
% 0.30/0.47 ( ( ord_less @ binDag_Mirabelle_dag )
% 0.30/0.47 = ( ^ [S3: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T2 @ S3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_dag_def
% 0.30/0.47 thf(fact_145_less__dag__Tip,axiom,
% 0.30/0.47 ! [X: binDag_Mirabelle_dag] :
% 0.30/0.47 ~ ( ord_less @ binDag_Mirabelle_dag @ X @ binDag_Mirabelle_Tip ) ).
% 0.30/0.47
% 0.30/0.47 % less_dag_Tip
% 0.30/0.47 thf(fact_146_bot__eq__sup__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ( bot_bot @ A )
% 0.30/0.47 = ( sup_sup @ A @ X @ Y ) )
% 0.30/0.47 = ( ( X
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 & ( Y
% 0.30/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_eq_sup_iff
% 0.30/0.47 thf(fact_147_sup__eq__bot__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ( sup_sup @ A @ X @ Y )
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 = ( ( X
% 0.30/0.47 = ( bot_bot @ A ) )
% 0.30/0.47 & ( Y
% 0.30/0.47 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_eq_bot_iff
% 0.30/0.47 thf(fact_148_sup__bot_Oleft__neutral,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.left_neutral
% 0.30/0.47 thf(fact_149_sup__bot_Oright__neutral,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.right_neutral
% 0.30/0.47 thf(fact_150_is__singleton__the__elem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( is_singleton @ A )
% 0.30/0.47 = ( ^ [A5: set @ A] :
% 0.30/0.47 ( A5
% 0.30/0.47 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_singleton_the_elem
% 0.30/0.47 thf(fact_151_sup_Oright__idem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.right_idem
% 0.30/0.47 thf(fact_152_sup__left__idem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.30/0.47 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_left_idem
% 0.30/0.47 thf(fact_153_sup__apply,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.30/0.47 => ( ( sup_sup @ ( A > B ) )
% 0.30/0.47 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_apply
% 0.30/0.47 thf(fact_154_sup_Oidem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( sup_sup @ A @ A2 @ A2 )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.idem
% 0.30/0.47 thf(fact_155_sup__idem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X @ X )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_idem
% 0.30/0.47 thf(fact_156_sup_Oleft__idem,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A] :
% 0.30/0.47 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.left_idem
% 0.30/0.47 thf(fact_157_the__elem__eq,axiom,
% 0.30/0.47 ! [A: $tType,X: A] :
% 0.30/0.47 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % the_elem_eq
% 0.30/0.47 thf(fact_158_not__psubset__empty,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_psubset_empty
% 0.30/0.47 thf(fact_159_inf__sup__aci_I8_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.30/0.47 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_aci(8)
% 0.30/0.47 thf(fact_160_inf__sup__aci_I7_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.30/0.47 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_aci(7)
% 0.30/0.47 thf(fact_161_inf__sup__aci_I6_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.30/0.47 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_aci(6)
% 0.30/0.47 thf(fact_162_inf__sup__aci_I5_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( sup_sup @ A )
% 0.30/0.47 = ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inf_sup_aci(5)
% 0.30/0.47 thf(fact_163_sup__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.30/0.47 => ( ( sup_sup @ ( A > B ) )
% 0.30/0.47 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_fun_def
% 0.30/0.47 thf(fact_164_sup_Oassoc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A,C: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.assoc
% 0.30/0.47 thf(fact_165_sup__assoc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.30/0.47 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_assoc
% 0.30/0.47 thf(fact_166_sup_Ocommute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( sup_sup @ A )
% 0.30/0.47 = ( ^ [A3: A,B2: A] : ( sup_sup @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.commute
% 0.30/0.47 thf(fact_167_sup__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( sup_sup @ A )
% 0.30/0.47 = ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_commute
% 0.30/0.47 thf(fact_168_sup_Oleft__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,A2: A,C: A] :
% 0.30/0.47 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C ) )
% 0.30/0.47 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.left_commute
% 0.30/0.47 thf(fact_169_sup__left__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.30/0.47 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_left_commute
% 0.30/0.47 thf(fact_170_less__DAG__set__of,axiom,
% 0.30/0.47 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
% 0.30/0.47 ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
% 0.30/0.47 => ( ( binDag_Mirabelle_DAG @ Y )
% 0.30/0.47 => ( ord_less @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_DAG_set_of
% 0.30/0.47 thf(fact_171_less__supI1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,A2: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ A2 )
% 0.30/0.47 => ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_supI1
% 0.30/0.47 thf(fact_172_less__supI2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,B3: A,A2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ B3 )
% 0.30/0.47 => ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_supI2
% 0.30/0.47 thf(fact_173_sup_Ostrict__boundedE,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,C: A,A2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C ) @ A2 )
% 0.30/0.47 => ~ ( ( ord_less @ A @ B3 @ A2 )
% 0.30/0.47 => ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.strict_boundedE
% 0.30/0.47 thf(fact_174_sup_Ostrict__order__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [B2: A,A3: A] :
% 0.30/0.47 ( ( A3
% 0.30/0.47 = ( sup_sup @ A @ A3 @ B2 ) )
% 0.30/0.47 & ( A3 != B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.strict_order_iff
% 0.30/0.47 thf(fact_175_sup_Ostrict__coboundedI1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [C: A,A2: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ C @ A2 )
% 0.30/0.47 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.strict_coboundedI1
% 0.30/0.47 thf(fact_176_sup_Ostrict__coboundedI2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [C: A,B3: A,A2: A] :
% 0.30/0.47 ( ( ord_less @ A @ C @ B3 )
% 0.30/0.47 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.strict_coboundedI2
% 0.30/0.47 thf(fact_177_sup__bot__left,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot_left
% 0.30/0.47 thf(fact_178_sup__bot__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot_right
% 0.30/0.47 thf(fact_179_sup__bot_Osemilattice__neutr__axioms,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.semilattice_neutr_axioms
% 0.30/0.47 thf(fact_180_sup__bot_Omonoid__axioms,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.monoid_axioms
% 0.30/0.47 thf(fact_181_minf_I11_J,axiom,
% 0.30/0.47 ! [C2: $tType,D2: $tType] :
% 0.30/0.47 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.30/0.47 => ! [F3: D2] :
% 0.30/0.47 ? [Z3: C2] :
% 0.30/0.47 ! [X5: C2] :
% 0.30/0.47 ( ( ord_less @ C2 @ X5 @ Z3 )
% 0.30/0.47 => ( F3 = F3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(11)
% 0.30/0.47 thf(fact_182_psubsetD,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( ( member @ A @ C @ A4 )
% 0.30/0.47 => ( member @ A @ C @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubsetD
% 0.30/0.47 thf(fact_183_psubset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_trans
% 0.30/0.47 thf(fact_184_pinf_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.30/0.47 ( ? [Z4: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.47 => ( ( P2 @ X4 )
% 0.30/0.47 = ( P3 @ X4 ) ) )
% 0.30/0.47 => ( ? [Z4: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.47 => ( ( Q2 @ X4 )
% 0.30/0.47 = ( Q3 @ X4 ) ) )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.47 => ( ( ( P2 @ X5 )
% 0.30/0.47 & ( Q2 @ X5 ) )
% 0.30/0.47 = ( ( P3 @ X5 )
% 0.30/0.47 & ( Q3 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(1)
% 0.30/0.47 thf(fact_185_pinf_I2_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.30/0.47 ( ? [Z4: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.47 => ( ( P2 @ X4 )
% 0.30/0.47 = ( P3 @ X4 ) ) )
% 0.30/0.47 => ( ? [Z4: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.47 => ( ( Q2 @ X4 )
% 0.30/0.47 = ( Q3 @ X4 ) ) )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.47 => ( ( ( P2 @ X5 )
% 0.30/0.47 | ( Q2 @ X5 ) )
% 0.30/0.47 = ( ( P3 @ X5 )
% 0.30/0.47 | ( Q3 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(2)
% 0.30/0.47 thf(fact_186_pinf_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.47 => ( X5 != T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(3)
% 0.30/0.47 thf(fact_187_pinf_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.47 => ( X5 != T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(4)
% 0.30/0.47 thf(fact_188_pinf_I5_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.47 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(5)
% 0.30/0.47 thf(fact_189_pinf_I7_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.47 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(7)
% 0.30/0.47 thf(fact_190_pinf_I11_J,axiom,
% 0.30/0.47 ! [C2: $tType,D2: $tType] :
% 0.30/0.47 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.30/0.47 => ! [F3: D2] :
% 0.30/0.47 ? [Z3: C2] :
% 0.30/0.47 ! [X5: C2] :
% 0.30/0.47 ( ( ord_less @ C2 @ Z3 @ X5 )
% 0.30/0.47 => ( F3 = F3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(11)
% 0.30/0.47 thf(fact_191_minf_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.30/0.47 ( ? [Z4: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.47 => ( ( P2 @ X4 )
% 0.30/0.47 = ( P3 @ X4 ) ) )
% 0.30/0.47 => ( ? [Z4: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.47 => ( ( Q2 @ X4 )
% 0.30/0.47 = ( Q3 @ X4 ) ) )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.47 => ( ( ( P2 @ X5 )
% 0.30/0.47 & ( Q2 @ X5 ) )
% 0.30/0.47 = ( ( P3 @ X5 )
% 0.30/0.47 & ( Q3 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(1)
% 0.30/0.47 thf(fact_192_minf_I2_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.30/0.47 ( ? [Z4: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.47 => ( ( P2 @ X4 )
% 0.30/0.47 = ( P3 @ X4 ) ) )
% 0.30/0.47 => ( ? [Z4: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.47 => ( ( Q2 @ X4 )
% 0.30/0.47 = ( Q3 @ X4 ) ) )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.47 => ( ( ( P2 @ X5 )
% 0.30/0.47 | ( Q2 @ X5 ) )
% 0.30/0.47 = ( ( P3 @ X5 )
% 0.30/0.47 | ( Q3 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(2)
% 0.30/0.47 thf(fact_193_minf_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.47 => ( X5 != T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(3)
% 0.30/0.47 thf(fact_194_minf_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.47 => ( X5 != T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(4)
% 0.30/0.47 thf(fact_195_minf_I5_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.47 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(5)
% 0.30/0.47 thf(fact_196_minf_I7_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.47 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(7)
% 0.30/0.47 thf(fact_197_ex__gt__or__lt,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ? [B7: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B7 )
% 0.30/0.47 | ( ord_less @ A @ B7 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ex_gt_or_lt
% 0.30/0.47 thf(fact_198_linorder__neqE__linordered__idom,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 => ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_neqE_linordered_idom
% 0.30/0.47 thf(fact_199_linordered__field__no__ub,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X5: A] :
% 0.30/0.47 ? [X1: A] : ( ord_less @ A @ X5 @ X1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % linordered_field_no_ub
% 0.30/0.47 thf(fact_200_linordered__field__no__lb,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X5: A] :
% 0.30/0.47 ? [Y3: A] : ( ord_less @ A @ Y3 @ X5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % linordered_field_no_lb
% 0.30/0.47 thf(fact_201_dependent__wellorder__choice,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P2: ( A > B ) > A > B > $o] :
% 0.30/0.47 ( ! [R4: B,F4: A > B,G3: A > B,X4: A] :
% 0.30/0.47 ( ! [Y4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Y4 @ X4 )
% 0.30/0.47 => ( ( F4 @ Y4 )
% 0.30/0.47 = ( G3 @ Y4 ) ) )
% 0.30/0.47 => ( ( P2 @ F4 @ X4 @ R4 )
% 0.30/0.47 = ( P2 @ G3 @ X4 @ R4 ) ) )
% 0.30/0.47 => ( ! [X4: A,F4: A > B] :
% 0.30/0.47 ( ! [Y4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Y4 @ X4 )
% 0.30/0.47 => ( P2 @ F4 @ Y4 @ ( F4 @ Y4 ) ) )
% 0.30/0.47 => ? [X12: B] : ( P2 @ F4 @ X4 @ X12 ) )
% 0.30/0.47 => ? [F4: A > B] :
% 0.30/0.47 ! [X5: A] : ( P2 @ F4 @ X5 @ ( F4 @ X5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dependent_wellorder_choice
% 0.30/0.47 thf(fact_202_in__inv__imagep,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( inv_imagep @ A @ B )
% 0.30/0.47 = ( ^ [R2: A > A > $o,F2: B > A,X2: B,Y2: B] : ( R2 @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_inv_imagep
% 0.30/0.47 thf(fact_203_pred__on_Ochain__extend,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,P2: A > A > $o,C3: set @ A,Z: A] :
% 0.30/0.47 ( ( pred_chain @ A @ A4 @ P2 @ C3 )
% 0.30/0.47 => ( ( member @ A @ Z @ A4 )
% 0.30/0.47 => ( ! [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ C3 )
% 0.30/0.47 => ( sup_sup @ ( A > A > $o ) @ P2
% 0.30/0.47 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 )
% 0.30/0.47 @ X4
% 0.30/0.47 @ Z ) )
% 0.30/0.47 => ( pred_chain @ A @ A4 @ P2 @ ( sup_sup @ ( set @ A ) @ ( insert @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_on.chain_extend
% 0.30/0.47 thf(fact_204_sup__bot_Ocomm__monoid__axioms,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_bot.comm_monoid_axioms
% 0.30/0.47 thf(fact_205_sup2CI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A4: A > B > $o] :
% 0.30/0.47 ( ( ~ ( B4 @ X @ Y )
% 0.30/0.47 => ( A4 @ X @ Y ) )
% 0.30/0.47 => ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup2CI
% 0.30/0.47 thf(fact_206_pred__on_Ochain__empty,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,P2: A > A > $o] : ( pred_chain @ A @ A4 @ P2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_on.chain_empty
% 0.30/0.47 thf(fact_207_subset_Ochain__total,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),X: set @ A,Y: set @ A] :
% 0.30/0.47 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 0.30/0.47 => ( ( member @ ( set @ A ) @ X @ C3 )
% 0.30/0.47 => ( ( member @ ( set @ A ) @ Y @ C3 )
% 0.30/0.47 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.47 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 0.30/0.47 @ X
% 0.30/0.47 @ Y )
% 0.30/0.47 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.47 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 0.30/0.47 @ Y
% 0.30/0.47 @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset.chain_total
% 0.30/0.47 thf(fact_208_pred__on_Ochain__total,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,P2: A > A > $o,C3: set @ A,X: A,Y: A] :
% 0.30/0.47 ( ( pred_chain @ A @ A4 @ P2 @ C3 )
% 0.30/0.47 => ( ( member @ A @ X @ C3 )
% 0.30/0.47 => ( ( member @ A @ Y @ C3 )
% 0.30/0.47 => ( ( sup_sup @ ( A > A > $o ) @ P2
% 0.30/0.47 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 )
% 0.30/0.47 @ X
% 0.30/0.47 @ Y )
% 0.30/0.47 | ( sup_sup @ ( A > A > $o ) @ P2
% 0.30/0.47 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 )
% 0.30/0.47 @ Y
% 0.30/0.47 @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_on.chain_total
% 0.30/0.47 thf(fact_209_subset_Ochain__empty,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset.chain_empty
% 0.30/0.47 thf(fact_210_subset_Ochain__extend,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),Z: set @ A] :
% 0.30/0.47 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 0.30/0.47 => ( ( member @ ( set @ A ) @ Z @ A4 )
% 0.30/0.47 => ( ! [X4: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ X4 @ C3 )
% 0.30/0.47 => ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.47 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 0.30/0.47 @ X4
% 0.30/0.47 @ Z ) )
% 0.30/0.47 => ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset.chain_extend
% 0.30/0.47 thf(fact_211_sup2I2,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A4: A > B > $o] :
% 0.30/0.47 ( ( B4 @ X @ Y )
% 0.30/0.47 => ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup2I2
% 0.30/0.47 thf(fact_212_sup2I1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: B,B4: A > B > $o] :
% 0.30/0.47 ( ( A4 @ X @ Y )
% 0.30/0.47 => ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup2I1
% 0.30/0.47 thf(fact_213_sup2E,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A4: A > B > $o,B4: A > B > $o,X: A,Y: B] :
% 0.30/0.47 ( ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y )
% 0.30/0.47 => ( ~ ( A4 @ X @ Y )
% 0.30/0.47 => ( B4 @ X @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup2E
% 0.30/0.47 thf(fact_214_semilattice__neutr_Oaxioms_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,F: A > A > A,Z: A] :
% 0.30/0.47 ( ( semilattice_neutr @ A @ F @ Z )
% 0.30/0.47 => ( comm_monoid @ A @ F @ Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % semilattice_neutr.axioms(2)
% 0.30/0.47 thf(fact_215_reflclp__idemp,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > A > $o] :
% 0.30/0.47 ( ( sup_sup @ ( A > A > $o )
% 0.30/0.47 @ ( sup_sup @ ( A > A > $o ) @ P2
% 0.30/0.47 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 ) )
% 0.30/0.47 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 ) )
% 0.30/0.47 = ( sup_sup @ ( A > A > $o ) @ P2
% 0.30/0.47 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % reflclp_idemp
% 0.30/0.47 thf(fact_216_pred__on_Omaxchain__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( pred_maxchain @ A )
% 0.30/0.47 = ( ^ [A5: set @ A,P4: A > A > $o,C4: set @ A] :
% 0.30/0.47 ( ( pred_chain @ A @ A5 @ P4 @ C4 )
% 0.30/0.47 & ~ ? [S2: set @ A] :
% 0.30/0.47 ( ( pred_chain @ A @ A5 @ P4 @ S2 )
% 0.30/0.47 & ( ord_less @ ( set @ A ) @ C4 @ S2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_on.maxchain_def
% 0.30/0.47 thf(fact_217_subset_Omaxchain__def,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
% 0.30/0.47 ( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 0.30/0.47 = ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 0.30/0.47 & ~ ? [S2: set @ ( set @ A )] :
% 0.30/0.47 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ S2 )
% 0.30/0.47 & ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ S2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset.maxchain_def
% 0.30/0.47 thf(fact_218_subset_Omaxchain__imp__chain,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
% 0.30/0.47 ( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 0.30/0.47 => ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset.maxchain_imp_chain
% 0.30/0.47 thf(fact_219_subset_OHausdorff,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ ( set @ A )] :
% 0.30/0.47 ? [X1: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X1 ) ).
% 0.30/0.47
% 0.30/0.47 % subset.Hausdorff
% 0.30/0.47 thf(fact_220_Pow__singleton__iff,axiom,
% 0.30/0.47 ! [A: $tType,X6: set @ A,Y6: set @ A] :
% 0.30/0.47 ( ( ( pow @ A @ X6 )
% 0.30/0.47 = ( insert @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
% 0.30/0.47 = ( ( X6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 & ( Y6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pow_singleton_iff
% 0.30/0.47 thf(fact_221_Pow__empty,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pow_empty
% 0.30/0.47 thf(fact_222_Pow__bottom,axiom,
% 0.30/0.47 ! [A: $tType,B4: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pow_bottom
% 0.30/0.47 thf(fact_223_Pow__not__empty,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( pow @ A @ A4 )
% 0.30/0.47 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pow_not_empty
% 0.30/0.47 thf(fact_224_Pow__top,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ A4 @ ( pow @ A @ A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pow_top
% 0.30/0.47 thf(fact_225_subset_Ochain__def,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
% 0.30/0.47 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 0.30/0.47 = ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A4 )
% 0.30/0.47 & ! [X2: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ X2 @ C3 )
% 0.30/0.47 => ! [Y2: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ Y2 @ C3 )
% 0.30/0.47 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.47 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 0.30/0.47 @ X2
% 0.30/0.47 @ Y2 )
% 0.30/0.47 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.47 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 0.30/0.47 @ Y2
% 0.30/0.47 @ X2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset.chain_def
% 0.30/0.47 thf(fact_226_subset_OchainI,axiom,
% 0.30/0.47 ! [A: $tType,C3: set @ ( set @ A ),A4: set @ ( set @ A )] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A4 )
% 0.30/0.47 => ( ! [X4: set @ A,Y3: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ X4 @ C3 )
% 0.30/0.47 => ( ( member @ ( set @ A ) @ Y3 @ C3 )
% 0.30/0.47 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.47 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 0.30/0.47 @ X4
% 0.30/0.47 @ Y3 )
% 0.30/0.47 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.47 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 0.30/0.47 @ Y3
% 0.30/0.47 @ X4 ) ) ) )
% 0.30/0.47 => ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset.chainI
% 0.30/0.47 thf(fact_227_order__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_refl
% 0.30/0.47 thf(fact_228_subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ! [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ A4 )
% 0.30/0.47 => ( member @ A @ X4 @ B4 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetI
% 0.30/0.47 thf(fact_229_subset__antisym,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.30/0.47 => ( A4 = B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_antisym
% 0.30/0.47 thf(fact_230_le__sup__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.30/0.47 = ( ( ord_less_eq @ A @ X @ Z )
% 0.30/0.47 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_sup_iff
% 0.30/0.47 thf(fact_231_sup_Obounded__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,C: A,A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C ) @ A2 )
% 0.30/0.47 = ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.30/0.47 & ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup.bounded_iff
% 0.30/0.47 thf(fact_232_empty__subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.30/0.47
% 0.30/0.47 % empty_subsetI
% 0.30/0.47 thf(fact_233_subset__empty,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( A4
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_empty
% 0.30/0.47 thf(fact_234_insert__subset,axiom,
% 0.30/0.47 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B4 )
% 0.30/0.47 = ( ( member @ A @ X @ B4 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % insert_subset
% 0.30/0.47 thf(fact_235_Un__subset__iff,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
% 0.30/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Un_subset_iff
% 0.30/0.47 thf(fact_236_psubsetI,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( ( A4 != B4 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubsetI
% 0.30/0.47 thf(fact_237_Pow__iff,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B4 ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pow_iff
% 0.30/0.47 thf(fact_238_PowI,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % PowI
% 0.30/0.47 thf(fact_239_singleton__insert__inj__eq,axiom,
% 0.30/0.47 ! [A: $tType,B3: A,A2: A,A4: set @ A] :
% 0.30/0.47 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( insert @ A @ A2 @ A4 ) )
% 0.30/0.47 = ( ( A2 = B3 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_insert_inj_eq
% 0.30/0.47 thf(fact_240_singleton__insert__inj__eq_H,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,A4: set @ A,B3: A] :
% 0.30/0.47 ( ( ( insert @ A @ A2 @ A4 )
% 0.30/0.47 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 = ( ( A2 = B3 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % singleton_insert_inj_eq'
% 0.30/0.47 thf(fact_241_subset__Zorn,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ ( set @ A )] :
% 0.30/0.47 ( ! [C5: set @ ( set @ A )] :
% 0.30/0.47 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
% 0.30/0.47 => ? [X5: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ X5 @ A4 )
% 0.30/0.47 & ! [Xa: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ Xa @ C5 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ Xa @ X5 ) ) ) )
% 0.30/0.47 => ? [X4: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ X4 @ A4 )
% 0.30/0.47 & ! [Xa2: set @ A] :
% 0.30/0.47 ( ( member @ ( set @ A ) @ Xa2 @ A4 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ Xa2 )
% 0.30/0.47 => ( Xa2 = X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_Zorn
% 0.30/0.47 thf(fact_242_psubsetE,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubsetE
% 0.30/0.47 thf(fact_243_psubset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.30/0.47 & ( A5 != B5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_eq
% 0.30/0.47 thf(fact_244_psubset__imp__subset,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_imp_subset
% 0.30/0.47 thf(fact_245_psubset__subset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_subset_trans
% 0.30/0.47 thf(fact_246_subset__not__subset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.30/0.47 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_not_subset_eq
% 0.30/0.47 thf(fact_247_subset__psubset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_psubset_trans
% 0.30/0.47 thf(fact_248_subset__iff__psubset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
% 0.30/0.47 | ( A5 = B5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_iff_psubset_eq
% 0.30/0.47 thf(fact_249_pinf_I6_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(6)
% 0.30/0.47 thf(fact_250_pinf_I8_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.47 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(8)
% 0.30/0.47 thf(fact_251_minf_I6_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.47 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(6)
% 0.30/0.47 thf(fact_252_minf_I8_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(8)
% 0.30/0.47 thf(fact_253_order_Onot__eq__order__implies__strict,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B3: A] :
% 0.30/0.47 ( ( A2 != B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.30/0.47 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.not_eq_order_implies_strict
% 0.30/0.47 thf(fact_254_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,A2: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.30/0.47 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_implies_order
% 0.30/0.47 thf(fact_255_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [B2: A,A3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.47 & ( A3 != B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_iff_order
% 0.30/0.47
% 0.30/0.47 %----Type constructors (34)
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.30/0.47 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.30/0.47 ! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( bounde1808546759up_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( bounded_lattice_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( semilattice_sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( order_bot @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( order_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( lattice @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.47 ! [A6: $tType,A7: $tType] :
% 0.30/0.47 ( ( bot @ A7 @ ( type2 @ A7 ) )
% 0.30/0.47 => ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
% 0.30/0.47 ! [A6: $tType] : ( bounde1808546759up_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
% 0.30/0.47 ! [A6: $tType] : ( bounded_lattice_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
% 0.30/0.47 ! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 0.30/0.47 ! [A6: $tType] : ( order_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.30/0.47 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
% 0.30/0.47 ! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.30/0.47 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 0.30/0.47 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 0.30/0.47 ! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12,axiom,
% 0.30/0.47 bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_13,axiom,
% 0.30/0.47 bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
% 0.30/0.47 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__bot_15,axiom,
% 0.30/0.47 order_bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
% 0.30/0.47 preorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.30/0.47 linorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
% 0.30/0.47 lattice @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
% 0.30/0.47 order @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
% 0.30/0.47 ord @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
% 0.30/0.47 bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_21,axiom,
% 0.30/0.47 preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_22,axiom,
% 0.30/0.47 order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_23,axiom,
% 0.30/0.47 ord @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.30/0.47
% 0.30/0.47 %----Helper facts (3)
% 0.30/0.47 thf(help_If_3_1_T,axiom,
% 0.30/0.47 ! [P2: $o] :
% 0.30/0.47 ( ( P2 = $true )
% 0.30/0.47 | ( P2 = $false ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_2_1_T,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_1_1_T,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 %----Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 ( ( binDag_Mirabelle_Dag @ p @ ( fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p ) @ r @ t )
% 0.30/0.47 = ( ( p = simpl_Null )
% 0.30/0.47 & ( t = binDag_Mirabelle_Tip ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.94/1.53 ---- Embedded in THF ---
% 2.94/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.94/1.53 %%% Generated on Tue Feb 24 04:15:43 EST 2026
% 2.94/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.94/1.53 %%% Logic specification used:
% 2.94/1.53 %%% thf(spec, logic, $$dhol).
% 2.94/1.53
% 2.94/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.vZxap9H0LD/DTF2DTF19524.p
% 2.94/1.53 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag, type, binDag_Mirabelle_dag: $tType).
% 2.94/1.53 thf(ty_t_Simpl__Heap_Oref, type, simpl_ref: $tType).
% 2.94/1.53 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.94/1.53 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.94/1.53 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde1808546759up_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.94/1.53 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG, type, binDag_Mirabelle_DAG: (binDag_Mirabelle_dag > $o)).
% 2.94/1.53 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODag, type, binDag_Mirabelle_Dag: (simpl_ref > ((simpl_ref > simpl_ref) > ((simpl_ref > simpl_ref) > (binDag_Mirabelle_dag > $o))))).
% 2.94/1.53 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode, type, binDag476092410e_Node: (binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > binDag_Mirabelle_dag)))).
% 2.94/1.53 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip, type, binDag_Mirabelle_Tip: binDag_Mirabelle_dag).
% 2.94/1.53 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag, type, binDag1297733282se_dag: (!> [A:$tType]: ((A > ((binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A))) > (binDag_Mirabelle_dag > A)))))).
% 2.94/1.53 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag, type, binDag1442713106ec_dag: (!> [A:$tType]: ((A > ((binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A))))) > (binDag_Mirabelle_dag > A)))))).
% 2.94/1.53 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of, type, binDag1380252983set_of: (binDag_Mirabelle_dag > (set @ simpl_ref))).
% 2.94/1.53 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag, type, binDag786255756subdag: (binDag_Mirabelle_dag > (binDag_Mirabelle_dag > $o))).
% 2.94/1.53 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))).
% 2.94/1.53 thf(sy_c_Fun_Ooverride__on, type, override_on: (!> [A:$tType,B:$tType]: (((A > B) > ((A > B) > ((set @ A) > (A > B))))))).
% 2.94/1.53 thf(sy_c_Groups_Ocomm__monoid, type, comm_monoid: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 2.94/1.53 thf(sy_c_Groups_Omonoid, type, monoid: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 2.94/1.53 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.94/1.53 thf(sy_c_Lattices_Osemilattice__neutr, type, semilattice_neutr: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 2.94/1.53 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.94/1.53 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.94/1.53 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.94/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.94/1.53 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.94/1.53 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.94/1.53 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.94/1.53 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.94/1.53 thf(sy_c_Set_Obind, type, bind: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ B)))))).
% 2.94/1.53 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.94/1.53 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.94/1.53 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.94/1.53 thf(sy_c_Set_Opairwise, type, pairwise: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.94/1.53 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.94/1.53 thf(sy_c_Simpl__Heap_ONull, type, simpl_Null: simpl_ref).
% 2.94/1.53 thf(sy_c_Zorn_Opred__on_Ochain, type, pred_chain: (!> [A:$tType]: (((set @ A) > ((A > (A > $o)) > ((set @ A) > $o)))))).
% 2.94/1.53 thf(sy_c_Zorn_Opred__on_Omaxchain, type, pred_maxchain: (!> [A:$tType]: (((set @ A) > ((A > (A > $o)) > ((set @ A) > $o)))))).
% 2.94/1.53 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.94/1.53 thf(sy_v_l, type, l: (simpl_ref > simpl_ref)).
% 2.94/1.53 thf(sy_v_p, type, p: simpl_ref).
% 2.94/1.53 thf(sy_v_r, type, r: (simpl_ref > simpl_ref)).
% 2.94/1.53 thf(sy_v_t, type, t: binDag_Mirabelle_dag).
% 2.94/1.53 thf(fact_0_Dag__Null, axiom, (! [L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ simpl_Null) @ L) @ R) @ T) = (T = binDag_Mirabelle_Tip))))).
% 2.94/1.53 thf(fact_1_Dag_Osimps_I1_J, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref)]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ binDag_Mirabelle_Tip) = (P = simpl_Null))))).
% 2.94/1.53 thf(fact_2_Dag__upd__same__l__lemma, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: (((P != simpl_Null) => (~ (((((binDag_Mirabelle_Dag @ P) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ L) @ P) @ P)) @ R) @ T))))))).
% 2.94/1.53 thf(fact_3_fun__upd__upd, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,Z:B]: (((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ X) @ Y)) @ X) @ Z) = (((((fun_upd @ A) @ B) @ F) @ X) @ Z))))).
% 2.94/1.53 thf(fact_4_fun__upd__triv, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: (((((((fun_upd @ A) @ B) @ F) @ X) @ (F @ X)) = F)))).
% 2.94/1.53 thf(fact_5_fun__upd__apply, axiom, (! [A:$tType,B:$tType]: ((((fun_upd @ B) @ A) = (^ [F2:(B > A),X2:B,Y2:A,Z2:B]: (((((if @ A) @ (Z2 = X2)) @ Y2) @ (F2 @ Z2)))))))).
% 2.94/1.53 thf(fact_6_notin__Dag__update__l, axiom, (! [Q:simpl_ref,T:binDag_Mirabelle_dag,P:simpl_ref,L:(simpl_ref > simpl_ref),Y:simpl_ref,R:(simpl_ref > simpl_ref)]: (((~ ((((member @ simpl_ref) @ Q) @ (binDag1380252983set_of @ T)))) => (((((binDag_Mirabelle_Dag @ P) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ L) @ Q) @ Y)) @ R) @ T) = ((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T)))))).
% 2.94/1.53 thf(fact_7_notin__Dag__update__r, axiom, (! [Q:simpl_ref,T:binDag_Mirabelle_dag,P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),Y:simpl_ref]: (((~ ((((member @ simpl_ref) @ Q) @ (binDag1380252983set_of @ T)))) => (((((binDag_Mirabelle_Dag @ P) @ L) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ R) @ Q) @ Y)) @ T) = ((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T)))))).
% 2.94/1.53 thf(fact_8_Null__notin__Dag, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T) => (~ ((((member @ simpl_ref) @ simpl_Null) @ (binDag1380252983set_of @ T)))))))).
% 2.94/1.53 thf(fact_9_Dag__Ref, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: (((P != simpl_Null) => (((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T) = (? [Lt:binDag_Mirabelle_dag,Rt:binDag_Mirabelle_dag]: (((T = (((binDag476092410e_Node @ Lt) @ P) @ Rt)) & (((((binDag_Mirabelle_Dag @ (L @ P)) @ L) @ R) @ Lt) & ((((binDag_Mirabelle_Dag @ (R @ P)) @ L) @ R) @ Rt)))))))))).
% 2.94/1.53 thf(fact_10_Dag_Osimps_I2_J, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),Lt2:binDag_Mirabelle_dag,A2:simpl_ref,Rt2:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ (((binDag476092410e_Node @ Lt2) @ A2) @ Rt2)) = ((P = A2) & ((P != simpl_Null) & (((((binDag_Mirabelle_Dag @ (L @ P)) @ L) @ R) @ Lt2) & ((((binDag_Mirabelle_Dag @ (R @ P)) @ L) @ R) @ Rt2)))))))).
% 2.94/1.53 thf(fact_11_fun__upd__def, axiom, (! [B:$tType,A:$tType]: ((((fun_upd @ A) @ B) = (^ [F2:(A > B),A3:A,B2:B,X2:A]: (((((if @ B) @ (X2 = A3)) @ B2) @ (F2 @ X2)))))))).
% 2.94/1.53 thf(fact_12_fun__upd__eqD, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,G:(A > B),Z:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = (((((fun_upd @ A) @ B) @ G) @ X) @ Z)) => (Y = Z))))).
% 2.94/1.53 thf(fact_13_fun__upd__idem, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B,Y:A]: ((((F @ X) = Y) => ((((((fun_upd @ B) @ A) @ F) @ X) @ Y) = F))))).
% 2.94/1.53 thf(fact_14_fun__upd__same, axiom, (! [B:$tType,A:$tType,F:(B > A),X:B,Y:A]: ((((((((fun_upd @ B) @ A) @ F) @ X) @ Y) @ X) = Y)))).
% 2.94/1.53 thf(fact_15_dag_Oinject, axiom, (! [X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag,Y21:binDag_Mirabelle_dag,Y22:simpl_ref,Y23:binDag_Mirabelle_dag]: ((((((binDag476092410e_Node @ X21) @ X22) @ X23) = (((binDag476092410e_Node @ Y21) @ Y22) @ Y23)) = ((X21 = Y21) & ((X22 = Y22) & (X23 = Y23))))))).
% 2.94/1.53 thf(fact_16_dag_Oexhaust, axiom, (! [Y:binDag_Mirabelle_dag]: (((Y != binDag_Mirabelle_Tip) => (~ ((! [X212:binDag_Mirabelle_dag,X222:simpl_ref,X232:binDag_Mirabelle_dag]: ((Y != (((binDag476092410e_Node @ X212) @ X222) @ X232)))))))))).
% 2.94/1.53 thf(fact_17_dag_Oinduct, axiom, (! [P2:(binDag_Mirabelle_dag > $o),Dag:binDag_Mirabelle_dag]: (((P2 @ binDag_Mirabelle_Tip) => ((! [X1:binDag_Mirabelle_dag,X24:simpl_ref,X3:binDag_Mirabelle_dag]: (((P2 @ X1) => ((P2 @ X3) => (P2 @ (((binDag476092410e_Node @ X1) @ X24) @ X3)))))) => (P2 @ Dag)))))).
% 2.94/1.53 thf(fact_18_dag_Odistinct_I1_J, axiom, (! [X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((binDag_Mirabelle_Tip != (((binDag476092410e_Node @ X21) @ X22) @ X23))))).
% 2.94/1.53 thf(fact_19_fun__upd__idem__iff, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = F) = ((F @ X) = Y))))).
% 2.94/1.53 thf(fact_20_fun__upd__twist, axiom, (! [A:$tType,B:$tType,A2:A,C:A,M:(A > B),B3:B,D:B]: (((A2 != C) => ((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M) @ A2) @ B3)) @ C) @ D) = (((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M) @ C) @ D)) @ A2) @ B3)))))).
% 2.94/1.53 thf(fact_21_fun__upd__other, axiom, (! [B:$tType,A:$tType,Z:A,X:A,F:(A > B),Y:B]: (((Z != X) => (((((((fun_upd @ A) @ B) @ F) @ X) @ Y) @ Z) = (F @ Z)))))).
% 2.94/1.53 thf(fact_22_DAG_Osimps_I2_J, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((binDag_Mirabelle_DAG @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((~ ((((member @ simpl_ref) @ A2) @ (binDag1380252983set_of @ L)))) & ((~ ((((member @ simpl_ref) @ A2) @ (binDag1380252983set_of @ R)))) & ((binDag_Mirabelle_DAG @ L) & (binDag_Mirabelle_DAG @ R)))))))).
% 2.94/1.53 thf(fact_23_in__set__of__decomp, axiom, (! [P:simpl_ref,T:binDag_Mirabelle_dag]: (((((member @ simpl_ref) @ P) @ (binDag1380252983set_of @ T)) = (? [L2:binDag_Mirabelle_dag,R2:binDag_Mirabelle_dag]: (((T = (((binDag476092410e_Node @ L2) @ P) @ R2)) | ((binDag786255756subdag @ T) @ (((binDag476092410e_Node @ L2) @ P) @ R2))))))))).
% 2.94/1.53 thf(fact_24_dag_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F22:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A)))]: ((((((binDag1297733282se_dag @ A) @ F1) @ F22) @ binDag_Mirabelle_Tip) = F1)))).
% 2.94/1.53 thf(fact_25_dag_Osimps_I6_J, axiom, (! [A:$tType,F1:A,F22:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A)))))]: ((((((binDag1442713106ec_dag @ A) @ F1) @ F22) @ binDag_Mirabelle_Tip) = F1)))).
% 2.94/1.53 thf(fact_26_DAG_Osimps_I1_J, axiom, (binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip)).
% 2.94/1.53 thf(fact_27_dag_Osimps_I7_J, axiom, (! [A:$tType,F1:A,F22:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A))))),X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((((((binDag1442713106ec_dag @ A) @ F1) @ F22) @ (((binDag476092410e_Node @ X21) @ X22) @ X23)) = (((((F22 @ X21) @ X22) @ X23) @ ((((binDag1442713106ec_dag @ A) @ F1) @ F22) @ X21)) @ ((((binDag1442713106ec_dag @ A) @ F1) @ F22) @ X23)))))).
% 2.94/1.53 thf(fact_28_dag_Osimps_I5_J, axiom, (! [A:$tType,F1:A,F22:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A))),X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((((((binDag1297733282se_dag @ A) @ F1) @ F22) @ (((binDag476092410e_Node @ X21) @ X22) @ X23)) = (((F22 @ X21) @ X22) @ X23))))).
% 2.94/1.53 thf(fact_29_set__of__Tip, axiom, ((binDag1380252983set_of @ binDag_Mirabelle_Tip) = (bot_bot @ (set @ simpl_ref)))).
% 2.94/1.53 thf(fact_30_subdag_Osimps_I1_J, axiom, (! [T:binDag_Mirabelle_dag]: ((~ (((binDag786255756subdag @ binDag_Mirabelle_Tip) @ T)))))).
% 2.94/1.53 thf(fact_31_subdag_Osimps_I2_J, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag,T:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ (((binDag476092410e_Node @ L) @ A2) @ R)) @ T) = ((T = L) | ((T = R) | (((binDag786255756subdag @ L) @ T) | ((binDag786255756subdag @ R) @ T)))))))).
% 2.94/1.53 thf(fact_32_subdag__not__sym, axiom, (! [S:binDag_Mirabelle_dag,T:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ S) @ T) => (~ (((binDag786255756subdag @ T) @ S))))))).
% 2.94/1.53 thf(fact_33_subdag__trans, axiom, (! [T:binDag_Mirabelle_dag,S:binDag_Mirabelle_dag,R:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ S) => (((binDag786255756subdag @ S) @ R) => ((binDag786255756subdag @ T) @ R)))))).
% 2.94/1.53 thf(fact_34_subdag__neq, axiom, (! [T:binDag_Mirabelle_dag,S:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ S) => (T != S))))).
% 2.94/1.53 thf(fact_35_subdag__NodeD, axiom, (! [T:binDag_Mirabelle_dag,Lt2:binDag_Mirabelle_dag,A2:simpl_ref,Rt2:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ (((binDag476092410e_Node @ Lt2) @ A2) @ Rt2)) => (((binDag786255756subdag @ T) @ Lt2) & ((binDag786255756subdag @ T) @ Rt2)))))).
% 2.94/1.53 thf(fact_36_empty__Collect__eq, axiom, (! [A:$tType,P2:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P2)) = (! [X2:A]: ((~ ((P2 @ X2))))))))).
% 2.94/1.53 thf(fact_37_Collect__empty__eq, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (! [X2:A]: ((~ ((P2 @ X2))))))))).
% 2.94/1.53 thf(fact_38_all__not__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A4))))) = (A4 = (bot_bot @ (set @ A))))))).
% 2.94/1.53 thf(fact_39_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 2.94/1.53 thf(fact_40_bot__apply, axiom, (! [C2:$tType,D2:$tType]: ((((bot @ C2) @ (type2 @ C2)) => ((bot_bot @ (D2 > C2)) = (^ [X2:D2]: ((bot_bot @ C2)))))))).
% 2.94/1.53 thf(fact_41_override__on__emptyset, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((((((override_on @ A) @ B) @ F) @ G) @ (bot_bot @ (set @ A))) = F)))).
% 2.94/1.53 thf(fact_42_ex__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ A4))) = (A4 != (bot_bot @ (set @ A))))))).
% 2.94/1.53 thf(fact_43_equals0I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A4))))) => (A4 = (bot_bot @ (set @ A))))))).
% 2.94/1.53 thf(fact_44_equals0D, axiom, (! [A:$tType,A4:(set @ A),A2:A]: (((A4 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A4))))))).
% 2.94/1.53 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 2.94/1.53 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A4)))) = A4)))).
% 2.94/1.53 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q2:(A > $o)]: (((! [X4:A]: (((P2 @ X4) = (Q2 @ X4)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q2)))))).
% 2.94/1.53 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.94/1.53 thf(fact_49_override__on__apply__in, axiom, (! [B:$tType,A:$tType,A2:A,A4:(set @ A),F:(A > B),G:(A > B)]: (((((member @ A) @ A2) @ A4) => (((((((override_on @ A) @ B) @ F) @ G) @ A4) @ A2) = (G @ A2)))))).
% 2.94/1.53 thf(fact_50_override__on__apply__notin, axiom, (! [B:$tType,A:$tType,A2:A,A4:(set @ A),F:(A > B),G:(A > B)]: (((~ ((((member @ A) @ A2) @ A4))) => (((((((override_on @ A) @ B) @ F) @ G) @ A4) @ A2) = (F @ A2)))))).
% 2.94/1.53 thf(fact_51_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.94/1.53 thf(fact_52_override__on__def, axiom, (! [B:$tType,A:$tType]: ((((override_on @ A) @ B) = (^ [F2:(A > B),G2:(A > B),A5:(set @ A),A3:A]: (((((if @ B) @ (((member @ A) @ A3) @ A5)) @ (G2 @ A3)) @ (F2 @ A3)))))))).
% 2.94/1.53 thf(fact_53_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X2:A]: ((bot_bot @ B)))))))).
% 2.94/1.53 thf(fact_54_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.94/1.53 thf(fact_55_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 2.94/1.53 thf(fact_56_Collect__empty__eq__bot, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (P2 = (bot_bot @ (A > $o))))))).
% 2.94/1.53 thf(fact_57_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.94/1.53 thf(fact_58_is__singletonI_H, axiom, (! [A:$tType,A4:(set @ A)]: (((A4 != (bot_bot @ (set @ A))) => ((! [X4:A,Y3:A]: (((((member @ A) @ X4) @ A4) => ((((member @ A) @ Y3) @ A4) => (X4 = Y3))))) => ((is_singleton @ A) @ A4)))))).
% 2.94/1.53 thf(fact_59_empty__bind, axiom, (! [B:$tType,A:$tType,F:(B > (set @ A))]: ((((((bind @ B) @ A) @ (bot_bot @ (set @ B))) @ F) = (bot_bot @ (set @ A)))))).
% 2.94/1.53 thf(fact_60_set__of__Node, axiom, (! [Lt2:binDag_Mirabelle_dag,A2:simpl_ref,Rt2:binDag_Mirabelle_dag]: (((binDag1380252983set_of @ (((binDag476092410e_Node @ Lt2) @ A2) @ Rt2)) = (((sup_sup @ (set @ simpl_ref)) @ (((sup_sup @ (set @ simpl_ref)) @ (((insert @ simpl_ref) @ A2) @ (bot_bot @ (set @ simpl_ref)))) @ (binDag1380252983set_of @ Lt2))) @ (binDag1380252983set_of @ Rt2)))))).
% 2.94/1.53 thf(fact_61_pairwise__empty, axiom, (! [A:$tType,P2:(A > (A > $o))]: ((((pairwise @ A) @ P2) @ (bot_bot @ (set @ A)))))).
% 2.94/1.53 thf(fact_62_DAG__less, axiom, (! [Y:binDag_Mirabelle_dag,X:binDag_Mirabelle_dag]: (((binDag_Mirabelle_DAG @ Y) => ((((ord_less @ binDag_Mirabelle_dag) @ X) @ Y) => (binDag_Mirabelle_DAG @ X)))))).
% 2.94/1.53 thf(fact_63_insertCI, axiom, (! [A:$tType,A2:A,B4:(set @ A),B3:A]: ((((~ ((((member @ A) @ A2) @ B4))) => (A2 = B3)) => (((member @ A) @ A2) @ (((insert @ A) @ B3) @ B4)))))).
% 2.94/1.53 thf(fact_64_insert__iff, axiom, (! [A:$tType,A2:A,B3:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B3) @ A4)) = ((A2 = B3) | (((member @ A) @ A2) @ A4)))))).
% 2.94/1.53 thf(fact_65_insert__absorb2, axiom, (! [A:$tType,X:A,A4:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A4)) = (((insert @ A) @ X) @ A4))))).
% 2.94/1.53 thf(fact_66_UnCI, axiom, (! [A:$tType,C:A,B4:(set @ A),A4:(set @ A)]: ((((~ ((((member @ A) @ C) @ B4))) => (((member @ A) @ C) @ A4)) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B4)))))).
% 2.94/1.53 thf(fact_67_Un__iff, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) = ((((member @ A) @ C) @ A4) | (((member @ A) @ C) @ B4)))))).
% 2.94/1.53 thf(fact_68_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.94/1.53 thf(fact_69_Un__empty, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A4) @ B4) = (bot_bot @ (set @ A))) = ((A4 = (bot_bot @ (set @ A))) & (B4 = (bot_bot @ (set @ A)))))))).
% 2.94/1.53 thf(fact_70_Un__insert__left, axiom, (! [A:$tType,A2:A,B4:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A2) @ B4)) @ C3) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ B4) @ C3)))))).
% 2.94/1.53 thf(fact_71_Un__insert__right, axiom, (! [A:$tType,A4:(set @ A),A2:A,B4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ B4)) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ A4) @ B4)))))).
% 2.94/1.53 thf(fact_72_is__singletonI, axiom, (! [A:$tType,X:A]: (((is_singleton @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))).
% 2.94/1.53 thf(fact_73_pairwise__singleton, axiom, (! [A:$tType,P2:(A > (A > $o)),A4:A]: ((((pairwise @ A) @ P2) @ (((insert @ A) @ A4) @ (bot_bot @ (set @ A))))))).
% 2.94/1.53 thf(fact_74_singleton__Un__iff, axiom, (! [A:$tType,X:A,A4:(set @ A),B4:(set @ A)]: ((((((insert @ A) @ X) @ (bot_bot @ (set @ A))) = (((sup_sup @ (set @ A)) @ A4) @ B4)) = (((A4 = (bot_bot @ (set @ A))) & (B4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) | (((A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) & (B4 = (bot_bot @ (set @ A)))) | ((A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) & (B4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))).
% 2.94/1.53 thf(fact_75_Un__singleton__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X:A]: ((((((sup_sup @ (set @ A)) @ A4) @ B4) = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (((A4 = (bot_bot @ (set @ A))) & (B4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) | (((A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) & (B4 = (bot_bot @ (set @ A)))) | ((A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) & (B4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))).
% 2.94/1.53 thf(fact_76_insert__is__Un, axiom, (! [A:$tType]: (((insert @ A) = (^ [A3:A]: (((sup_sup @ (set @ A)) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))))))))).
% 2.94/1.53 thf(fact_77_UnE, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) => ((~ ((((member @ A) @ C) @ A4))) => (((member @ A) @ C) @ B4)))))).
% 2.94/1.53 thf(fact_78_UnI1, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ A4) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B4)))))).
% 2.94/1.53 thf(fact_79_UnI2, axiom, (! [A:$tType,C:A,B4:(set @ A),A4:(set @ A)]: (((((member @ A) @ C) @ B4) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A4) @ B4)))))).
% 2.94/1.53 thf(fact_80_bex__Un, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),P2:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) & (P2 @ X2)))) = ((? [X2:A]: (((((member @ A) @ X2) @ A4) & (P2 @ X2)))) | (? [X2:A]: (((((member @ A) @ X2) @ B4) & (P2 @ X2))))))))).
% 2.94/1.53 thf(fact_81_ball__Un, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),P2:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) => (P2 @ X2)))) = ((! [X2:A]: (((((member @ A) @ X2) @ A4) => (P2 @ X2)))) & (! [X2:A]: (((((member @ A) @ X2) @ B4) => (P2 @ X2))))))))).
% 2.94/1.53 thf(fact_82_insertE, axiom, (! [A:$tType,A2:A,B3:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B3) @ A4)) => ((A2 != B3) => (((member @ A) @ A2) @ A4)))))).
% 2.94/1.53 thf(fact_83_Un__assoc, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) @ C3) = (((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B4) @ C3)))))).
% 2.94/1.53 thf(fact_84_insertI1, axiom, (! [A:$tType,A2:A,B4:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B4))))).
% 2.94/1.53 thf(fact_85_insertI2, axiom, (! [A:$tType,A2:A,B4:(set @ A),B3:A]: (((((member @ A) @ A2) @ B4) => (((member @ A) @ A2) @ (((insert @ A) @ B3) @ B4)))))).
% 2.94/1.53 thf(fact_86_Un__absorb, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ A4) = A4)))).
% 2.94/1.53 thf(fact_87_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((((sup_sup @ (set @ A)) @ B5) @ A5))))))).
% 2.94/1.53 thf(fact_88_Set_Oset__insert, axiom, (! [A:$tType,X:A,A4:(set @ A)]: (((((member @ A) @ X) @ A4) => (~ ((! [B6:(set @ A)]: (((A4 = (((insert @ A) @ X) @ B6)) => (((member @ A) @ X) @ B6)))))))))).
% 2.94/1.53 thf(fact_89_insert__ident, axiom, (! [A:$tType,X:A,A4:(set @ A),B4:(set @ A)]: (((~ ((((member @ A) @ X) @ A4))) => ((~ ((((member @ A) @ X) @ B4))) => (((((insert @ A) @ X) @ A4) = (((insert @ A) @ X) @ B4)) = (A4 = B4))))))).
% 2.94/1.53 thf(fact_90_pairwise__def, axiom, (! [A:$tType]: (((pairwise @ A) = (^ [R3:(A > (A > $o)),S2:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ S2) => (! [Y2:A]: (((((member @ A) @ Y2) @ S2) => ((X2 != Y2) => ((R3 @ X2) @ Y2)))))))))))))).
% 2.94/1.53 thf(fact_91_insert__absorb, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => ((((insert @ A) @ A2) @ A4) = A4))))).
% 2.94/1.53 thf(fact_92_insert__eq__iff, axiom, (! [A:$tType,A2:A,A4:(set @ A),B3:A,B4:(set @ A)]: (((~ ((((member @ A) @ A2) @ A4))) => ((~ ((((member @ A) @ B3) @ B4))) => (((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B3) @ B4)) = (((A2 = B3) => (A4 = B4)) & ((A2 != B3) => (? [C4:(set @ A)]: (((A4 = (((insert @ A) @ B3) @ C4)) & ((~ ((((member @ A) @ B3) @ C4))) & ((B4 = (((insert @ A) @ A2) @ C4)) & (~ ((((member @ A) @ A2) @ C4)))))))))))))))).
% 2.94/1.53 thf(fact_93_Un__left__absorb, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) = (((sup_sup @ (set @ A)) @ A4) @ B4))))).
% 2.94/1.53 thf(fact_94_insert__commute, axiom, (! [A:$tType,X:A,Y:A,A4:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ Y) @ A4)) = (((insert @ A) @ Y) @ (((insert @ A) @ X) @ A4)))))).
% 2.94/1.53 thf(fact_95_Un__left__commute, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B4) @ C3)) = (((sup_sup @ (set @ A)) @ B4) @ (((sup_sup @ (set @ A)) @ A4) @ C3)))))).
% 2.94/1.53 thf(fact_96_pairwise__insert, axiom, (! [A:$tType,R:(A > (A > $o)),X:A,S:(set @ A)]: (((((pairwise @ A) @ R) @ (((insert @ A) @ X) @ S)) = ((! [Y2:A]: ((((((member @ A) @ Y2) @ S) & (Y2 != X)) => (((R @ X) @ Y2) & ((R @ Y2) @ X))))) & (((pairwise @ A) @ R) @ S)))))).
% 2.94/1.53 thf(fact_97_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => (? [B6:(set @ A)]: (((A4 = (((insert @ A) @ A2) @ B6)) & (~ ((((member @ A) @ A2) @ B6)))))))))).
% 2.94/1.53 thf(fact_98_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B3:B,C:B]: (((A2 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X4:B,Y3:B]: (((((ord_less @ B) @ X4) @ Y3) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.94/1.53 thf(fact_99_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B3) => (((F @ B3) = C) => ((! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (((ord_less @ B) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 2.94/1.53 thf(fact_100_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X4:B,Y3:B]: (((((ord_less @ B) @ X4) @ Y3) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.94/1.53 thf(fact_101_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ C2) @ (F @ B3)) @ C) => ((! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (((ord_less @ C2) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.94/1.53 thf(fact_102_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X))))))))).
% 2.94/1.53 thf(fact_103_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.94/1.53 thf(fact_104_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.94/1.53 thf(fact_105_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.94/1.53 thf(fact_106_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.94/1.53 thf(fact_107_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y))))))))))).
% 2.94/1.53 thf(fact_108_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.94/1.53 thf(fact_109_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.94/1.53 thf(fact_110_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.94/1.53 thf(fact_111_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.94/1.53 thf(fact_112_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.94/1.53 thf(fact_113_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.94/1.53 thf(fact_114_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((A2 = B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.94/1.53 thf(fact_115_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => ((B3 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.94/1.53 thf(fact_116_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (~ ((((ord_less @ A) @ A2) @ B3)))))))))).
% 2.94/1.53 thf(fact_117_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.94/1.53 thf(fact_118_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.94/1.53 thf(fact_119_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),A2:A]: (((! [X4:A]: (((! [Y4:A]: (((((ord_less @ A) @ Y4) @ X4) => (P2 @ Y4)))) => (P2 @ X4)))) => (P2 @ A2)))))))).
% 2.94/1.53 thf(fact_120_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.94/1.53 thf(fact_121_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 2.94/1.53 thf(fact_122_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,P2:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P2)))))))).
% 2.94/1.53 thf(fact_123_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.94/1.53 thf(fact_124_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.94/1.53 thf(fact_125_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.94/1.53 thf(fact_126_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.94/1.53 thf(fact_127_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C:A]: (((((ord_less @ A) @ B3) @ A2) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.94/1.53 thf(fact_128_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 2.94/1.53 thf(fact_129_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (A2 != B3)))))))).
% 2.94/1.53 thf(fact_130_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (A2 != B3)))))))).
% 2.94/1.53 thf(fact_131_is__singletonE, axiom, (! [A:$tType,A4:(set @ A)]: ((((is_singleton @ A) @ A4) => (~ ((! [X4:A]: ((A4 != (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))))).
% 2.94/1.53 thf(fact_132_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((? [X2:A]: ((A5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))).
% 2.94/1.53 thf(fact_133_Un__empty__left, axiom, (! [A:$tType,B4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B4) = B4)))).
% 2.94/1.53 thf(fact_134_Un__empty__right, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = A4)))).
% 2.94/1.53 thf(fact_135_singletonD, axiom, (! [A:$tType,B3:A,A2:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B3 = A2))))).
% 2.94/1.53 thf(fact_136_singleton__iff, axiom, (! [A:$tType,B3:A,A2:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B3 = A2))))).
% 2.94/1.53 thf(fact_137_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B3:A,C:A,D:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C) @ (((insert @ A) @ D) @ (bot_bot @ (set @ A))))) = (((A2 = C) & (B3 = D)) | ((A2 = D) & (B3 = C))))))).
% 2.94/1.53 thf(fact_138_insert__not__empty, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((insert @ A) @ A2) @ A4) != (bot_bot @ (set @ A)))))).
% 2.94/1.53 thf(fact_139_singleton__inject, axiom, (! [A:$tType,A2:A,B3:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) => (A2 = B3))))).
% 2.94/1.53 thf(fact_140_bot_Oextremum__strict, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ (bot_bot @ A)))))))))).
% 2.94/1.53 thf(fact_141_bot_Onot__eq__extremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((A2 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A2)))))))).
% 2.94/1.53 thf(fact_142_less__dag__Node_H, axiom, (! [X:binDag_Mirabelle_dag,L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X) @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((X = L) | ((X = R) | ((((ord_less @ binDag_Mirabelle_dag) @ X) @ L) | (((ord_less @ binDag_Mirabelle_dag) @ X) @ R)))))))).
% 2.94/1.53 thf(fact_143_less__Node__dag, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag,X:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ (((binDag476092410e_Node @ L) @ A2) @ R)) @ X) => ((((ord_less @ binDag_Mirabelle_dag) @ L) @ X) & (((ord_less @ binDag_Mirabelle_dag) @ R) @ X)))))).
% 2.94/1.53 thf(fact_144_less__dag__def, axiom, ((ord_less @ binDag_Mirabelle_dag) = (^ [S3:binDag_Mirabelle_dag,T2:binDag_Mirabelle_dag]: (((binDag786255756subdag @ T2) @ S3))))).
% 2.94/1.53 thf(fact_145_less__dag__Tip, axiom, (! [X:binDag_Mirabelle_dag]: ((~ ((((ord_less @ binDag_Mirabelle_dag) @ X) @ binDag_Mirabelle_Tip)))))).
% 2.94/1.53 thf(fact_146_bot__eq__sup__iff, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X) @ Y)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.94/1.53 thf(fact_147_sup__eq__bot__iff, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((((sup_sup @ A) @ X) @ Y) = (bot_bot @ A)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.94/1.53 thf(fact_148_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A2) = A2))))))).
% 2.94/1.53 thf(fact_149_sup__bot_Oright__neutral, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ (bot_bot @ A)) = A2))))))).
% 2.94/1.53 thf(fact_150_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((A5 = (((insert @ A) @ ((the_elem @ A) @ A5)) @ (bot_bot @ (set @ A)))))))))).
% 2.94/1.53 thf(fact_151_sup_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B3)) @ B3) = (((sup_sup @ A) @ A2) @ B3)))))))).
% 2.94/1.53 thf(fact_152_sup__left__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.94/1.53 thf(fact_153_sup__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 2.94/1.53 thf(fact_154_sup_Oidem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 2.94/1.53 thf(fact_155_sup__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 2.94/1.53 thf(fact_156_sup_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B3)) = (((sup_sup @ A) @ A2) @ B3)))))))).
% 2.94/1.53 thf(fact_157_the__elem__eq, axiom, (! [A:$tType,X:A]: ((((the_elem @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X)))).
% 2.94/1.53 thf(fact_158_not__psubset__empty, axiom, (! [A:$tType,A4:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A4) @ (bot_bot @ (set @ A)))))))).
% 2.94/1.53 thf(fact_159_inf__sup__aci_I8_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.94/1.53 thf(fact_160_inf__sup__aci_I7_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.94/1.53 thf(fact_161_inf__sup__aci_I6_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.94/1.53 thf(fact_162_inf__sup__aci_I5_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X2:A,Y2:A]: ((((sup_sup @ A) @ Y2) @ X2)))))))).
% 2.94/1.53 thf(fact_163_sup__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 2.94/1.53 thf(fact_164_sup_Oassoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B3)) @ C) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B3) @ C))))))))).
% 2.94/1.53 thf(fact_165_sup__assoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.94/1.53 thf(fact_166_sup_Ocommute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [A3:A,B2:A]: ((((sup_sup @ A) @ B2) @ A3)))))))).
% 2.94/1.53 thf(fact_167_sup__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X2:A,Y2:A]: ((((sup_sup @ A) @ Y2) @ X2)))))))).
% 2.94/1.53 thf(fact_168_sup_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C:A]: (((((sup_sup @ A) @ B3) @ (((sup_sup @ A) @ A2) @ C)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B3) @ C))))))))).
% 2.94/1.53 thf(fact_169_sup__left__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.94/1.53 thf(fact_170_less__DAG__set__of, axiom, (! [X:binDag_Mirabelle_dag,Y:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X) @ Y) => ((binDag_Mirabelle_DAG @ Y) => (((ord_less @ (set @ simpl_ref)) @ (binDag1380252983set_of @ X)) @ (binDag1380252983set_of @ Y))))))).
% 2.94/1.53 thf(fact_171_less__supI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,A2:A,B3:A]: (((((ord_less @ A) @ X) @ A2) => (((ord_less @ A) @ X) @ (((sup_sup @ A) @ A2) @ B3))))))))).
% 2.94/1.53 thf(fact_172_less__supI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,B3:A,A2:A]: (((((ord_less @ A) @ X) @ B3) => (((ord_less @ A) @ X) @ (((sup_sup @ A) @ A2) @ B3))))))))).
% 2.94/1.53 thf(fact_173_sup_Ostrict__boundedE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,C:A,A2:A]: (((((ord_less @ A) @ (((sup_sup @ A) @ B3) @ C)) @ A2) => (~ (((((ord_less @ A) @ B3) @ A2) => (~ ((((ord_less @ A) @ C) @ A2))))))))))))).
% 2.94/1.53 thf(fact_174_sup_Ostrict__order__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B2:A,A3:A]: (((A3 = (((sup_sup @ A) @ A3) @ B2)) & (A3 != B2))))))))).
% 2.94/1.53 thf(fact_175_sup_Ostrict__coboundedI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C:A,A2:A,B3:A]: (((((ord_less @ A) @ C) @ A2) => (((ord_less @ A) @ C) @ (((sup_sup @ A) @ A2) @ B3))))))))).
% 2.94/1.53 thf(fact_176_sup_Ostrict__coboundedI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C:A,B3:A,A2:A]: (((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ (((sup_sup @ A) @ A2) @ B3))))))))).
% 2.94/1.53 thf(fact_177_sup__bot__left, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X) = X))))))).
% 2.94/1.53 thf(fact_178_sup__bot__right, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ (bot_bot @ A)) = X))))))).
% 2.94/1.53 thf(fact_179_sup__bot_Osemilattice__neutr__axioms, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (((semilattice_neutr @ A) @ (sup_sup @ A)) @ (bot_bot @ A)))))).
% 2.94/1.53 thf(fact_180_sup__bot_Omonoid__axioms, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (((monoid @ A) @ (sup_sup @ A)) @ (bot_bot @ A)))))).
% 2.94/1.53 thf(fact_181_minf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D2]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ X5) @ Z3) => (F3 = F3)))))))))))).
% 2.94/1.53 thf(fact_182_psubsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C) @ A4) => (((member @ A) @ C) @ B4)))))).
% 2.94/1.53 thf(fact_183_psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B4) => ((((ord_less @ (set @ A)) @ B4) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 2.94/1.53 thf(fact_184_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P3:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((P2 @ X4) = (P3 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((Q2 @ X4) = (Q3 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P2 @ X5) & (Q2 @ X5)) = ((P3 @ X5) & (Q3 @ X5)))))))))))))))).
% 2.94/1.53 thf(fact_185_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P3:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((P2 @ X4) = (P3 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => ((Q2 @ X4) = (Q3 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P2 @ X5) | (Q2 @ X5)) = ((P3 @ X5) | (Q3 @ X5)))))))))))))))).
% 2.94/1.53 thf(fact_186_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.94/1.53 thf(fact_187_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.94/1.53 thf(fact_188_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less @ A) @ X5) @ T)))))))))))))).
% 2.94/1.53 thf(fact_189_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less @ A) @ T) @ X5)))))))))))).
% 2.94/1.53 thf(fact_190_pinf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D2]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ Z3) @ X5) => (F3 = F3)))))))))))).
% 2.94/1.53 thf(fact_191_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P3:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((P2 @ X4) = (P3 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((Q2 @ X4) = (Q3 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P2 @ X5) & (Q2 @ X5)) = ((P3 @ X5) & (Q3 @ X5)))))))))))))))).
% 2.94/1.53 thf(fact_192_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P3:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((P2 @ X4) = (P3 @ X4))))))) => ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => ((Q2 @ X4) = (Q3 @ X4))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P2 @ X5) | (Q2 @ X5)) = ((P3 @ X5) | (Q3 @ X5)))))))))))))))).
% 2.94/1.53 thf(fact_193_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.94/1.53 thf(fact_194_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.94/1.53 thf(fact_195_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less @ A) @ X5) @ T)))))))))))).
% 2.94/1.53 thf(fact_196_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less @ A) @ T) @ X5)))))))))))))).
% 2.94/1.53 thf(fact_197_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B7:A]: (((((ord_less @ A) @ A2) @ B7) | (((ord_less @ A) @ B7) @ A2)))))))))).
% 2.94/1.53 thf(fact_198_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.94/1.53 thf(fact_199_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X5:A]: ((? [X1:A]: ((((ord_less @ A) @ X5) @ X1))))))))).
% 2.94/1.53 thf(fact_200_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X5:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X5))))))))).
% 2.94/1.53 thf(fact_201_dependent__wellorder__choice, axiom, (! [B:$tType,A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P2:((A > B) > (A > (B > $o)))]: (((! [R4:B,F4:(A > B),G3:(A > B),X4:A]: (((! [Y4:A]: (((((ord_less @ A) @ Y4) @ X4) => ((F4 @ Y4) = (G3 @ Y4))))) => ((((P2 @ F4) @ X4) @ R4) = (((P2 @ G3) @ X4) @ R4))))) => ((! [X4:A,F4:(A > B)]: (((! [Y4:A]: (((((ord_less @ A) @ Y4) @ X4) => (((P2 @ F4) @ Y4) @ (F4 @ Y4))))) => (? [X12:B]: ((((P2 @ F4) @ X4) @ X12)))))) => (? [F4:(A > B)]: ((! [X5:A]: ((((P2 @ F4) @ X5) @ (F4 @ X5)))))))))))))).
% 2.94/1.53 thf(fact_202_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R2:(A > (A > $o)),F2:(B > A),X2:B,Y2:B]: (((R2 @ (F2 @ X2)) @ (F2 @ Y2)))))))).
% 2.94/1.53 thf(fact_203_pred__on_Ochain__extend, axiom, (! [A:$tType,A4:(set @ A),P2:(A > (A > $o)),C3:(set @ A),Z:A]: ((((((pred_chain @ A) @ A4) @ P2) @ C3) => ((((member @ A) @ Z) @ A4) => ((! [X4:A]: (((((member @ A) @ X4) @ C3) => (((((sup_sup @ (A > (A > $o))) @ P2) @ (^ [Y5:A,Z5:A]: ((Y5 = Z5)))) @ X4) @ Z)))) => ((((pred_chain @ A) @ A4) @ P2) @ (((sup_sup @ (set @ A)) @ (((insert @ A) @ Z) @ (bot_bot @ (set @ A)))) @ C3)))))))).
% 2.94/1.53 thf(fact_204_sup__bot_Ocomm__monoid__axioms, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (((comm_monoid @ A) @ (sup_sup @ A)) @ (bot_bot @ A)))))).
% 2.94/1.53 thf(fact_205_sup2CI, axiom, (! [A:$tType,B:$tType,B4:(A > (B > $o)),X:A,Y:B,A4:(A > (B > $o))]: ((((~ (((B4 @ X) @ Y))) => ((A4 @ X) @ Y)) => (((((sup_sup @ (A > (B > $o))) @ A4) @ B4) @ X) @ Y))))).
% 2.94/1.53 thf(fact_206_pred__on_Ochain__empty, axiom, (! [A:$tType,A4:(set @ A),P2:(A > (A > $o))]: (((((pred_chain @ A) @ A4) @ P2) @ (bot_bot @ (set @ A)))))).
% 2.94/1.53 thf(fact_207_subset_Ochain__total, axiom, (! [A:$tType,A4:(set @ (set @ A)),C3:(set @ (set @ A)),X:(set @ A),Y:(set @ A)]: ((((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C3) => ((((member @ (set @ A)) @ X) @ C3) => ((((member @ (set @ A)) @ Y) @ C3) => ((((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y5:(set @ A),Z5:(set @ A)]: ((Y5 = Z5)))) @ X) @ Y) | (((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y5:(set @ A),Z5:(set @ A)]: ((Y5 = Z5)))) @ Y) @ X)))))))).
% 2.94/1.53 thf(fact_208_pred__on_Ochain__total, axiom, (! [A:$tType,A4:(set @ A),P2:(A > (A > $o)),C3:(set @ A),X:A,Y:A]: ((((((pred_chain @ A) @ A4) @ P2) @ C3) => ((((member @ A) @ X) @ C3) => ((((member @ A) @ Y) @ C3) => ((((((sup_sup @ (A > (A > $o))) @ P2) @ (^ [Y5:A,Z5:A]: ((Y5 = Z5)))) @ X) @ Y) | (((((sup_sup @ (A > (A > $o))) @ P2) @ (^ [Y5:A,Z5:A]: ((Y5 = Z5)))) @ Y) @ X)))))))).
% 2.94/1.53 thf(fact_209_subset_Ochain__empty, axiom, (! [A:$tType,A4:(set @ (set @ A))]: (((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ (bot_bot @ (set @ (set @ A))))))).
% 2.94/1.53 thf(fact_210_subset_Ochain__extend, axiom, (! [A:$tType,A4:(set @ (set @ A)),C3:(set @ (set @ A)),Z:(set @ A)]: ((((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C3) => ((((member @ (set @ A)) @ Z) @ A4) => ((! [X4:(set @ A)]: (((((member @ (set @ A)) @ X4) @ C3) => (((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y5:(set @ A),Z5:(set @ A)]: ((Y5 = Z5)))) @ X4) @ Z)))) => ((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ (((sup_sup @ (set @ (set @ A))) @ (((insert @ (set @ A)) @ Z) @ (bot_bot @ (set @ (set @ A))))) @ C3)))))))).
% 2.94/1.53 thf(fact_211_sup2I2, axiom, (! [A:$tType,B:$tType,B4:(A > (B > $o)),X:A,Y:B,A4:(A > (B > $o))]: ((((B4 @ X) @ Y) => (((((sup_sup @ (A > (B > $o))) @ A4) @ B4) @ X) @ Y))))).
% 2.94/1.53 thf(fact_212_sup2I1, axiom, (! [A:$tType,B:$tType,A4:(A > (B > $o)),X:A,Y:B,B4:(A > (B > $o))]: ((((A4 @ X) @ Y) => (((((sup_sup @ (A > (B > $o))) @ A4) @ B4) @ X) @ Y))))).
% 2.94/1.53 thf(fact_213_sup2E, axiom, (! [A:$tType,B:$tType,A4:(A > (B > $o)),B4:(A > (B > $o)),X:A,Y:B]: (((((((sup_sup @ (A > (B > $o))) @ A4) @ B4) @ X) @ Y) => ((~ (((A4 @ X) @ Y))) => ((B4 @ X) @ Y)))))).
% 2.94/1.53 thf(fact_214_semilattice__neutr_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z:A]: (((((semilattice_neutr @ A) @ F) @ Z) => (((comm_monoid @ A) @ F) @ Z))))).
% 2.94/1.53 thf(fact_215_reflclp__idemp, axiom, (! [A:$tType,P2:(A > (A > $o))]: (((((sup_sup @ (A > (A > $o))) @ (((sup_sup @ (A > (A > $o))) @ P2) @ (^ [Y5:A,Z5:A]: ((Y5 = Z5))))) @ (^ [Y5:A,Z5:A]: ((Y5 = Z5)))) = (((sup_sup @ (A > (A > $o))) @ P2) @ (^ [Y5:A,Z5:A]: ((Y5 = Z5)))))))).
% 2.94/1.53 thf(fact_216_pred__on_Omaxchain__def, axiom, (! [A:$tType]: (((pred_maxchain @ A) = (^ [A5:(set @ A),P4:(A > (A > $o)),C4:(set @ A)]: ((((((pred_chain @ A) @ A5) @ P4) @ C4) & (~ ((? [S2:(set @ A)]: ((((((pred_chain @ A) @ A5) @ P4) @ S2) & (((ord_less @ (set @ A)) @ C4) @ S2))))))))))))).
% 2.94/1.53 thf(fact_217_subset_Omaxchain__def, axiom, (! [A:$tType,A4:(set @ (set @ A)),C3:(set @ (set @ A))]: ((((((pred_maxchain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C3) = (((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C3) & (~ ((? [S2:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ S2) & (((ord_less @ (set @ (set @ A))) @ C3) @ S2))))))))))).
% 2.94/1.53 thf(fact_218_subset_Omaxchain__imp__chain, axiom, (! [A:$tType,A4:(set @ (set @ A)),C3:(set @ (set @ A))]: ((((((pred_maxchain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C3) => ((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C3))))).
% 2.94/1.53 thf(fact_219_subset_OHausdorff, axiom, (! [A:$tType,A4:(set @ (set @ A))]: ((? [X1:(set @ (set @ A))]: (((((pred_maxchain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ X1)))))).
% 2.94/1.53 thf(fact_220_Pow__singleton__iff, axiom, (! [A:$tType,X6:(set @ A),Y6:(set @ A)]: (((((pow @ A) @ X6) = (((insert @ (set @ A)) @ Y6) @ (bot_bot @ (set @ (set @ A))))) = ((X6 = (bot_bot @ (set @ A))) & (Y6 = (bot_bot @ (set @ A)))))))).
% 2.94/1.53 thf(fact_221_Pow__empty, axiom, (! [A:$tType]: ((((pow @ A) @ (bot_bot @ (set @ A))) = (((insert @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ (set @ A)))))))).
% 2.94/1.53 thf(fact_222_Pow__bottom, axiom, (! [A:$tType,B4:(set @ A)]: ((((member @ (set @ A)) @ (bot_bot @ (set @ A))) @ ((pow @ A) @ B4))))).
% 2.94/1.53 thf(fact_223_Pow__not__empty, axiom, (! [A:$tType,A4:(set @ A)]: ((((pow @ A) @ A4) != (bot_bot @ (set @ (set @ A))))))).
% 2.94/1.53 thf(fact_224_Pow__top, axiom, (! [A:$tType,A4:(set @ A)]: ((((member @ (set @ A)) @ A4) @ ((pow @ A) @ A4))))).
% 2.94/1.53 thf(fact_225_subset_Ochain__def, axiom, (! [A:$tType,A4:(set @ (set @ A)),C3:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C3) = ((((ord_less_eq @ (set @ (set @ A))) @ C3) @ A4) & (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ C3) => (! [Y2:(set @ A)]: (((((member @ (set @ A)) @ Y2) @ C3) => ((((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y5:(set @ A),Z5:(set @ A)]: ((Y5 = Z5)))) @ X2) @ Y2) | (((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y5:(set @ A),Z5:(set @ A)]: ((Y5 = Z5)))) @ Y2) @ X2))))))))))))).
% 2.94/1.53 thf(fact_226_subset_OchainI, axiom, (! [A:$tType,C3:(set @ (set @ A)),A4:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ C3) @ A4) => ((! [X4:(set @ A),Y3:(set @ A)]: (((((member @ (set @ A)) @ X4) @ C3) => ((((member @ (set @ A)) @ Y3) @ C3) => ((((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y5:(set @ A),Z5:(set @ A)]: ((Y5 = Z5)))) @ X4) @ Y3) | (((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y5:(set @ A),Z5:(set @ A)]: ((Y5 = Z5)))) @ Y3) @ X4)))))) => ((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C3)))))).
% 2.94/1.53 thf(fact_227_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.94/1.53 thf(fact_228_subsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A4) => (((member @ A) @ X4) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.94/1.53 thf(fact_229_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A4) => (A4 = B4)))))).
% 2.94/1.53 thf(fact_230_le__sup__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = ((((ord_less_eq @ A) @ X) @ Z) & (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.94/1.53 thf(fact_231_sup_Obounded__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,C:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C)) @ A2) = ((((ord_less_eq @ A) @ B3) @ A2) & (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.94/1.53 thf(fact_232_empty__subsetI, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A4)))).
% 2.94/1.53 thf(fact_233_subset__empty, axiom, (! [A:$tType,A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = (A4 = (bot_bot @ (set @ A))))))).
% 2.94/1.53 thf(fact_234_insert__subset, axiom, (! [A:$tType,X:A,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A4)) @ B4) = ((((member @ A) @ X) @ B4) & (((ord_less_eq @ (set @ A)) @ A4) @ B4)))))).
% 2.94/1.53 thf(fact_235_Un__subset__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) @ C3) = ((((ord_less_eq @ (set @ A)) @ A4) @ C3) & (((ord_less_eq @ (set @ A)) @ B4) @ C3)))))).
% 2.94/1.53 thf(fact_236_psubsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((A4 != B4) => (((ord_less @ (set @ A)) @ A4) @ B4)))))).
% 2.94/1.53 thf(fact_237_Pow__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((member @ (set @ A)) @ A4) @ ((pow @ A) @ B4)) = (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.94/1.53 thf(fact_238_PowI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((member @ (set @ A)) @ A4) @ ((pow @ A) @ B4)))))).
% 2.94/1.53 thf(fact_239_singleton__insert__inj__eq, axiom, (! [A:$tType,B3:A,A2:A,A4:(set @ A)]: ((((((insert @ A) @ B3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A4)) = ((A2 = B3) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.94/1.53 thf(fact_240_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A4:(set @ A),B3:A]: ((((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = ((A2 = B3) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.94/1.53 thf(fact_241_subset__Zorn, axiom, (! [A:$tType,A4:(set @ (set @ A))]: (((! [C5:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A4) @ (ord_less @ (set @ A))) @ C5) => (? [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ A4) & (! [Xa:(set @ A)]: (((((member @ (set @ A)) @ Xa) @ C5) => (((ord_less_eq @ (set @ A)) @ Xa) @ X5)))))))))) => (? [X4:(set @ A)]: (((((member @ (set @ A)) @ X4) @ A4) & (! [Xa2:(set @ A)]: (((((member @ (set @ A)) @ Xa2) @ A4) => ((((ord_less_eq @ (set @ A)) @ X4) @ Xa2) => (Xa2 = X4)))))))))))).
% 2.94/1.53 thf(fact_242_psubsetE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B4) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))).
% 2.94/1.53 thf(fact_243_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (A5 != B5)))))))).
% 2.94/1.53 thf(fact_244_psubset__imp__subset, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.94/1.53 thf(fact_245_psubset__subset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 2.94/1.53 thf(fact_246_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))))).
% 2.94/1.53 thf(fact_247_subset__psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less @ (set @ A)) @ B4) @ C3) => (((ord_less @ (set @ A)) @ A4) @ C3)))))).
% 2.94/1.53 thf(fact_248_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B5) | (A5 = B5)))))))).
% 2.94/1.53 thf(fact_249_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ T)))))))))))))).
% 2.94/1.53 thf(fact_250_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less_eq @ A) @ T) @ X5)))))))))))).
% 2.94/1.53 thf(fact_251_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less_eq @ A) @ X5) @ T)))))))))))).
% 2.94/1.53 thf(fact_252_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X5)))))))))))))).
% 2.94/1.53 thf(fact_253_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((A2 != B3) => ((((ord_less_eq @ A) @ A2) @ B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.94/1.53 thf(fact_254_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (((ord_less_eq @ A) @ B3) @ A2)))))))).
% 2.94/1.53 thf(fact_255_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) & (A3 != B2))))))))).
% 2.94/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, ((bounded_lattice @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A6:$tType]: (((bounded_lattice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A6:$tType,A7:$tType]: ((((bounded_lattice @ A7) @ (type2 @ A7)) => ((bounded_lattice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A6:$tType,A7:$tType]: ((((bounded_lattice @ A7) @ (type2 @ A7)) => ((bounde1808546759up_bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A6:$tType,A7:$tType]: ((((bounded_lattice @ A7) @ (type2 @ A7)) => ((bounded_lattice_bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A6:$tType,A7:$tType]: ((((semilattice_sup @ A7) @ (type2 @ A7)) => ((semilattice_sup @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A6:$tType,A7:$tType]: ((((order_bot @ A7) @ (type2 @ A7)) => ((order_bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Lattices_Olattice, axiom, (! [A6:$tType,A7:$tType]: ((((lattice @ A7) @ (type2 @ A7)) => ((lattice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_fun___Orderings_Obot, axiom, (! [A6:$tType,A7:$tType]: ((((bot @ A7) @ (type2 @ A7)) => ((bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.94/1.53 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3, axiom, (! [A6:$tType]: (((bounde1808546759up_bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4, axiom, (! [A6:$tType]: (((bounded_lattice_bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5, axiom, (! [A6:$tType]: (((semilattice_sup @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_Set_Oset___Orderings_Oorder__bot_6, axiom, (! [A6:$tType]: (((order_bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A6:$tType]: (((preorder @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_Set_Oset___Lattices_Olattice_8, axiom, (! [A6:$tType]: (((lattice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_Set_Oset___Orderings_Oorder_9, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_Set_Oset___Orderings_Oord_10, axiom, (! [A6:$tType]: (((ord @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_Set_Oset___Orderings_Obot_11, axiom, (! [A6:$tType]: (((bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.94/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12, axiom, ((bounde1808546759up_bot @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_13, axiom, ((bounded_lattice_bot @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Orderings_Oorder__bot_15, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_16, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Lattices_Olattice_17, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_18, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Orderings_Oord_19, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_HOL_Obool___Orderings_Obot_20, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.94/1.53 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_21, axiom, ((preorder @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.94/1.53 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_22, axiom, ((order @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.94/1.53 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_23, axiom, ((ord @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.94/1.53 thf(help_If_3_1_T, axiom, (! [P2:$o]: (((P2 = $true) | (P2 = $false))))).
% 2.94/1.53 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.94/1.53 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.94/1.53 thf(conj_0, conjecture, (((((binDag_Mirabelle_Dag @ p) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ l) @ p) @ p)) @ r) @ t) = ((p = simpl_Null) & (t = binDag_Mirabelle_Tip)))).
% 2.94/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.vZxap9H0LD/DTF2DTF19524.p
% 2.94/1.53 ------------------------
% 2.94/1.57 ---- Cleaned THF ---
% 2.94/1.58 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
% 2.94/1.58 binDag_Mirabelle_dag: $tType ).
% 2.94/1.58
% 2.94/1.58 thf(ty_t_Simpl__Heap_Oref,type,
% 2.94/1.58 simpl_ref: $tType ).
% 2.94/1.58
% 2.94/1.58 thf(ty_t_Set_Oset,type,
% 2.94/1.58 set: $tType > $tType ).
% 2.94/1.58
% 2.94/1.58 thf(ty_t_itself,type,
% 2.94/1.58 itself: $tType > $tType ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.94/1.58 bounded_lattice:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_HOL_Otype,type,
% 2.94/1.58 type:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Obot,type,
% 2.94/1.58 bot:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Oord,type,
% 2.94/1.58 ord:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Oorder,type,
% 2.94/1.58 order:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Lattices_Olattice,type,
% 2.94/1.58 lattice:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Ono__bot,type,
% 2.94/1.58 no_bot:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Ono__top,type,
% 2.94/1.58 no_top:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Olinorder,type,
% 2.94/1.58 linorder:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Opreorder,type,
% 2.94/1.58 preorder:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.94/1.58 order_bot:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Owellorder,type,
% 2.94/1.58 wellorder:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Orderings_Odense__order,type,
% 2.94/1.58 dense_order:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.94/1.58 linordered_idom:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Fields_Olinordered__field,type,
% 2.94/1.58 linordered_field:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.94/1.58 semilattice_sup:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 2.94/1.58 bounded_lattice_bot:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 2.94/1.58 bounde1808546759up_bot:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.94/1.58 condit1656338222tinuum:
% 2.94/1.58 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
% 2.94/1.58 binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODag,type,
% 2.94/1.58 binDag_Mirabelle_Dag: simpl_ref > ( simpl_ref > simpl_ref ) > ( simpl_ref > simpl_ref ) > binDag_Mirabelle_dag > $o ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
% 2.94/1.58 binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
% 2.94/1.58 binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
% 2.94/1.58 binDag1297733282se_dag:
% 2.94/1.58 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
% 2.94/1.58 binDag1442713106ec_dag:
% 2.94/1.58 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
% 2.94/1.58 binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
% 2.94/1.58 binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Fun_Ofun__upd,type,
% 2.94/1.58 fun_upd:
% 2.94/1.58 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Fun_Ooverride__on,type,
% 2.94/1.58 override_on:
% 2.94/1.58 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Groups_Ocomm__monoid,type,
% 2.94/1.58 comm_monoid:
% 2.94/1.58 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Groups_Omonoid,type,
% 2.94/1.58 monoid:
% 2.94/1.58 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_If,type,
% 2.94/1.58 if:
% 2.94/1.58 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Lattices_Osemilattice__neutr,type,
% 2.94/1.58 semilattice_neutr:
% 2.94/1.58 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.94/1.58 sup_sup:
% 2.94/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.94/1.58 bot_bot:
% 2.94/1.58 !>[A: $tType] : A ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.94/1.58 ord_less:
% 2.94/1.58 !>[A: $tType] : ( A > A > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.94/1.58 ord_less_eq:
% 2.94/1.58 !>[A: $tType] : ( A > A > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Pure_Otype,type,
% 2.94/1.58 type2:
% 2.94/1.58 !>[A: $tType] : ( itself @ A ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Relation_Oinv__imagep,type,
% 2.94/1.58 inv_imagep:
% 2.94/1.58 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Set_OCollect,type,
% 2.94/1.58 collect:
% 2.94/1.58 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Set_OPow,type,
% 2.94/1.58 pow:
% 2.94/1.58 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Set_Obind,type,
% 2.94/1.58 bind:
% 2.94/1.58 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Set_Oinsert,type,
% 2.94/1.58 insert:
% 2.94/1.58 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Set_Ois__empty,type,
% 2.94/1.58 is_empty:
% 2.94/1.58 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Set_Ois__singleton,type,
% 2.94/1.58 is_singleton:
% 2.94/1.58 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Set_Opairwise,type,
% 2.94/1.58 pairwise:
% 2.94/1.58 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Set_Othe__elem,type,
% 2.94/1.58 the_elem:
% 2.94/1.58 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Simpl__Heap_ONull,type,
% 2.94/1.58 simpl_Null: simpl_ref ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Zorn_Opred__on_Ochain,type,
% 2.94/1.58 pred_chain:
% 2.94/1.58 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_Zorn_Opred__on_Omaxchain,type,
% 2.94/1.58 pred_maxchain:
% 2.94/1.58 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_c_member,type,
% 2.94/1.58 member:
% 2.94/1.58 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(sy_v_l,type,
% 2.94/1.58 l: simpl_ref > simpl_ref ).
% 2.94/1.58
% 2.94/1.58 thf(sy_v_p,type,
% 2.94/1.58 p: simpl_ref ).
% 2.94/1.58
% 2.94/1.58 thf(sy_v_r,type,
% 2.94/1.58 r: simpl_ref > simpl_ref ).
% 2.94/1.58
% 2.94/1.58 thf(sy_v_t,type,
% 2.94/1.58 t: binDag_Mirabelle_dag ).
% 2.94/1.58
% 2.94/1.58 thf(fact_0_Dag__Null,axiom,
% 2.94/1.58 ! [L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag_Mirabelle_Dag @ simpl_Null @ L @ R @ T )
% 2.94/1.58 = ( T = binDag_Mirabelle_Tip ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_1_Dag_Osimps_I1_J,axiom,
% 2.94/1.58 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref] :
% 2.94/1.58 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ binDag_Mirabelle_Tip )
% 2.94/1.58 = ( P = simpl_Null ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_2_Dag__upd__same__l__lemma,axiom,
% 2.94/1.58 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( P != simpl_Null )
% 2.94/1.58 => ~ ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ P @ P ) @ R @ T ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_3_fun__upd__upd,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z: B] :
% 2.94/1.58 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z )
% 2.94/1.58 = ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_4_fun__upd__triv,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 2.94/1.58 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 2.94/1.58 = F ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_5_fun__upd__apply,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType] :
% 2.94/1.58 ( ( fun_upd @ B @ A )
% 2.94/1.58 = ( ^ [F2: B > A,X2: B,Y2: A,Z2: B] : ( if @ A @ ( Z2 = X2 ) @ Y2 @ ( F2 @ Z2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_6_notin__Dag__update__l,axiom,
% 2.94/1.58 ! [Q: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,Y: simpl_ref,R: simpl_ref > simpl_ref] :
% 2.94/1.58 ( ~ ( member @ simpl_ref @ Q @ ( binDag1380252983set_of @ T ) )
% 2.94/1.58 => ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q @ Y ) @ R @ T )
% 2.94/1.58 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_7_notin__Dag__update__r,axiom,
% 2.94/1.58 ! [Q: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Y: simpl_ref] :
% 2.94/1.58 ( ~ ( member @ simpl_ref @ Q @ ( binDag1380252983set_of @ T ) )
% 2.94/1.58 => ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q @ Y ) @ T )
% 2.94/1.58 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_8_Null__notin__Dag,axiom,
% 2.94/1.58 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 2.94/1.58 => ~ ( member @ simpl_ref @ simpl_Null @ ( binDag1380252983set_of @ T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_9_Dag__Ref,axiom,
% 2.94/1.58 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( P != simpl_Null )
% 2.94/1.58 => ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 2.94/1.58 = ( ? [Lt: binDag_Mirabelle_dag,Rt: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( T
% 2.94/1.58 = ( binDag476092410e_Node @ Lt @ P @ Rt ) )
% 2.94/1.58 & ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt )
% 2.94/1.58 & ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_10_Dag_Osimps_I2_J,axiom,
% 2.94/1.58 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 2.94/1.58 = ( ( P = A2 )
% 2.94/1.58 & ( P != simpl_Null )
% 2.94/1.58 & ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt2 )
% 2.94/1.58 & ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_11_fun__upd__def,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType] :
% 2.94/1.58 ( ( fun_upd @ A @ B )
% 2.94/1.58 = ( ^ [F2: A > B,A3: A,B2: B,X2: A] : ( if @ B @ ( X2 = A3 ) @ B2 @ ( F2 @ X2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_12_fun__upd__eqD,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z: B] :
% 2.94/1.58 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 2.94/1.58 = ( fun_upd @ A @ B @ G @ X @ Z ) )
% 2.94/1.58 => ( Y = Z ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_13_fun__upd__idem,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 2.94/1.58 ( ( ( F @ X )
% 2.94/1.58 = Y )
% 2.94/1.58 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 2.94/1.58 = F ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_14_fun__upd__same,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 2.94/1.58 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 2.94/1.58 = Y ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_15_dag_Oinject,axiom,
% 2.94/1.58 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
% 2.94/1.58 = ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
% 2.94/1.58 = ( ( X21 = Y21 )
% 2.94/1.58 & ( X22 = Y22 )
% 2.94/1.58 & ( X23 = Y23 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_16_dag_Oexhaust,axiom,
% 2.94/1.58 ! [Y: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( Y != binDag_Mirabelle_Tip )
% 2.94/1.58 => ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
% 2.94/1.58 ( Y
% 2.94/1.58 != ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_17_dag_Oinduct,axiom,
% 2.94/1.58 ! [P2: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( P2 @ binDag_Mirabelle_Tip )
% 2.94/1.58 => ( ! [X1: binDag_Mirabelle_dag,X24: simpl_ref,X3: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( P2 @ X1 )
% 2.94/1.58 => ( ( P2 @ X3 )
% 2.94/1.58 => ( P2 @ ( binDag476092410e_Node @ X1 @ X24 @ X3 ) ) ) )
% 2.94/1.58 => ( P2 @ Dag ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_18_dag_Odistinct_I1_J,axiom,
% 2.94/1.58 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.94/1.58 ( binDag_Mirabelle_Tip
% 2.94/1.58 != ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_19_fun__upd__idem__iff,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 2.94/1.58 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 2.94/1.58 = F )
% 2.94/1.58 = ( ( F @ X )
% 2.94/1.58 = Y ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_20_fun__upd__twist,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,A2: A,C: A,M: A > B,B3: B,D: B] :
% 2.94/1.58 ( ( A2 != C )
% 2.94/1.58 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A2 @ B3 ) @ C @ D )
% 2.94/1.58 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C @ D ) @ A2 @ B3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_21_fun__upd__other,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y: B] :
% 2.94/1.58 ( ( Z != X )
% 2.94/1.58 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z )
% 2.94/1.58 = ( F @ Z ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_22_DAG_Osimps_I2_J,axiom,
% 2.94/1.58 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.94/1.58 = ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
% 2.94/1.58 & ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
% 2.94/1.58 & ( binDag_Mirabelle_DAG @ L )
% 2.94/1.58 & ( binDag_Mirabelle_DAG @ R ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_23_in__set__of__decomp,axiom,
% 2.94/1.58 ! [P: simpl_ref,T: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( member @ simpl_ref @ P @ ( binDag1380252983set_of @ T ) )
% 2.94/1.58 = ( ? [L2: binDag_Mirabelle_dag,R2: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( T
% 2.94/1.58 = ( binDag476092410e_Node @ L2 @ P @ R2 ) )
% 2.94/1.58 | ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ L2 @ P @ R2 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_24_dag_Osimps_I4_J,axiom,
% 2.94/1.58 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
% 2.94/1.58 ( ( binDag1297733282se_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
% 2.94/1.58 = F1 ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_25_dag_Osimps_I6_J,axiom,
% 2.94/1.58 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
% 2.94/1.58 ( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
% 2.94/1.58 = F1 ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_26_DAG_Osimps_I1_J,axiom,
% 2.94/1.58 binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% 2.94/1.58
% 2.94/1.58 thf(fact_27_dag_Osimps_I7_J,axiom,
% 2.94/1.58 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 2.94/1.58 = ( F22 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X23 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_28_dag_Osimps_I5_J,axiom,
% 2.94/1.58 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag1297733282se_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 2.94/1.58 = ( F22 @ X21 @ X22 @ X23 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_29_set__of__Tip,axiom,
% 2.94/1.58 ( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
% 2.94/1.58 = ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_30_subdag_Osimps_I1_J,axiom,
% 2.94/1.58 ! [T: binDag_Mirabelle_dag] :
% 2.94/1.58 ~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_31_subdag_Osimps_I2_J,axiom,
% 2.94/1.58 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T )
% 2.94/1.58 = ( ( T = L )
% 2.94/1.58 | ( T = R )
% 2.94/1.58 | ( binDag786255756subdag @ L @ T )
% 2.94/1.58 | ( binDag786255756subdag @ R @ T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_32_subdag__not__sym,axiom,
% 2.94/1.58 ! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag786255756subdag @ S @ T )
% 2.94/1.58 => ~ ( binDag786255756subdag @ T @ S ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_33_subdag__trans,axiom,
% 2.94/1.58 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag786255756subdag @ T @ S )
% 2.94/1.58 => ( ( binDag786255756subdag @ S @ R )
% 2.94/1.58 => ( binDag786255756subdag @ T @ R ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_34_subdag__neq,axiom,
% 2.94/1.58 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag786255756subdag @ T @ S )
% 2.94/1.58 => ( T != S ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_35_subdag__NodeD,axiom,
% 2.94/1.58 ! [T: binDag_Mirabelle_dag,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 2.94/1.58 => ( ( binDag786255756subdag @ T @ Lt2 )
% 2.94/1.58 & ( binDag786255756subdag @ T @ Rt2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_36_empty__Collect__eq,axiom,
% 2.94/1.58 ! [A: $tType,P2: A > $o] :
% 2.94/1.58 ( ( ( bot_bot @ ( set @ A ) )
% 2.94/1.58 = ( collect @ A @ P2 ) )
% 2.94/1.58 = ( ! [X2: A] :
% 2.94/1.58 ~ ( P2 @ X2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_37_Collect__empty__eq,axiom,
% 2.94/1.58 ! [A: $tType,P2: A > $o] :
% 2.94/1.58 ( ( ( collect @ A @ P2 )
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = ( ! [X2: A] :
% 2.94/1.58 ~ ( P2 @ X2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_38_all__not__in__conv,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( ! [X2: A] :
% 2.94/1.58 ~ ( member @ A @ X2 @ A4 ) )
% 2.94/1.58 = ( A4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_39_empty__iff,axiom,
% 2.94/1.58 ! [A: $tType,C: A] :
% 2.94/1.58 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_40_bot__apply,axiom,
% 2.94/1.58 ! [C2: $tType,D2: $tType] :
% 2.94/1.58 ( ( bot @ C2 @ ( type2 @ C2 ) )
% 2.94/1.58 => ( ( bot_bot @ ( D2 > C2 ) )
% 2.94/1.58 = ( ^ [X2: D2] : ( bot_bot @ C2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_41_override__on__emptyset,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.94/1.58 ( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = F ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_42_ex__in__conv,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( ? [X2: A] : ( member @ A @ X2 @ A4 ) )
% 2.94/1.58 = ( A4
% 2.94/1.58 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_43_equals0I,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ! [Y3: A] :
% 2.94/1.58 ~ ( member @ A @ Y3 @ A4 )
% 2.94/1.58 => ( A4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_44_equals0D,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,A2: A] :
% 2.94/1.58 ( ( A4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_45_mem__Collect__eq,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,P2: A > $o] :
% 2.94/1.58 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 2.94/1.58 = ( P2 @ A2 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_46_Collect__mem__eq,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( collect @ A
% 2.94/1.58 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 2.94/1.58 = A4 ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_47_Collect__cong,axiom,
% 2.94/1.58 ! [A: $tType,P2: A > $o,Q2: A > $o] :
% 2.94/1.58 ( ! [X4: A] :
% 2.94/1.58 ( ( P2 @ X4 )
% 2.94/1.58 = ( Q2 @ X4 ) )
% 2.94/1.58 => ( ( collect @ A @ P2 )
% 2.94/1.58 = ( collect @ A @ Q2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_48_ext,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.94/1.58 ( ! [X4: A] :
% 2.94/1.58 ( ( F @ X4 )
% 2.94/1.58 = ( G @ X4 ) )
% 2.94/1.58 => ( F = G ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_49_override__on__apply__in,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType,A2: A,A4: set @ A,F: A > B,G: A > B] :
% 2.94/1.58 ( ( member @ A @ A2 @ A4 )
% 2.94/1.58 => ( ( override_on @ A @ B @ F @ G @ A4 @ A2 )
% 2.94/1.58 = ( G @ A2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_50_override__on__apply__notin,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType,A2: A,A4: set @ A,F: A > B,G: A > B] :
% 2.94/1.58 ( ~ ( member @ A @ A2 @ A4 )
% 2.94/1.58 => ( ( override_on @ A @ B @ F @ G @ A4 @ A2 )
% 2.94/1.58 = ( F @ A2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_51_bot__set__def,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bot_bot @ ( set @ A ) )
% 2.94/1.58 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_52_override__on__def,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType] :
% 2.94/1.58 ( ( override_on @ A @ B )
% 2.94/1.58 = ( ^ [F2: A > B,G2: A > B,A5: set @ A,A3: A] : ( if @ B @ ( member @ A @ A3 @ A5 ) @ ( G2 @ A3 ) @ ( F2 @ A3 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_53_bot__fun__def,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType] :
% 2.94/1.58 ( ( bot @ B @ ( type2 @ B ) )
% 2.94/1.58 => ( ( bot_bot @ ( A > B ) )
% 2.94/1.58 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_54_emptyE,axiom,
% 2.94/1.58 ! [A: $tType,A2: A] :
% 2.94/1.58 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_55_Set_Ois__empty__def,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( is_empty @ A )
% 2.94/1.58 = ( ^ [A5: set @ A] :
% 2.94/1.58 ( A5
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_56_Collect__empty__eq__bot,axiom,
% 2.94/1.58 ! [A: $tType,P2: A > $o] :
% 2.94/1.58 ( ( ( collect @ A @ P2 )
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = ( P2
% 2.94/1.58 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_57_bot__empty__eq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bot_bot @ ( A > $o ) )
% 2.94/1.58 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_58_is__singletonI_H,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( A4
% 2.94/1.58 != ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 => ( ! [X4: A,Y3: A] :
% 2.94/1.58 ( ( member @ A @ X4 @ A4 )
% 2.94/1.58 => ( ( member @ A @ Y3 @ A4 )
% 2.94/1.58 => ( X4 = Y3 ) ) )
% 2.94/1.58 => ( is_singleton @ A @ A4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_59_empty__bind,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType,F: B > ( set @ A )] :
% 2.94/1.58 ( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_60_set__of__Node,axiom,
% 2.94/1.58 ! [Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag1380252983set_of @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 2.94/1.58 = ( sup_sup @ ( set @ simpl_ref ) @ ( sup_sup @ ( set @ simpl_ref ) @ ( insert @ simpl_ref @ A2 @ ( bot_bot @ ( set @ simpl_ref ) ) ) @ ( binDag1380252983set_of @ Lt2 ) ) @ ( binDag1380252983set_of @ Rt2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_61_pairwise__empty,axiom,
% 2.94/1.58 ! [A: $tType,P2: A > A > $o] : ( pairwise @ A @ P2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_62_DAG__less,axiom,
% 2.94/1.58 ! [Y: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( binDag_Mirabelle_DAG @ Y )
% 2.94/1.58 => ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
% 2.94/1.58 => ( binDag_Mirabelle_DAG @ X ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_63_insertCI,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,B4: set @ A,B3: A] :
% 2.94/1.58 ( ( ~ ( member @ A @ A2 @ B4 )
% 2.94/1.58 => ( A2 = B3 ) )
% 2.94/1.58 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_64_insert__iff,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,B3: A,A4: set @ A] :
% 2.94/1.58 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
% 2.94/1.58 = ( ( A2 = B3 )
% 2.94/1.58 | ( member @ A @ A2 @ A4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_65_insert__absorb2,axiom,
% 2.94/1.58 ! [A: $tType,X: A,A4: set @ A] :
% 2.94/1.58 ( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
% 2.94/1.58 = ( insert @ A @ X @ A4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_66_UnCI,axiom,
% 2.94/1.58 ! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
% 2.94/1.58 ( ( ~ ( member @ A @ C @ B4 )
% 2.94/1.58 => ( member @ A @ C @ A4 ) )
% 2.94/1.58 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_67_Un__iff,axiom,
% 2.94/1.58 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 2.94/1.58 = ( ( member @ A @ C @ A4 )
% 2.94/1.58 | ( member @ A @ C @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_68_singletonI,axiom,
% 2.94/1.58 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_69_Un__empty,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = ( ( A4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 & ( B4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_70_Un__insert__left,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,B4: set @ A,C3: set @ A] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B4 ) @ C3 )
% 2.94/1.58 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_71_Un__insert__right,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,A2: A,B4: set @ A] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B4 ) )
% 2.94/1.58 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_72_is__singletonI,axiom,
% 2.94/1.58 ! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_73_pairwise__singleton,axiom,
% 2.94/1.58 ! [A: $tType,P2: A > A > $o,A4: A] : ( pairwise @ A @ P2 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_74_singleton__Un__iff,axiom,
% 2.94/1.58 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 2.94/1.58 = ( ( ( A4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 & ( B4
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.58 | ( ( A4
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 & ( B4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 | ( ( A4
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 & ( B4
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_75_Un__singleton__iff,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 2.94/1.58 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 = ( ( ( A4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 & ( B4
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.58 | ( ( A4
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 & ( B4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 | ( ( A4
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 & ( B4
% 2.94/1.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_76_insert__is__Un,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( insert @ A )
% 2.94/1.58 = ( ^ [A3: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_77_UnE,axiom,
% 2.94/1.58 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 2.94/1.58 => ( ~ ( member @ A @ C @ A4 )
% 2.94/1.58 => ( member @ A @ C @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_78_UnI1,axiom,
% 2.94/1.58 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( member @ A @ C @ A4 )
% 2.94/1.58 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_79_UnI2,axiom,
% 2.94/1.58 ! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
% 2.94/1.58 ( ( member @ A @ C @ B4 )
% 2.94/1.58 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_80_bex__Un,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,P2: A > $o] :
% 2.94/1.58 ( ( ? [X2: A] :
% 2.94/1.58 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 2.94/1.58 & ( P2 @ X2 ) ) )
% 2.94/1.58 = ( ? [X2: A] :
% 2.94/1.58 ( ( member @ A @ X2 @ A4 )
% 2.94/1.58 & ( P2 @ X2 ) )
% 2.94/1.58 | ? [X2: A] :
% 2.94/1.58 ( ( member @ A @ X2 @ B4 )
% 2.94/1.58 & ( P2 @ X2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_81_ball__Un,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,P2: A > $o] :
% 2.94/1.58 ( ( ! [X2: A] :
% 2.94/1.58 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 2.94/1.58 => ( P2 @ X2 ) ) )
% 2.94/1.58 = ( ! [X2: A] :
% 2.94/1.58 ( ( member @ A @ X2 @ A4 )
% 2.94/1.58 => ( P2 @ X2 ) )
% 2.94/1.58 & ! [X2: A] :
% 2.94/1.58 ( ( member @ A @ X2 @ B4 )
% 2.94/1.58 => ( P2 @ X2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_82_insertE,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,B3: A,A4: set @ A] :
% 2.94/1.58 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
% 2.94/1.58 => ( ( A2 != B3 )
% 2.94/1.58 => ( member @ A @ A2 @ A4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_83_Un__assoc,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
% 2.94/1.58 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_84_insertI1,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_85_insertI2,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,B4: set @ A,B3: A] :
% 2.94/1.58 ( ( member @ A @ A2 @ B4 )
% 2.94/1.58 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_86_Un__absorb,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 2.94/1.58 = A4 ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_87_Un__commute,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) )
% 2.94/1.58 = ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_88_Set_Oset__insert,axiom,
% 2.94/1.58 ! [A: $tType,X: A,A4: set @ A] :
% 2.94/1.58 ( ( member @ A @ X @ A4 )
% 2.94/1.58 => ~ ! [B6: set @ A] :
% 2.94/1.58 ( ( A4
% 2.94/1.58 = ( insert @ A @ X @ B6 ) )
% 2.94/1.58 => ( member @ A @ X @ B6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_89_insert__ident,axiom,
% 2.94/1.58 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ~ ( member @ A @ X @ A4 )
% 2.94/1.58 => ( ~ ( member @ A @ X @ B4 )
% 2.94/1.58 => ( ( ( insert @ A @ X @ A4 )
% 2.94/1.58 = ( insert @ A @ X @ B4 ) )
% 2.94/1.58 = ( A4 = B4 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_90_pairwise__def,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( pairwise @ A )
% 2.94/1.58 = ( ^ [R3: A > A > $o,S2: set @ A] :
% 2.94/1.58 ! [X2: A] :
% 2.94/1.58 ( ( member @ A @ X2 @ S2 )
% 2.94/1.58 => ! [Y2: A] :
% 2.94/1.58 ( ( member @ A @ Y2 @ S2 )
% 2.94/1.58 => ( ( X2 != Y2 )
% 2.94/1.58 => ( R3 @ X2 @ Y2 ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_91_insert__absorb,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,A4: set @ A] :
% 2.94/1.58 ( ( member @ A @ A2 @ A4 )
% 2.94/1.58 => ( ( insert @ A @ A2 @ A4 )
% 2.94/1.58 = A4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_92_insert__eq__iff,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,A4: set @ A,B3: A,B4: set @ A] :
% 2.94/1.58 ( ~ ( member @ A @ A2 @ A4 )
% 2.94/1.58 => ( ~ ( member @ A @ B3 @ B4 )
% 2.94/1.58 => ( ( ( insert @ A @ A2 @ A4 )
% 2.94/1.58 = ( insert @ A @ B3 @ B4 ) )
% 2.94/1.58 = ( ( ( A2 = B3 )
% 2.94/1.58 => ( A4 = B4 ) )
% 2.94/1.58 & ( ( A2 != B3 )
% 2.94/1.58 => ? [C4: set @ A] :
% 2.94/1.58 ( ( A4
% 2.94/1.58 = ( insert @ A @ B3 @ C4 ) )
% 2.94/1.58 & ~ ( member @ A @ B3 @ C4 )
% 2.94/1.58 & ( B4
% 2.94/1.58 = ( insert @ A @ A2 @ C4 ) )
% 2.94/1.58 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_93_Un__left__absorb,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
% 2.94/1.58 = ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_94_insert__commute,axiom,
% 2.94/1.58 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 2.94/1.58 ( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
% 2.94/1.58 = ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_95_Un__left__commute,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
% 2.94/1.58 = ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_96_pairwise__insert,axiom,
% 2.94/1.58 ! [A: $tType,R: A > A > $o,X: A,S: set @ A] :
% 2.94/1.58 ( ( pairwise @ A @ R @ ( insert @ A @ X @ S ) )
% 2.94/1.58 = ( ! [Y2: A] :
% 2.94/1.58 ( ( ( member @ A @ Y2 @ S )
% 2.94/1.58 & ( Y2 != X ) )
% 2.94/1.58 => ( ( R @ X @ Y2 )
% 2.94/1.58 & ( R @ Y2 @ X ) ) )
% 2.94/1.58 & ( pairwise @ A @ R @ S ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_97_mk__disjoint__insert,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,A4: set @ A] :
% 2.94/1.58 ( ( member @ A @ A2 @ A4 )
% 2.94/1.58 => ? [B6: set @ A] :
% 2.94/1.58 ( ( A4
% 2.94/1.58 = ( insert @ A @ A2 @ B6 ) )
% 2.94/1.58 & ~ ( member @ A @ A2 @ B6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_98_ord__eq__less__subst,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType] :
% 2.94/1.58 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.94/1.58 & ( ord @ A @ ( type2 @ A ) ) )
% 2.94/1.58 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.94/1.58 ( ( A2
% 2.94/1.58 = ( F @ B3 ) )
% 2.94/1.58 => ( ( ord_less @ B @ B3 @ C )
% 2.94/1.58 => ( ! [X4: B,Y3: B] :
% 2.94/1.58 ( ( ord_less @ B @ X4 @ Y3 )
% 2.94/1.58 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.94/1.58 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_99_ord__less__eq__subst,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType] :
% 2.94/1.58 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.94/1.58 & ( ord @ A @ ( type2 @ A ) ) )
% 2.94/1.58 => ! [A2: A,B3: A,F: A > B,C: B] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ B3 )
% 2.94/1.58 => ( ( ( F @ B3 )
% 2.94/1.58 = C )
% 2.94/1.58 => ( ! [X4: A,Y3: A] :
% 2.94/1.58 ( ( ord_less @ A @ X4 @ Y3 )
% 2.94/1.58 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.94/1.58 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_100_order__less__subst1,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType] :
% 2.94/1.58 ( ( ( order @ B @ ( type2 @ B ) )
% 2.94/1.58 & ( order @ A @ ( type2 @ A ) ) )
% 2.94/1.58 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 2.94/1.58 => ( ( ord_less @ B @ B3 @ C )
% 2.94/1.58 => ( ! [X4: B,Y3: B] :
% 2.94/1.58 ( ( ord_less @ B @ X4 @ Y3 )
% 2.94/1.58 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.94/1.58 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_101_order__less__subst2,axiom,
% 2.94/1.58 ! [A: $tType,C2: $tType] :
% 2.94/1.58 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.94/1.58 & ( order @ A @ ( type2 @ A ) ) )
% 2.94/1.58 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ B3 )
% 2.94/1.58 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 2.94/1.58 => ( ! [X4: A,Y3: A] :
% 2.94/1.58 ( ( ord_less @ A @ X4 @ Y3 )
% 2.94/1.58 => ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.94/1.58 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_102_lt__ex,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A] :
% 2.94/1.58 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_103_gt__ex,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( no_top @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A] :
% 2.94/1.58 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_104_neqE,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( X != Y )
% 2.94/1.58 => ( ~ ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_105_neq__iff,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( X != Y )
% 2.94/1.58 = ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_106_order_Oasym,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ B3 )
% 2.94/1.58 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_107_dense,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ? [Z3: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Z3 )
% 2.94/1.58 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_108_less__imp__neq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ( X != Y ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_109_less__asym,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_110_less__asym_H,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ B3 )
% 2.94/1.58 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_111_less__trans,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A,Z: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ( ( ord_less @ A @ Y @ Z )
% 2.94/1.58 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_112_less__linear,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 | ( X = Y )
% 2.94/1.58 | ( ord_less @ A @ Y @ X ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_113_less__irrefl,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A] :
% 2.94/1.58 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_114_ord__eq__less__trans,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( ord @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A,C: A] :
% 2.94/1.58 ( ( A2 = B3 )
% 2.94/1.58 => ( ( ord_less @ A @ B3 @ C )
% 2.94/1.58 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_115_ord__less__eq__trans,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( ord @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A,C: A] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ B3 )
% 2.94/1.58 => ( ( B3 = C )
% 2.94/1.58 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_116_dual__order_Oasym,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [B3: A,A2: A] :
% 2.94/1.58 ( ( ord_less @ A @ B3 @ A2 )
% 2.94/1.58 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_117_less__imp__not__eq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ( X != Y ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_118_less__not__sym,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_119_less__induct,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [P2: A > $o,A2: A] :
% 2.94/1.58 ( ! [X4: A] :
% 2.94/1.58 ( ! [Y4: A] :
% 2.94/1.58 ( ( ord_less @ A @ Y4 @ X4 )
% 2.94/1.58 => ( P2 @ Y4 ) )
% 2.94/1.58 => ( P2 @ X4 ) )
% 2.94/1.58 => ( P2 @ A2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_120_antisym__conv3,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [Y: A,X: A] :
% 2.94/1.58 ( ~ ( ord_less @ A @ Y @ X )
% 2.94/1.58 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.94/1.58 = ( X = Y ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_121_less__imp__not__eq2,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ( Y != X ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_122_less__imp__triv,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A,P2: $o] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ( ( ord_less @ A @ Y @ X )
% 2.94/1.58 => P2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_123_linorder__cases,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ~ ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ( ( X != Y )
% 2.94/1.58 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_124_dual__order_Oirrefl,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A] :
% 2.94/1.58 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_125_order_Ostrict__trans,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A,C: A] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ B3 )
% 2.94/1.58 => ( ( ord_less @ A @ B3 @ C )
% 2.94/1.58 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_126_less__imp__not__less,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_127_dual__order_Ostrict__trans,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [B3: A,A2: A,C: A] :
% 2.94/1.58 ( ( ord_less @ A @ B3 @ A2 )
% 2.94/1.58 => ( ( ord_less @ A @ C @ B3 )
% 2.94/1.58 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_128_not__less__iff__gr__or__eq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.94/1.58 = ( ( ord_less @ A @ Y @ X )
% 2.94/1.58 | ( X = Y ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_129_order_Ostrict__implies__not__eq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ B3 )
% 2.94/1.58 => ( A2 != B3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_130_dual__order_Ostrict__implies__not__eq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [B3: A,A2: A] :
% 2.94/1.58 ( ( ord_less @ A @ B3 @ A2 )
% 2.94/1.58 => ( A2 != B3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_131_is__singletonE,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( is_singleton @ A @ A4 )
% 2.94/1.58 => ~ ! [X4: A] :
% 2.94/1.58 ( A4
% 2.94/1.58 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_132_is__singleton__def,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( is_singleton @ A )
% 2.94/1.58 = ( ^ [A5: set @ A] :
% 2.94/1.58 ? [X2: A] :
% 2.94/1.58 ( A5
% 2.94/1.58 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_133_Un__empty__left,axiom,
% 2.94/1.58 ! [A: $tType,B4: set @ A] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
% 2.94/1.58 = B4 ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_134_Un__empty__right,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = A4 ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_135_singletonD,axiom,
% 2.94/1.58 ! [A: $tType,B3: A,A2: A] :
% 2.94/1.58 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 => ( B3 = A2 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_136_singleton__iff,axiom,
% 2.94/1.58 ! [A: $tType,B3: A,A2: A] :
% 2.94/1.58 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 = ( B3 = A2 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_137_doubleton__eq__iff,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,B3: A,C: A,D: A] :
% 2.94/1.58 ( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 = ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.58 = ( ( ( A2 = C )
% 2.94/1.58 & ( B3 = D ) )
% 2.94/1.58 | ( ( A2 = D )
% 2.94/1.58 & ( B3 = C ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_138_insert__not__empty,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,A4: set @ A] :
% 2.94/1.58 ( ( insert @ A @ A2 @ A4 )
% 2.94/1.58 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_139_singleton__inject,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,B3: A] :
% 2.94/1.58 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 => ( A2 = B3 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_140_bot_Oextremum__strict,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A] :
% 2.94/1.58 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_141_bot_Onot__eq__extremum,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A] :
% 2.94/1.58 ( ( A2
% 2.94/1.58 != ( bot_bot @ A ) )
% 2.94/1.58 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_142_less__dag__Node_H,axiom,
% 2.94/1.58 ! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.94/1.58 = ( ( X = L )
% 2.94/1.58 | ( X = R )
% 2.94/1.58 | ( ord_less @ binDag_Mirabelle_dag @ X @ L )
% 2.94/1.58 | ( ord_less @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_143_less__Node__dag,axiom,
% 2.94/1.58 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ X )
% 2.94/1.58 => ( ( ord_less @ binDag_Mirabelle_dag @ L @ X )
% 2.94/1.58 & ( ord_less @ binDag_Mirabelle_dag @ R @ X ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_144_less__dag__def,axiom,
% 2.94/1.58 ( ( ord_less @ binDag_Mirabelle_dag )
% 2.94/1.58 = ( ^ [S3: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T2 @ S3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_145_less__dag__Tip,axiom,
% 2.94/1.58 ! [X: binDag_Mirabelle_dag] :
% 2.94/1.58 ~ ( ord_less @ binDag_Mirabelle_dag @ X @ binDag_Mirabelle_Tip ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_146_bot__eq__sup__iff,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ( bot_bot @ A )
% 2.94/1.58 = ( sup_sup @ A @ X @ Y ) )
% 2.94/1.58 = ( ( X
% 2.94/1.58 = ( bot_bot @ A ) )
% 2.94/1.58 & ( Y
% 2.94/1.58 = ( bot_bot @ A ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_147_sup__eq__bot__iff,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( ( sup_sup @ A @ X @ Y )
% 2.94/1.58 = ( bot_bot @ A ) )
% 2.94/1.58 = ( ( X
% 2.94/1.58 = ( bot_bot @ A ) )
% 2.94/1.58 & ( Y
% 2.94/1.58 = ( bot_bot @ A ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_148_sup__bot_Oleft__neutral,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A] :
% 2.94/1.58 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 2.94/1.58 = A2 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_149_sup__bot_Oright__neutral,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A] :
% 2.94/1.58 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 2.94/1.58 = A2 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_150_is__singleton__the__elem,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( is_singleton @ A )
% 2.94/1.58 = ( ^ [A5: set @ A] :
% 2.94/1.58 ( A5
% 2.94/1.58 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_151_sup_Oright__idem,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A] :
% 2.94/1.58 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
% 2.94/1.58 = ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_152_sup__left__idem,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.94/1.58 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_153_sup__apply,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.94/1.58 => ( ( sup_sup @ ( A > B ) )
% 2.94/1.58 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_154_sup_Oidem,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A] :
% 2.94/1.58 ( ( sup_sup @ A @ A2 @ A2 )
% 2.94/1.58 = A2 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_155_sup__idem,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A] :
% 2.94/1.58 ( ( sup_sup @ A @ X @ X )
% 2.94/1.58 = X ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_156_sup_Oleft__idem,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A] :
% 2.94/1.58 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
% 2.94/1.58 = ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_157_the__elem__eq,axiom,
% 2.94/1.58 ! [A: $tType,X: A] :
% 2.94/1.58 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 = X ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_158_not__psubset__empty,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_159_inf__sup__aci_I8_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( lattice @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.94/1.58 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_160_inf__sup__aci_I7_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( lattice @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A,Z: A] :
% 2.94/1.58 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.94/1.58 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_161_inf__sup__aci_I6_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( lattice @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A,Z: A] :
% 2.94/1.58 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.94/1.58 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_162_inf__sup__aci_I5_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( lattice @ A @ ( type2 @ A ) )
% 2.94/1.58 => ( ( sup_sup @ A )
% 2.94/1.58 = ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_163_sup__fun__def,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.94/1.58 => ( ( sup_sup @ ( A > B ) )
% 2.94/1.58 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_164_sup_Oassoc,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A,C: A] :
% 2.94/1.58 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C )
% 2.94/1.58 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_165_sup__assoc,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A,Z: A] :
% 2.94/1.58 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.94/1.58 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_166_sup_Ocommute,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ( ( sup_sup @ A )
% 2.94/1.58 = ( ^ [A3: A,B2: A] : ( sup_sup @ A @ B2 @ A3 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_167_sup__commute,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ( ( sup_sup @ A )
% 2.94/1.58 = ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_168_sup_Oleft__commute,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [B3: A,A2: A,C: A] :
% 2.94/1.58 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C ) )
% 2.94/1.58 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_169_sup__left__commute,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A,Z: A] :
% 2.94/1.58 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.94/1.58 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_170_less__DAG__set__of,axiom,
% 2.94/1.58 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
% 2.94/1.58 ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
% 2.94/1.58 => ( ( binDag_Mirabelle_DAG @ Y )
% 2.94/1.58 => ( ord_less @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_171_less__supI1,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,A2: A,B3: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ A2 )
% 2.94/1.58 => ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_172_less__supI2,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,B3: A,A2: A] :
% 2.94/1.58 ( ( ord_less @ A @ X @ B3 )
% 2.94/1.58 => ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_173_sup_Ostrict__boundedE,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [B3: A,C: A,A2: A] :
% 2.94/1.58 ( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C ) @ A2 )
% 2.94/1.58 => ~ ( ( ord_less @ A @ B3 @ A2 )
% 2.94/1.58 => ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_174_sup_Ostrict__order__iff,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ( ( ord_less @ A )
% 2.94/1.58 = ( ^ [B2: A,A3: A] :
% 2.94/1.58 ( ( A3
% 2.94/1.58 = ( sup_sup @ A @ A3 @ B2 ) )
% 2.94/1.58 & ( A3 != B2 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_175_sup_Ostrict__coboundedI1,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [C: A,A2: A,B3: A] :
% 2.94/1.58 ( ( ord_less @ A @ C @ A2 )
% 2.94/1.58 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_176_sup_Ostrict__coboundedI2,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [C: A,B3: A,A2: A] :
% 2.94/1.58 ( ( ord_less @ A @ C @ B3 )
% 2.94/1.58 => ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_177_sup__bot__left,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A] :
% 2.94/1.58 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 2.94/1.58 = X ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_178_sup__bot__right,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A] :
% 2.94/1.58 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 2.94/1.58 = X ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_179_sup__bot_Osemilattice__neutr__axioms,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_180_sup__bot_Omonoid__axioms,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_181_minf_I11_J,axiom,
% 2.94/1.58 ! [C2: $tType,D2: $tType] :
% 2.94/1.58 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.94/1.58 => ! [F3: D2] :
% 2.94/1.58 ? [Z3: C2] :
% 2.94/1.58 ! [X5: C2] :
% 2.94/1.58 ( ( ord_less @ C2 @ X5 @ Z3 )
% 2.94/1.58 => ( F3 = F3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_182_psubsetD,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
% 2.94/1.58 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( ( member @ A @ C @ A4 )
% 2.94/1.58 => ( member @ A @ C @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_183_psubset__trans,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.94/1.58 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
% 2.94/1.58 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_184_pinf_I1_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.94/1.58 ( ? [Z4: A] :
% 2.94/1.58 ! [X4: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z4 @ X4 )
% 2.94/1.58 => ( ( P2 @ X4 )
% 2.94/1.58 = ( P3 @ X4 ) ) )
% 2.94/1.58 => ( ? [Z4: A] :
% 2.94/1.58 ! [X4: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z4 @ X4 )
% 2.94/1.58 => ( ( Q2 @ X4 )
% 2.94/1.58 = ( Q3 @ X4 ) ) )
% 2.94/1.58 => ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z3 @ X5 )
% 2.94/1.58 => ( ( ( P2 @ X5 )
% 2.94/1.58 & ( Q2 @ X5 ) )
% 2.94/1.58 = ( ( P3 @ X5 )
% 2.94/1.58 & ( Q3 @ X5 ) ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_185_pinf_I2_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.94/1.58 ( ? [Z4: A] :
% 2.94/1.58 ! [X4: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z4 @ X4 )
% 2.94/1.58 => ( ( P2 @ X4 )
% 2.94/1.58 = ( P3 @ X4 ) ) )
% 2.94/1.58 => ( ? [Z4: A] :
% 2.94/1.58 ! [X4: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z4 @ X4 )
% 2.94/1.58 => ( ( Q2 @ X4 )
% 2.94/1.58 = ( Q3 @ X4 ) ) )
% 2.94/1.58 => ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z3 @ X5 )
% 2.94/1.58 => ( ( ( P2 @ X5 )
% 2.94/1.58 | ( Q2 @ X5 ) )
% 2.94/1.58 = ( ( P3 @ X5 )
% 2.94/1.58 | ( Q3 @ X5 ) ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_186_pinf_I3_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z3 @ X5 )
% 2.94/1.58 => ( X5 != T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_187_pinf_I4_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z3 @ X5 )
% 2.94/1.58 => ( X5 != T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_188_pinf_I5_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z3 @ X5 )
% 2.94/1.58 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_189_pinf_I7_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z3 @ X5 )
% 2.94/1.58 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_190_pinf_I11_J,axiom,
% 2.94/1.58 ! [C2: $tType,D2: $tType] :
% 2.94/1.58 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.94/1.58 => ! [F3: D2] :
% 2.94/1.58 ? [Z3: C2] :
% 2.94/1.58 ! [X5: C2] :
% 2.94/1.58 ( ( ord_less @ C2 @ Z3 @ X5 )
% 2.94/1.58 => ( F3 = F3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_191_minf_I1_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.94/1.58 ( ? [Z4: A] :
% 2.94/1.58 ! [X4: A] :
% 2.94/1.58 ( ( ord_less @ A @ X4 @ Z4 )
% 2.94/1.58 => ( ( P2 @ X4 )
% 2.94/1.58 = ( P3 @ X4 ) ) )
% 2.94/1.58 => ( ? [Z4: A] :
% 2.94/1.58 ! [X4: A] :
% 2.94/1.58 ( ( ord_less @ A @ X4 @ Z4 )
% 2.94/1.58 => ( ( Q2 @ X4 )
% 2.94/1.58 = ( Q3 @ X4 ) ) )
% 2.94/1.58 => ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ X5 @ Z3 )
% 2.94/1.58 => ( ( ( P2 @ X5 )
% 2.94/1.58 & ( Q2 @ X5 ) )
% 2.94/1.58 = ( ( P3 @ X5 )
% 2.94/1.58 & ( Q3 @ X5 ) ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_192_minf_I2_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.94/1.58 ( ? [Z4: A] :
% 2.94/1.58 ! [X4: A] :
% 2.94/1.58 ( ( ord_less @ A @ X4 @ Z4 )
% 2.94/1.58 => ( ( P2 @ X4 )
% 2.94/1.58 = ( P3 @ X4 ) ) )
% 2.94/1.58 => ( ? [Z4: A] :
% 2.94/1.58 ! [X4: A] :
% 2.94/1.58 ( ( ord_less @ A @ X4 @ Z4 )
% 2.94/1.58 => ( ( Q2 @ X4 )
% 2.94/1.58 = ( Q3 @ X4 ) ) )
% 2.94/1.58 => ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ X5 @ Z3 )
% 2.94/1.58 => ( ( ( P2 @ X5 )
% 2.94/1.58 | ( Q2 @ X5 ) )
% 2.94/1.58 = ( ( P3 @ X5 )
% 2.94/1.58 | ( Q3 @ X5 ) ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_193_minf_I3_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ X5 @ Z3 )
% 2.94/1.58 => ( X5 != T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_194_minf_I4_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ X5 @ Z3 )
% 2.94/1.58 => ( X5 != T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_195_minf_I5_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ X5 @ Z3 )
% 2.94/1.58 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_196_minf_I7_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ X5 @ Z3 )
% 2.94/1.58 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_197_ex__gt__or__lt,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A] :
% 2.94/1.58 ? [B7: A] :
% 2.94/1.58 ( ( ord_less @ A @ A2 @ B7 )
% 2.94/1.58 | ( ord_less @ A @ B7 @ A2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_198_linorder__neqE__linordered__idom,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A] :
% 2.94/1.58 ( ( X != Y )
% 2.94/1.58 => ( ~ ( ord_less @ A @ X @ Y )
% 2.94/1.58 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_199_linordered__field__no__ub,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X5: A] :
% 2.94/1.58 ? [X1: A] : ( ord_less @ A @ X5 @ X1 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_200_linordered__field__no__lb,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X5: A] :
% 2.94/1.58 ? [Y3: A] : ( ord_less @ A @ Y3 @ X5 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_201_dependent__wellorder__choice,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType] :
% 2.94/1.58 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [P2: ( A > B ) > A > B > $o] :
% 2.94/1.58 ( ! [R4: B,F4: A > B,G3: A > B,X4: A] :
% 2.94/1.58 ( ! [Y4: A] :
% 2.94/1.58 ( ( ord_less @ A @ Y4 @ X4 )
% 2.94/1.58 => ( ( F4 @ Y4 )
% 2.94/1.58 = ( G3 @ Y4 ) ) )
% 2.94/1.58 => ( ( P2 @ F4 @ X4 @ R4 )
% 2.94/1.58 = ( P2 @ G3 @ X4 @ R4 ) ) )
% 2.94/1.58 => ( ! [X4: A,F4: A > B] :
% 2.94/1.58 ( ! [Y4: A] :
% 2.94/1.58 ( ( ord_less @ A @ Y4 @ X4 )
% 2.94/1.58 => ( P2 @ F4 @ Y4 @ ( F4 @ Y4 ) ) )
% 2.94/1.58 => ? [X12: B] : ( P2 @ F4 @ X4 @ X12 ) )
% 2.94/1.58 => ? [F4: A > B] :
% 2.94/1.58 ! [X5: A] : ( P2 @ F4 @ X5 @ ( F4 @ X5 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_202_in__inv__imagep,axiom,
% 2.94/1.58 ! [B: $tType,A: $tType] :
% 2.94/1.58 ( ( inv_imagep @ A @ B )
% 2.94/1.58 = ( ^ [R2: A > A > $o,F2: B > A,X2: B,Y2: B] : ( R2 @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_203_pred__on_Ochain__extend,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,P2: A > A > $o,C3: set @ A,Z: A] :
% 2.94/1.58 ( ( pred_chain @ A @ A4 @ P2 @ C3 )
% 2.94/1.58 => ( ( member @ A @ Z @ A4 )
% 2.94/1.58 => ( ! [X4: A] :
% 2.94/1.58 ( ( member @ A @ X4 @ C3 )
% 2.94/1.58 => ( sup_sup @ ( A > A > $o ) @ P2
% 2.94/1.58 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 )
% 2.94/1.58 @ X4
% 2.94/1.58 @ Z ) )
% 2.94/1.58 => ( pred_chain @ A @ A4 @ P2 @ ( sup_sup @ ( set @ A ) @ ( insert @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) @ C3 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_204_sup__bot_Ocomm__monoid__axioms,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.94/1.58 => ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_205_sup2CI,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A4: A > B > $o] :
% 2.94/1.58 ( ( ~ ( B4 @ X @ Y )
% 2.94/1.58 => ( A4 @ X @ Y ) )
% 2.94/1.58 => ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_206_pred__on_Ochain__empty,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,P2: A > A > $o] : ( pred_chain @ A @ A4 @ P2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_207_subset_Ochain__total,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),X: set @ A,Y: set @ A] :
% 2.94/1.58 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 2.94/1.58 => ( ( member @ ( set @ A ) @ X @ C3 )
% 2.94/1.58 => ( ( member @ ( set @ A ) @ Y @ C3 )
% 2.94/1.58 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 2.94/1.58 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 2.94/1.58 @ X
% 2.94/1.58 @ Y )
% 2.94/1.58 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 2.94/1.58 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 2.94/1.58 @ Y
% 2.94/1.58 @ X ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_208_pred__on_Ochain__total,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,P2: A > A > $o,C3: set @ A,X: A,Y: A] :
% 2.94/1.58 ( ( pred_chain @ A @ A4 @ P2 @ C3 )
% 2.94/1.58 => ( ( member @ A @ X @ C3 )
% 2.94/1.58 => ( ( member @ A @ Y @ C3 )
% 2.94/1.58 => ( ( sup_sup @ ( A > A > $o ) @ P2
% 2.94/1.58 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 )
% 2.94/1.58 @ X
% 2.94/1.58 @ Y )
% 2.94/1.58 | ( sup_sup @ ( A > A > $o ) @ P2
% 2.94/1.58 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 )
% 2.94/1.58 @ Y
% 2.94/1.58 @ X ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_209_subset_Ochain__empty,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_210_subset_Ochain__extend,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),Z: set @ A] :
% 2.94/1.58 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 2.94/1.58 => ( ( member @ ( set @ A ) @ Z @ A4 )
% 2.94/1.58 => ( ! [X4: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ X4 @ C3 )
% 2.94/1.58 => ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 2.94/1.58 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 2.94/1.58 @ X4
% 2.94/1.58 @ Z ) )
% 2.94/1.58 => ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_211_sup2I2,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A4: A > B > $o] :
% 2.94/1.58 ( ( B4 @ X @ Y )
% 2.94/1.58 => ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_212_sup2I1,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: B,B4: A > B > $o] :
% 2.94/1.58 ( ( A4 @ X @ Y )
% 2.94/1.58 => ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_213_sup2E,axiom,
% 2.94/1.58 ! [A: $tType,B: $tType,A4: A > B > $o,B4: A > B > $o,X: A,Y: B] :
% 2.94/1.58 ( ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y )
% 2.94/1.58 => ( ~ ( A4 @ X @ Y )
% 2.94/1.58 => ( B4 @ X @ Y ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_214_semilattice__neutr_Oaxioms_I2_J,axiom,
% 2.94/1.58 ! [A: $tType,F: A > A > A,Z: A] :
% 2.94/1.58 ( ( semilattice_neutr @ A @ F @ Z )
% 2.94/1.58 => ( comm_monoid @ A @ F @ Z ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_215_reflclp__idemp,axiom,
% 2.94/1.58 ! [A: $tType,P2: A > A > $o] :
% 2.94/1.58 ( ( sup_sup @ ( A > A > $o )
% 2.94/1.58 @ ( sup_sup @ ( A > A > $o ) @ P2
% 2.94/1.58 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 ) )
% 2.94/1.58 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 ) )
% 2.94/1.58 = ( sup_sup @ ( A > A > $o ) @ P2
% 2.94/1.58 @ ^ [Y5: A,Z5: A] : ( Y5 = Z5 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_216_pred__on_Omaxchain__def,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( pred_maxchain @ A )
% 2.94/1.58 = ( ^ [A5: set @ A,P4: A > A > $o,C4: set @ A] :
% 2.94/1.58 ( ( pred_chain @ A @ A5 @ P4 @ C4 )
% 2.94/1.58 & ~ ? [S2: set @ A] :
% 2.94/1.58 ( ( pred_chain @ A @ A5 @ P4 @ S2 )
% 2.94/1.58 & ( ord_less @ ( set @ A ) @ C4 @ S2 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_217_subset_Omaxchain__def,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
% 2.94/1.58 ( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 2.94/1.58 = ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 2.94/1.58 & ~ ? [S2: set @ ( set @ A )] :
% 2.94/1.58 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ S2 )
% 2.94/1.58 & ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ S2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_218_subset_Omaxchain__imp__chain,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
% 2.94/1.58 ( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 2.94/1.58 => ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_219_subset_OHausdorff,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ ( set @ A )] :
% 2.94/1.58 ? [X1: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X1 ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_220_Pow__singleton__iff,axiom,
% 2.94/1.58 ! [A: $tType,X6: set @ A,Y6: set @ A] :
% 2.94/1.58 ( ( ( pow @ A @ X6 )
% 2.94/1.58 = ( insert @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
% 2.94/1.58 = ( ( X6
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 & ( Y6
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_221_Pow__empty,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_222_Pow__bottom,axiom,
% 2.94/1.58 ! [A: $tType,B4: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_223_Pow__not__empty,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( pow @ A @ A4 )
% 2.94/1.58 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_224_Pow__top,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ A4 @ ( pow @ A @ A4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_225_subset_Ochain__def,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
% 2.94/1.58 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
% 2.94/1.58 = ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A4 )
% 2.94/1.58 & ! [X2: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ X2 @ C3 )
% 2.94/1.58 => ! [Y2: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ Y2 @ C3 )
% 2.94/1.58 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 2.94/1.58 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 2.94/1.58 @ X2
% 2.94/1.58 @ Y2 )
% 2.94/1.58 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 2.94/1.58 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 2.94/1.58 @ Y2
% 2.94/1.58 @ X2 ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_226_subset_OchainI,axiom,
% 2.94/1.58 ! [A: $tType,C3: set @ ( set @ A ),A4: set @ ( set @ A )] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A4 )
% 2.94/1.58 => ( ! [X4: set @ A,Y3: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ X4 @ C3 )
% 2.94/1.58 => ( ( member @ ( set @ A ) @ Y3 @ C3 )
% 2.94/1.58 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 2.94/1.58 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 2.94/1.58 @ X4
% 2.94/1.58 @ Y3 )
% 2.94/1.58 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 2.94/1.58 @ ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 )
% 2.94/1.58 @ Y3
% 2.94/1.58 @ X4 ) ) ) )
% 2.94/1.58 => ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_227_order__refl,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_228_subsetI,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ! [X4: A] :
% 2.94/1.58 ( ( member @ A @ X4 @ A4 )
% 2.94/1.58 => ( member @ A @ X4 @ B4 ) )
% 2.94/1.58 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_229_subset__antisym,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.94/1.58 => ( A4 = B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_230_le__sup__iff,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [X: A,Y: A,Z: A] :
% 2.94/1.58 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.94/1.58 = ( ( ord_less_eq @ A @ X @ Z )
% 2.94/1.58 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_231_sup_Obounded__iff,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [B3: A,C: A,A2: A] :
% 2.94/1.58 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C ) @ A2 )
% 2.94/1.58 = ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.94/1.58 & ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_232_empty__subsetI,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_233_subset__empty,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = ( A4
% 2.94/1.58 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_234_insert__subset,axiom,
% 2.94/1.58 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B4 )
% 2.94/1.58 = ( ( member @ A @ X @ B4 )
% 2.94/1.58 & ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_235_Un__subset__iff,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
% 2.94/1.58 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 2.94/1.58 & ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_236_psubsetI,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( ( A4 != B4 )
% 2.94/1.58 => ( ord_less @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_237_Pow__iff,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B4 ) )
% 2.94/1.58 = ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_238_PowI,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_239_singleton__insert__inj__eq,axiom,
% 2.94/1.58 ! [A: $tType,B3: A,A2: A,A4: set @ A] :
% 2.94/1.58 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.58 = ( insert @ A @ A2 @ A4 ) )
% 2.94/1.58 = ( ( A2 = B3 )
% 2.94/1.58 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_240_singleton__insert__inj__eq_H,axiom,
% 2.94/1.58 ! [A: $tType,A2: A,A4: set @ A,B3: A] :
% 2.94/1.58 ( ( ( insert @ A @ A2 @ A4 )
% 2.94/1.58 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.58 = ( ( A2 = B3 )
% 2.94/1.58 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_241_subset__Zorn,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ ( set @ A )] :
% 2.94/1.58 ( ! [C5: set @ ( set @ A )] :
% 2.94/1.58 ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
% 2.94/1.58 => ? [X5: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ X5 @ A4 )
% 2.94/1.58 & ! [Xa: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ Xa @ C5 )
% 2.94/1.58 => ( ord_less_eq @ ( set @ A ) @ Xa @ X5 ) ) ) )
% 2.94/1.58 => ? [X4: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ X4 @ A4 )
% 2.94/1.58 & ! [Xa2: set @ A] :
% 2.94/1.58 ( ( member @ ( set @ A ) @ Xa2 @ A4 )
% 2.94/1.58 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ Xa2 )
% 2.94/1.58 => ( Xa2 = X4 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_242_psubsetE,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_243_psubset__eq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( ord_less @ ( set @ A ) )
% 2.94/1.58 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.94/1.58 & ( A5 != B5 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_244_psubset__imp__subset,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.94/1.58 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_245_psubset__subset__trans,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.94/1.58 ( ( ord_less @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 2.94/1.58 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_246_subset__not__subset__eq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( ord_less @ ( set @ A ) )
% 2.94/1.58 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.94/1.58 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_247_subset__psubset__trans,axiom,
% 2.94/1.58 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.94/1.58 => ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
% 2.94/1.58 => ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_248_subset__iff__psubset__eq,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( ord_less_eq @ ( set @ A ) )
% 2.94/1.58 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.94/1.58 ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
% 2.94/1.58 | ( A5 = B5 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_249_pinf_I6_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z3 @ X5 )
% 2.94/1.58 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_250_pinf_I8_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ Z3 @ X5 )
% 2.94/1.58 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_251_minf_I6_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ X5 @ Z3 )
% 2.94/1.58 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_252_minf_I8_J,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [T: A] :
% 2.94/1.58 ? [Z3: A] :
% 2.94/1.58 ! [X5: A] :
% 2.94/1.58 ( ( ord_less @ A @ X5 @ Z3 )
% 2.94/1.58 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_253_order_Onot__eq__order__implies__strict,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [A2: A,B3: A] :
% 2.94/1.58 ( ( A2 != B3 )
% 2.94/1.58 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.94/1.58 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_254_dual__order_Ostrict__implies__order,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ! [B3: A,A2: A] :
% 2.94/1.58 ( ( ord_less @ A @ B3 @ A2 )
% 2.94/1.58 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(fact_255_dual__order_Ostrict__iff__order,axiom,
% 2.94/1.58 ! [A: $tType] :
% 2.94/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.94/1.58 => ( ( ord_less @ A )
% 2.94/1.58 = ( ^ [B2: A,A3: A] :
% 2.94/1.58 ( ( ord_less_eq @ A @ B2 @ A3 )
% 2.94/1.58 & ( A3 != B2 ) ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.94/1.58 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.94/1.58 ! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( bounde1808546759up_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( bounded_lattice_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( semilattice_sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( order_bot @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( order_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( lattice @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( order @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Orderings_Oord,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_fun___Orderings_Obot,axiom,
% 2.94/1.58 ! [A6: $tType,A7: $tType] :
% 2.94/1.58 ( ( bot @ A7 @ ( type2 @ A7 ) )
% 2.94/1.58 => ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
% 2.94/1.58 ! [A6: $tType] : ( bounde1808546759up_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
% 2.94/1.58 ! [A6: $tType] : ( bounded_lattice_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
% 2.94/1.58 ! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 2.94/1.58 ! [A6: $tType] : ( order_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 2.94/1.58 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
% 2.94/1.58 ! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 2.94/1.58 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 2.94/1.58 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 2.94/1.58 ! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12,axiom,
% 2.94/1.58 bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_13,axiom,
% 2.94/1.58 bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
% 2.94/1.58 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Orderings_Oorder__bot_15,axiom,
% 2.94/1.58 order_bot @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
% 2.94/1.58 preorder @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.94/1.58 linorder @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
% 2.94/1.58 lattice @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
% 2.94/1.58 order @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
% 2.94/1.58 ord @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
% 2.94/1.58 bot @ $o @ ( type2 @ $o ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_21,axiom,
% 2.94/1.58 preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_22,axiom,
% 2.94/1.58 order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.94/1.58
% 2.94/1.58 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_23,axiom,
% 2.94/1.58 ord @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.94/1.58
% 2.94/1.58 thf(help_If_3_1_T,axiom,
% 2.94/1.58 ! [P2: $o] :
% 2.94/1.58 ( ( P2 = $true )
% 2.94/1.58 | ( P2 = $false ) ) ).
% 2.94/1.58
% 2.94/1.58 thf(help_If_2_1_T,axiom,
% 2.94/1.58 ! [A: $tType,X: A,Y: A] :
% 2.94/1.58 ( ( if @ A @ $false @ X @ Y )
% 2.94/1.58 = Y ) ).
% 2.94/1.58
% 2.94/1.58 thf(help_If_1_1_T,axiom,
% 2.94/1.58 ! [A: $tType,X: A,Y: A] :
% 2.94/1.58 ( ( if @ A @ $true @ X @ Y )
% 2.94/1.58 = X ) ).
% 2.94/1.58
% 2.94/1.58 thf(conj_0,conjecture,
% 2.94/1.58 ( ( binDag_Mirabelle_Dag @ p @ ( fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p ) @ r @ t )
% 2.94/1.58 = ( ( p = simpl_Null )
% 2.94/1.58 & ( t = binDag_Mirabelle_Tip ) ) ) ).
% 2.94/1.58 ------------------------
% 3.06/1.60 % (19718)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/183Mi)
% 3.06/1.60 % (19719)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_19524 for (2999ds/4Mi)
% 3.06/1.60 % (19719)Instruction limit reached!
% 3.06/1.60 % (19719)------------------------------
% 3.06/1.60 % (19719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.61 % (19719)Termination reason: Unknown
% 3.06/1.61 % (19720)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_19524 for (2999ds/27Mi)
% 3.06/1.61 % (19719)Termination phase: shuffling
% 3.06/1.61
% 3.06/1.61 % (19719)Memory used [KB]: 1407
% 3.06/1.61 % (19719)Time elapsed: 0.003 s
% 3.06/1.61 % (19719)Instructions burned: 6 (million)
% 3.06/1.61 % (19719)------------------------------
% 3.06/1.61 % (19719)------------------------------
% 3.06/1.61 % (19722)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_19524 for (2999ds/2Mi)
% 3.06/1.61 % (19721)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/2Mi)
% 3.06/1.61 % (19723)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_19524 for (2999ds/275Mi)
% 3.06/1.61 % (19721)Instruction limit reached!
% 3.06/1.61 % (19721)------------------------------
% 3.06/1.61 % (19721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.61 % (19721)Termination reason: Unknown
% 3.06/1.61 % (19721)Termination phase: shuffling
% 3.06/1.61
% 3.06/1.61 % (19721)Memory used [KB]: 1407
% 3.06/1.61 % (19722)Instruction limit reached!
% 3.06/1.61 % (19722)------------------------------
% 3.06/1.61 % (19722)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.61 % (19721)Time elapsed: 0.003 s
% 3.06/1.61 % (19721)Instructions burned: 2 (million)
% 3.06/1.61 % (19721)------------------------------
% 3.06/1.61 % (19721)------------------------------
% 3.06/1.61 % (19722)Termination reason: Unknown
% 3.06/1.61 % (19722)Termination phase: shuffling
% 3.06/1.61
% 3.06/1.61 % (19722)Memory used [KB]: 1407
% 3.06/1.61 % (19722)Time elapsed: 0.003 s
% 3.06/1.61 % (19722)Instructions burned: 3 (million)
% 3.06/1.61 % (19722)------------------------------
% 3.06/1.61 % (19722)------------------------------
% 3.06/1.61 % (19724)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/18Mi)
% 3.06/1.61 % (19725)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/3Mi)
% 3.06/1.62 % (19725)Instruction limit reached!
% 3.06/1.62 % (19725)------------------------------
% 3.06/1.62 % (19725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.62 % (19725)Termination reason: Unknown
% 3.06/1.62 % (19725)Termination phase: shuffling
% 3.06/1.62
% 3.06/1.62 % (19725)Memory used [KB]: 1407
% 3.06/1.62 % (19725)Time elapsed: 0.003 s
% 3.06/1.62 % (19725)Instructions burned: 5 (million)
% 3.06/1.62 % (19725)------------------------------
% 3.06/1.62 % (19725)------------------------------
% 3.06/1.62 % (19724)Instruction limit reached!
% 3.06/1.62 % (19724)------------------------------
% 3.06/1.62 % (19724)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.62 % (19724)Termination reason: Unknown
% 3.06/1.62 % (19724)Termination phase: shuffling
% 3.06/1.62
% 3.06/1.62 % (19724)Memory used [KB]: 1663
% 3.06/1.62 % (19724)Time elapsed: 0.012 s
% 3.06/1.62 % (19724)Instructions burned: 19 (million)
% 3.06/1.62 % (19724)------------------------------
% 3.06/1.62 % (19724)------------------------------
% 3.06/1.62 % (19720)Instruction limit reached!
% 3.06/1.62 % (19720)------------------------------
% 3.06/1.62 % (19720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.62 % (19720)Termination reason: Unknown
% 3.06/1.62 % (19720)Termination phase: Property scanning
% 3.06/1.62
% 3.06/1.62 % (19720)Memory used [KB]: 1791
% 3.06/1.62 % (19720)Time elapsed: 0.019 s
% 3.06/1.62 % (19720)Instructions burned: 28 (million)
% 3.06/1.62 % (19720)------------------------------
% 3.06/1.62 % (19720)------------------------------
% 3.06/1.62 % (19726)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_19524 for (2999ds/37Mi)
% 3.06/1.63 % (19728)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_19524 for (2999ds/3Mi)
% 3.06/1.63 % (19728)Instruction limit reached!
% 3.06/1.63 % (19728)------------------------------
% 3.06/1.63 % (19728)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.63 % (19728)Termination reason: Unknown
% 3.06/1.63 % (19728)Termination phase: shuffling
% 3.06/1.63
% 3.06/1.63 % (19728)Memory used [KB]: 1407
% 3.06/1.63 % (19728)Time elapsed: 0.003 s
% 3.06/1.63 % (19728)Instructions burned: 5 (million)
% 3.06/1.63 % (19728)------------------------------
% 3.06/1.63 % (19728)------------------------------
% 3.06/1.63 % (19727)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_19524 for (2999ds/15Mi)
% 3.06/1.63 % (19723)First to succeed.
% 3.06/1.63 % (19729)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_19524 for (2999ds/1041Mi)
% 3.06/1.63 % (19723)Refutation found. Thanks to Tanya!
% 3.06/1.63 % SZS status Theorem for DTF2THF_19524
% 3.06/1.63 % SZS output start Proof for DTF2THF_19524
% 3.06/1.63 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.06/1.63 thf(type_def_7, type, binDag_Mirabelle_dag: $tType).
% 3.06/1.63 thf(type_def_8, type, simpl_ref: $tType).
% 3.06/1.63 thf(type_def_9, type, set: ($tType) > $tType).
% 3.06/1.63 thf(func_def_0, type, binDag_Mirabelle_dag: $tType).
% 3.06/1.63 thf(func_def_1, type, simpl_ref: $tType).
% 3.06/1.63 thf(func_def_2, type, set: ($tType) > $tType).
% 3.06/1.63 thf(func_def_3, type, itself: ($tType) > $tType).
% 3.06/1.63 thf(func_def_4, type, bounded_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_6, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_9, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_10, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_11, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_12, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_13, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_14, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_15, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_16, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_17, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_18, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_19, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_20, type, bounded_lattice_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_21, type, bounde1808546759up_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_22, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.06/1.63 thf(func_def_23, type, binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o).
% 3.06/1.63 thf(func_def_24, type, binDag_Mirabelle_Dag: simpl_ref > (simpl_ref > simpl_ref) > (simpl_ref > simpl_ref) > binDag_Mirabelle_dag > $o).
% 3.06/1.63 thf(func_def_25, type, binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_26, type, binDag_Mirabelle_Tip: binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_27, type, binDag1297733282se_dag: !>[X0: $tType]:(X0 > (binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > X0) > binDag_Mirabelle_dag > X0)).
% 3.06/1.63 thf(func_def_28, type, binDag1442713106ec_dag: !>[X0: $tType]:(X0 > (binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > X0 > X0 > X0) > binDag_Mirabelle_dag > X0)).
% 3.06/1.63 thf(func_def_29, type, binDag1380252983set_of: binDag_Mirabelle_dag > set @ simpl_ref).
% 3.06/1.63 thf(func_def_30, type, binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o).
% 3.06/1.63 thf(func_def_31, type, fun_upd: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1 > X0 > X1)).
% 3.06/1.63 thf(func_def_32, type, override_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > X1) > set @ X0 > X0 > X1)).
% 3.06/1.63 thf(func_def_33, type, comm_monoid: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 3.06/1.63 thf(func_def_34, type, monoid: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 3.06/1.63 thf(func_def_35, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.06/1.63 thf(func_def_36, type, semilattice_neutr: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 3.06/1.63 thf(func_def_37, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.06/1.63 thf(func_def_38, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.06/1.63 thf(func_def_39, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.06/1.63 thf(func_def_40, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.06/1.63 thf(func_def_41, type, type2: !>[X0: $tType]:(itself @ X0)).
% 3.06/1.63 thf(func_def_42, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 3.06/1.63 thf(func_def_43, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.06/1.63 thf(func_def_44, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 3.06/1.63 thf(func_def_45, type, bind: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ X1)).
% 3.06/1.63 thf(func_def_46, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.06/1.63 thf(func_def_47, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 3.06/1.63 thf(func_def_48, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 3.06/1.63 thf(func_def_49, type, pairwise: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 3.06/1.63 thf(func_def_50, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 3.06/1.63 thf(func_def_51, type, simpl_Null: simpl_ref).
% 3.06/1.63 thf(func_def_52, type, pred_chain: !>[X0: $tType]:(set @ X0 > (X0 > X0 > $o) > set @ X0 > $o)).
% 3.06/1.63 thf(func_def_53, type, pred_maxchain: !>[X0: $tType]:(set @ X0 > (X0 > X0 > $o) > set @ X0 > $o)).
% 3.06/1.63 thf(func_def_54, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.06/1.63 thf(func_def_55, type, l: simpl_ref > simpl_ref).
% 3.06/1.63 thf(func_def_56, type, p: simpl_ref).
% 3.06/1.63 thf(func_def_57, type, r: simpl_ref > simpl_ref).
% 3.06/1.63 thf(func_def_58, type, t: binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_74, type, sK0: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.06/1.63 thf(func_def_75, type, sK1: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.06/1.63 thf(func_def_76, type, sK2: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 3.06/1.63 thf(func_def_77, type, sK3: !>[X0: $tType]:(set @ X0 > X0)).
% 3.06/1.63 thf(func_def_78, type, sK4: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_79, type, sK5: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_80, type, sK6: !>[X0: $tType]:(set @ X0 > X0)).
% 3.06/1.63 thf(func_def_81, type, sK7: (binDag_Mirabelle_dag > $o) > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_82, type, sK8: (binDag_Mirabelle_dag > $o) > simpl_ref).
% 3.06/1.63 thf(func_def_83, type, sK9: (binDag_Mirabelle_dag > $o) > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_84, type, sK10: binDag_Mirabelle_dag > (simpl_ref > simpl_ref) > (simpl_ref > simpl_ref) > simpl_ref > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_85, type, sK11: binDag_Mirabelle_dag > (simpl_ref > simpl_ref) > (simpl_ref > simpl_ref) > simpl_ref > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_86, type, sK12: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.06/1.63 thf(func_def_87, type, sK13: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.06/1.63 thf(func_def_88, type, sK14: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 3.06/1.63 thf(func_def_89, type, sK15: binDag_Mirabelle_dag > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_90, type, sK16: binDag_Mirabelle_dag > simpl_ref).
% 3.06/1.63 thf(func_def_91, type, sK17: binDag_Mirabelle_dag > binDag_Mirabelle_dag).
% 3.06/1.63 thf(func_def_92, type, sK18: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.06/1.63 thf(func_def_93, type, sK19: !>[X0: $tType]:(set @ X0 > X0)).
% 3.06/1.63 thf(func_def_95, type, ph21: !>[X0: $tType]:(X0)).
% 3.06/1.63 thf(f1041,plain,(
% 3.06/1.63 $false),
% 3.06/1.63 inference(avatar_sat_refutation,[status(thm)],[f1025,f1037,f1040])).
% 3.06/1.63 thf(f1040,plain,(
% 3.06/1.63 ~spl20_1),
% 3.06/1.63 inference(avatar_contradiction_clause,[status(thm)],[f1039])).
% 3.06/1.63 thf(f1039,plain,(
% 3.06/1.63 $false | ~spl20_1),
% 3.06/1.63 inference(subsumption_resolution,[status(thm)],[f1038,f1029])).
% 3.06/1.63 thf(f1029,plain,(
% 3.06/1.63 ($true != (binDag_Mirabelle_Dag @ simpl_Null @ (fun_upd @ simpl_ref @ simpl_ref @ l @ simpl_Null @ simpl_Null) @ r @ t))),
% 3.06/1.63 inference(subsumption_resolution,[status(thm)],[f1028,f934])).
% 3.06/1.63 thf(f934,plain,(
% 3.06/1.63 ( ! [X2 : binDag_Mirabelle_dag,X0 : simpl_ref > simpl_ref,X1 : simpl_ref > simpl_ref] : (((binDag_Mirabelle_Dag @ simpl_Null @ X0 @ X1 @ X2) != $true) | (binDag_Mirabelle_Tip = X2)) )),
% 3.06/1.63 inference(cnf_transformation,[status(thm)],[f873])).
% 3.06/1.63 thf(f873,plain,(
% 3.06/1.63 ! [X0 : simpl_ref > simpl_ref,X1 : simpl_ref > simpl_ref,X2 : binDag_Mirabelle_dag] : (((binDag_Mirabelle_Tip = X2) | ((binDag_Mirabelle_Dag @ simpl_Null @ X0 @ X1 @ X2) != $true)) & (((binDag_Mirabelle_Dag @ simpl_Null @ X0 @ X1 @ X2) = $true) | (binDag_Mirabelle_Tip != X2)))),
% 3.06/1.63 inference(nnf_transformation,[status(thm)],[f650])).
% 3.06/1.63 thf(f650,plain,(
% 3.06/1.63 ! [X0 : simpl_ref > simpl_ref,X1 : simpl_ref > simpl_ref,X2 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_Tip = X2) <=> ((binDag_Mirabelle_Dag @ simpl_Null @ X0 @ X1 @ X2) = $true))),
% 3.06/1.63 inference(fool_elimination,[status(thm)],[f649])).
% 3.06/1.63 thf(f649,plain,(
% 3.06/1.63 ! [X0 : simpl_ref > simpl_ref,X1 : simpl_ref > simpl_ref,X2 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_Dag @ simpl_Null @ X0 @ X1 @ X2) = (binDag_Mirabelle_Tip = X2))),
% 3.06/1.63 inference(rectify,[status(thm)],[f1])).
% 3.06/1.63 thf(f1,axiom,(
% 3.06/1.63 ! [X2 : simpl_ref > simpl_ref,X3 : simpl_ref > simpl_ref,X4 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_Dag @ simpl_Null @ X2 @ X3 @ X4) = (binDag_Mirabelle_Tip = X4))),
% 3.06/1.63 file('/export/starexec/sandbox/tmp/tmp.vZxap9H0LD/DTF2THF_19524.p',fact_0_Dag__Null)).
% 3.06/1.63 thf(f1028,plain,(
% 3.06/1.63 (binDag_Mirabelle_Tip != t) | ($true != (binDag_Mirabelle_Dag @ simpl_Null @ (fun_upd @ simpl_ref @ simpl_ref @ l @ simpl_Null @ simpl_Null) @ r @ t))),
% 3.06/1.63 inference(forward_demodulation,[status(thm)],[f1027,f1026])).
% 3.06/1.63 thf(f1026,plain,(
% 3.06/1.63 (simpl_Null = p)),
% 3.06/1.63 inference(subsumption_resolution,[status(thm)],[f976,f992])).
% 3.06/1.63 thf(f992,plain,(
% 3.06/1.63 ( ! [X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag,X0 : simpl_ref,X1 : simpl_ref > simpl_ref] : (($true != (binDag_Mirabelle_Dag @ X0 @ (fun_upd @ simpl_ref @ simpl_ref @ X1 @ X0 @ X0) @ X2 @ X3)) | (simpl_Null = X0)) )),
% 3.06/1.63 inference(cnf_transformation,[status(thm)],[f844])).
% 3.06/1.63 thf(f844,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag] : (($true != (binDag_Mirabelle_Dag @ X0 @ (fun_upd @ simpl_ref @ simpl_ref @ X1 @ X0 @ X0) @ X2 @ X3)) | (simpl_Null = X0))),
% 3.06/1.63 inference(ennf_transformation,[status(thm)],[f820])).
% 3.06/1.63 thf(f820,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag] : ((simpl_Null != X0) => ($true != (binDag_Mirabelle_Dag @ X0 @ (fun_upd @ simpl_ref @ simpl_ref @ X1 @ X0 @ X0) @ X2 @ X3)))),
% 3.06/1.63 inference(flattening,[status(thm)],[f749])).
% 3.06/1.63 thf(f749,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag] : ((simpl_Null != X0) => ~($true = (binDag_Mirabelle_Dag @ X0 @ (fun_upd @ simpl_ref @ simpl_ref @ X1 @ X0 @ X0) @ X2 @ X3)))),
% 3.06/1.63 inference(fool_elimination,[status(thm)],[f748])).
% 3.06/1.63 thf(f748,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag] : ((simpl_Null != X0) => ~(binDag_Mirabelle_Dag @ X0 @ (fun_upd @ simpl_ref @ simpl_ref @ X1 @ X0 @ X0) @ X2 @ X3))),
% 3.06/1.63 inference(rectify,[status(thm)],[f3])).
% 3.06/1.63 thf(f3,axiom,(
% 3.06/1.63 ! [X5 : simpl_ref,X2 : simpl_ref > simpl_ref,X3 : simpl_ref > simpl_ref,X4 : binDag_Mirabelle_dag] : ((simpl_Null != X5) => ~(binDag_Mirabelle_Dag @ X5 @ (fun_upd @ simpl_ref @ simpl_ref @ X2 @ X5 @ X5) @ X3 @ X4))),
% 3.06/1.63 file('/export/starexec/sandbox/tmp/tmp.vZxap9H0LD/DTF2THF_19524.p',fact_2_Dag__upd__same__l__lemma)).
% 3.06/1.63 thf(f976,plain,(
% 3.06/1.63 ((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) = $true) | (simpl_Null = p)),
% 3.06/1.63 inference(cnf_transformation,[status(thm)],[f901])).
% 3.06/1.63 thf(f901,plain,(
% 3.06/1.63 (((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) != $true) | (simpl_Null != p) | (binDag_Mirabelle_Tip != t)) & (((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) = $true) | ((simpl_Null = p) & (binDag_Mirabelle_Tip = t)))),
% 3.06/1.63 inference(flattening,[status(thm)],[f900])).
% 3.06/1.63 thf(f900,plain,(
% 3.06/1.63 (((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) != $true) | ((simpl_Null != p) | (binDag_Mirabelle_Tip != t))) & (((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) = $true) | ((simpl_Null = p) & (binDag_Mirabelle_Tip = t)))),
% 3.06/1.63 inference(nnf_transformation,[status(thm)],[f858])).
% 3.06/1.63 thf(f858,plain,(
% 3.06/1.63 ((simpl_Null = p) & (binDag_Mirabelle_Tip = t)) <~> ((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) = $true)),
% 3.06/1.63 inference(ennf_transformation,[status(thm)],[f683])).
% 3.06/1.63 thf(f683,plain,(
% 3.06/1.63 ~(((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) = $true) <=> ((simpl_Null = p) & (binDag_Mirabelle_Tip = t)))),
% 3.06/1.63 inference(fool_elimination,[status(thm)],[f295])).
% 3.06/1.63 thf(f295,negated_conjecture,(
% 3.06/1.63 ~((binDag_Mirabelle_Tip = t) & (simpl_Null = p) = (binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t))),
% 3.06/1.63 inference(negated_conjecture,[status(cth)],[f294])).
% 3.06/1.63 thf(f294,conjecture,(
% 3.06/1.63 ((binDag_Mirabelle_Tip = t) & (simpl_Null = p) = (binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t))),
% 3.06/1.63 file('/export/starexec/sandbox/tmp/tmp.vZxap9H0LD/DTF2THF_19524.p',conj_0)).
% 3.06/1.63 thf(f1027,plain,(
% 3.06/1.63 (binDag_Mirabelle_Tip != t) | ((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) != $true)),
% 3.06/1.63 inference(subsumption_resolution,[status(thm)],[f977,f1026])).
% 3.06/1.63 thf(f977,plain,(
% 3.06/1.63 ((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) != $true) | (binDag_Mirabelle_Tip != t) | (simpl_Null != p)),
% 3.06/1.63 inference(cnf_transformation,[status(thm)],[f901])).
% 3.06/1.63 thf(f1038,plain,(
% 3.06/1.63 ($true = (binDag_Mirabelle_Dag @ simpl_Null @ (fun_upd @ simpl_ref @ simpl_ref @ l @ simpl_Null @ simpl_Null) @ r @ t)) | ~spl20_1),
% 3.06/1.63 inference(forward_demodulation,[status(thm)],[f1020,f1026])).
% 3.06/1.63 thf(f1020,plain,(
% 3.06/1.63 ((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) = $true) | ~spl20_1),
% 3.06/1.63 inference(avatar_component_clause,[status(thm)],[f1018])).
% 3.06/1.63 thf(f1018,definition,(
% 3.06/1.63 spl20_1 <=> ((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) = $true)),
% 3.06/1.63 introduced(definition,[new_symbols(naming,[spl20_1])],[avatar_definition])).
% 3.06/1.63 thf(f1037,plain,(
% 3.06/1.63 ~spl20_2),
% 3.06/1.63 inference(avatar_contradiction_clause,[status(thm)],[f1036])).
% 3.06/1.63 thf(f1036,plain,(
% 3.06/1.63 $false | ~spl20_2),
% 3.06/1.63 inference(subsumption_resolution,[status(thm)],[f1035,f1002])).
% 3.06/1.63 thf(f1002,plain,(
% 3.06/1.63 ( ! [X0 : simpl_ref > simpl_ref,X1 : simpl_ref > simpl_ref] : (($true = (binDag_Mirabelle_Dag @ simpl_Null @ X0 @ X1 @ binDag_Mirabelle_Tip))) )),
% 3.06/1.63 inference(equality_resolution,[status(thm)],[f933])).
% 3.06/1.63 thf(f933,plain,(
% 3.06/1.63 ( ! [X2 : binDag_Mirabelle_dag,X0 : simpl_ref > simpl_ref,X1 : simpl_ref > simpl_ref] : (((binDag_Mirabelle_Dag @ simpl_Null @ X0 @ X1 @ X2) = $true) | (binDag_Mirabelle_Tip != X2)) )),
% 3.06/1.63 inference(cnf_transformation,[status(thm)],[f873])).
% 3.06/1.63 thf(f1035,plain,(
% 3.06/1.63 ((binDag_Mirabelle_Dag @ simpl_Null @ l @ r @ binDag_Mirabelle_Tip) != $true) | ~spl20_2),
% 3.06/1.63 inference(forward_demodulation,[status(thm)],[f1034,f1024])).
% 3.06/1.63 thf(f1024,plain,(
% 3.06/1.63 (binDag_Mirabelle_Tip = t) | ~spl20_2),
% 3.06/1.63 inference(avatar_component_clause,[status(thm)],[f1022])).
% 3.06/1.63 thf(f1022,definition,(
% 3.06/1.63 spl20_2 <=> (binDag_Mirabelle_Tip = t)),
% 3.06/1.63 introduced(definition,[new_symbols(naming,[spl20_2])],[avatar_definition])).
% 3.06/1.63 thf(f1034,plain,(
% 3.06/1.63 ($true != (binDag_Mirabelle_Dag @ simpl_Null @ l @ r @ t))),
% 3.06/1.63 inference(subsumption_resolution,[status(thm)],[f1031,f944])).
% 3.06/1.63 thf(f944,plain,(
% 3.06/1.63 ( ! [X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag,X0 : simpl_ref,X1 : simpl_ref > simpl_ref] : (($true != (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3)) | ($true != (member @ simpl_ref @ simpl_Null @ (binDag1380252983set_of @ X3)))) )),
% 3.06/1.63 inference(cnf_transformation,[status(thm)],[f847])).
% 3.06/1.63 thf(f847,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag] : (($true != (member @ simpl_ref @ simpl_Null @ (binDag1380252983set_of @ X3))) | ($true != (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3)))),
% 3.06/1.63 inference(ennf_transformation,[status(thm)],[f821])).
% 3.06/1.63 thf(f821,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag] : (($true = (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3)) => ($true != (member @ simpl_ref @ simpl_Null @ (binDag1380252983set_of @ X3))))),
% 3.06/1.63 inference(flattening,[status(thm)],[f500])).
% 3.06/1.63 thf(f500,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag] : (($true = (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3)) => ~($true = (member @ simpl_ref @ simpl_Null @ (binDag1380252983set_of @ X3))))),
% 3.06/1.63 inference(fool_elimination,[status(thm)],[f499])).
% 3.06/1.63 thf(f499,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3) => ~(member @ simpl_ref @ simpl_Null @ (binDag1380252983set_of @ X3)))),
% 3.06/1.63 inference(rectify,[status(thm)],[f9])).
% 3.06/1.63 thf(f9,axiom,(
% 3.06/1.63 ! [X5 : simpl_ref,X2 : simpl_ref > simpl_ref,X3 : simpl_ref > simpl_ref,X4 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_Dag @ X5 @ X2 @ X3 @ X4) => ~(member @ simpl_ref @ simpl_Null @ (binDag1380252983set_of @ X4)))),
% 3.06/1.63 file('/export/starexec/sandbox/tmp/tmp.vZxap9H0LD/DTF2THF_19524.p',fact_8_Null__notin__Dag)).
% 3.06/1.63 thf(f1031,plain,(
% 3.06/1.63 ((member @ simpl_ref @ simpl_Null @ (binDag1380252983set_of @ t)) = $true) | ($true != (binDag_Mirabelle_Dag @ simpl_Null @ l @ r @ t))),
% 3.06/1.63 inference(superposition,[status(thm)],[f1029,f984])).
% 3.06/1.63 thf(f984,plain,(
% 3.06/1.63 ( ! [X2 : simpl_ref,X3 : simpl_ref > simpl_ref,X0 : simpl_ref,X1 : binDag_Mirabelle_dag,X4 : simpl_ref,X5 : simpl_ref > simpl_ref] : (((binDag_Mirabelle_Dag @ X2 @ X3 @ X5 @ X1) = (binDag_Mirabelle_Dag @ X2 @ (fun_upd @ simpl_ref @ simpl_ref @ X3 @ X0 @ X4) @ X5 @ X1)) | ((member @ simpl_ref @ X0 @ (binDag1380252983set_of @ X1)) = $true)) )),
% 3.06/1.63 inference(cnf_transformation,[status(thm)],[f852])).
% 3.06/1.63 thf(f852,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : binDag_Mirabelle_dag,X2 : simpl_ref,X3 : simpl_ref > simpl_ref,X4 : simpl_ref,X5 : simpl_ref > simpl_ref] : (((member @ simpl_ref @ X0 @ (binDag1380252983set_of @ X1)) = $true) | ((binDag_Mirabelle_Dag @ X2 @ X3 @ X5 @ X1) = (binDag_Mirabelle_Dag @ X2 @ (fun_upd @ simpl_ref @ simpl_ref @ X3 @ X0 @ X4) @ X5 @ X1)))),
% 3.06/1.63 inference(ennf_transformation,[status(thm)],[f827])).
% 3.06/1.63 thf(f827,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : binDag_Mirabelle_dag,X2 : simpl_ref,X3 : simpl_ref > simpl_ref,X4 : simpl_ref,X5 : simpl_ref > simpl_ref] : (((member @ simpl_ref @ X0 @ (binDag1380252983set_of @ X1)) != $true) => ((binDag_Mirabelle_Dag @ X2 @ X3 @ X5 @ X1) = (binDag_Mirabelle_Dag @ X2 @ (fun_upd @ simpl_ref @ simpl_ref @ X3 @ X0 @ X4) @ X5 @ X1)))),
% 3.06/1.63 inference(flattening,[status(thm)],[f436])).
% 3.06/1.63 thf(f436,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : binDag_Mirabelle_dag,X2 : simpl_ref,X3 : simpl_ref > simpl_ref,X4 : simpl_ref,X5 : simpl_ref > simpl_ref] : (~((member @ simpl_ref @ X0 @ (binDag1380252983set_of @ X1)) = $true) => ((binDag_Mirabelle_Dag @ X2 @ X3 @ X5 @ X1) = (binDag_Mirabelle_Dag @ X2 @ (fun_upd @ simpl_ref @ simpl_ref @ X3 @ X0 @ X4) @ X5 @ X1)))),
% 3.06/1.63 inference(fool_elimination,[status(thm)],[f435])).
% 3.06/1.63 thf(f435,plain,(
% 3.06/1.63 ! [X0 : simpl_ref,X1 : binDag_Mirabelle_dag,X2 : simpl_ref,X3 : simpl_ref > simpl_ref,X4 : simpl_ref,X5 : simpl_ref > simpl_ref] : (~(member @ simpl_ref @ X0 @ (binDag1380252983set_of @ X1)) => ((binDag_Mirabelle_Dag @ X2 @ X3 @ X5 @ X1) = (binDag_Mirabelle_Dag @ X2 @ (fun_upd @ simpl_ref @ simpl_ref @ X3 @ X0 @ X4) @ X5 @ X1)))),
% 3.06/1.63 inference(rectify,[status(thm)],[f7])).
% 3.06/1.63 thf(f7,axiom,(
% 3.06/1.63 ! [X14 : simpl_ref,X4 : binDag_Mirabelle_dag,X5 : simpl_ref,X2 : simpl_ref > simpl_ref,X8 : simpl_ref,X3 : simpl_ref > simpl_ref] : (~(member @ simpl_ref @ X14 @ (binDag1380252983set_of @ X4)) => ((binDag_Mirabelle_Dag @ X5 @ (fun_upd @ simpl_ref @ simpl_ref @ X2 @ X14 @ X8) @ X3 @ X4) = (binDag_Mirabelle_Dag @ X5 @ X2 @ X3 @ X4)))),
% 3.06/1.63 file('/export/starexec/sandbox/tmp/tmp.vZxap9H0LD/DTF2THF_19524.p',fact_6_notin__Dag__update__l)).
% 3.06/1.63 thf(f1025,plain,(
% 3.06/1.63 spl20_1 | spl20_2),
% 3.06/1.63 inference(avatar_split_clause,[status(thm)],[f975,f1022,f1018])).
% 3.06/1.63 thf(f975,plain,(
% 3.06/1.63 ((binDag_Mirabelle_Dag @ p @ (fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p) @ r @ t) = $true) | (binDag_Mirabelle_Tip = t)),
% 3.06/1.63 inference(cnf_transformation,[status(thm)],[f901])).
% 3.06/1.63 % SZS output end Proof for DTF2THF_19524
% 3.06/1.63 % (19723)------------------------------
% 3.06/1.63 % (19723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.63 % (19723)Termination reason: Refutation
% 3.06/1.63
% 3.06/1.63 % (19723)Memory used [KB]: 6652
% 3.06/1.63 % (19723)Time elapsed: 0.026 s
% 3.06/1.63 % (19723)Instructions burned: 59 (million)
% 3.06/1.63 % (19723)------------------------------
% 3.06/1.63 % (19723)------------------------------
% 3.06/1.63 % (19717)Success in time 0.043 s
%------------------------------------------------------------------------------